O {lattice models of planar aggregation Vittoria Silvestri · Vittoria Silvestri. December 7, 2016. Contents Chapter 1. Complex Analysis preliminaries 5 1. Holomorphic functions and

Off–lattice models of planar aggregation

Vittoria Silvestri

December 7, 2016

Contents

Chapter 1. Complex Analysis preliminaries 5

1. Holomorphic functions and Cauchy integration 5

2. The complex logarithm 7

3. Conformal maps 8

4. Conformal invariance of Brownian motion 10

5. Density of harmonic measure 14

Chapter 2. Hastings–Levitov models 17

1. Discrete aggregation models 17

2. A continuum model: Hastings–Levitov growth 19

Chapter 3. Small particle limit 27

1. Preliminary estimates 30

2. The scaling limit result 34

3. Scaling limit of regularised HL(α) clusters 40

Chapter 4. Fluctuations 43

1. Preliminary estimates 45

2. Pointwise fluctuations 46

3. Global fluctuations 53

Bibliography 61

3

CHAPTER 1

Complex Analysis preliminaries

In this first chapter we recall, for convenience of the reader and to fix the notation,

some basic concepts from Complex Analysis that we will use throughout the course.

1. Holomorphic functions and Cauchy integration

Definition 1.1. A domain in the complex plane C is a non-empty connected set. A

domain D is simply connected if it is path connected and every loop in D is homotopic

to a point. Equivalently, D is simply connected if C ∪ ∞ \D is connected.

Here are some examples.

• D = z ∈ C : |z| < 1 is simply connected.

• H = z ∈ C : Im(z) > 0 is simply connected.

• C ∪ ∞ without a strip is simply connected.

• C ∪ ∞ without an annulus is not simply connected.

Definition 1.2. A function f : C → C is said to be holomorphic (or analytic) in

a domain D iff for all z ∈ D the limit

limh→0

f(z + h)− f(z)

h

exists in C. In this case we define f ′(z) as the value of such limit.

We remark that, identifying C with R2, we see that f is holomorphic iff f is differen-

tiable, and the Cauchy-Riemann (C-R) equations hold:

(C-R)

∂ Re(f)

∂x=∂ Im(f)

∂y∂ Re(f)

∂y= −∂ Im(f)

∂x.

Moreover, if f is holomorphic then Re(f) and Im(f) are real-harmonic functions (the

reader should check this as a straightforward exercise).

1.1. The Cauchy integral theorem. In this section we discuss integration of com-

plex valued functions along piecewise differentiable curves in the complex plane.

Definition 1.3. A curve is a continuous map γ : [a, b] → C, a < b, from a compact

real interval into the complex plane. A curve is closed if it starts and ends at the same

points, i.e. γ(a) = γ(b). A curve is piecewise differentiable if there exists a finite

partition of [a, b] such that in each interval of this partition the function γ is differentiable.

5

6 1. COMPLEX ANALYSIS PRELIMINARIES

For example, a rectangle is a closed curve which is piecewise differentiable but not

differentiable. With a slight abuse of notation, we often write γ in place of γ([a, b]),

regarding it as a closed subset of C. From now on we work with differentiable curves for

simplicity. Everything extends trivially to the piecewise differentiable case.

Definition 1.4 (Contour integral). Let γ be a differentiable curve contained in a

domain D ⊂ C, and let f : D → C be a continuous function. Then the expression∫γf(z)dz :=

∫ b

af(γ(t))γ′(t)dt

is called contour integral of f on γ.

Note that, by the fundamental theorem of calculus, if f had a primitive F on D (i.e.

F is differentiable with F ′ = f) then∫γf(z)dz = F (γ(b))− F (γ(a)).

In particular, the integral of f along any closed curve vanishes. On connected open

domains this condition turns out to be sufficient for the existence of a primitive.

Proposition 1.1 (cf. [Car95], Chapter II). Let D be a connected open domain in C.

Then D is arcwise connected, i.e. for any two points z0, z1 in D there exists a piecewise

differentiable curve from z0 to z1 contained in D. Moreover, if f : D → C is a continuous

function, then the following statements are equivalent:

• f has a primitive on D,

•∫γ f(z)dz = 0 for any closed curve γ contained in D,

• the integral of f along any curve in D depends only on the starting and end points

of the curve.

We can finally state Cauchy Integral Formula.

Theorem 1.1 (cf. [BF09], Theorem II.3.2). Let D be open, and let f : D → C be

holomorphic on D. For z0 ∈ D take r small enough so that the closed disc Br(z0) = z :

|z − z0| ≤ r is contained in D. Then for each point z ∈ Br(z0) it holds

f(z) =1

2πi

∫∂Br(z0)

f(ζ)

ζ − zdζ,

where ∂Br(z0) denotes the boundary of Br(z0) oriented anticlockwise.

Remark 1.1. In fact, the above result can be generalised to points z on the boundary

∂D of D, assuming that f extends continuously to ∂D.

As an application, for any z0 ∈ C let γk denote the following curve winding around z0

exactly k times: γk(t) = z0 + re2πikt, t ∈ [0, 1]. Then the above theorem gives

1

2πi

∫γk

dζ

ζ − z =

k, if |z − z0| < r,

0, otherwise.

This suggests a way of measuring the winding of a curve γ around a given point.

2. THE COMPLEX LOGARITHM 7

Definition 1.5 (Winding number). Let γ be a closed, piecewise differentiable curve.

Then for z /∈ γ we define the winding number of γ around z as

χγ(z) :=1

2πi

∫γ

dζ

ζ − z .

Definition 1.6 (Residue). Let f : Br(z0) \ z0 → C be a holomorphic function in a

punctured neighbourhood of z0, with a singularity at z0. Let

f(z) =+∞∑−∞

ak(z − z0)k

be its power series expansion around z0 for z ∈ Br(z0) \ z0. The coefficient a−1 in this

expansion is called Residue of f at z0, and it is denotes by Res(f, z0).

It is left to the reader to verify that if z0 is a pole of order 1 then Res(f, z0) =

limz→z0(z − z0)f(z). In general, if z0 is a pole of order k ≥ 1 then

Res(f, z0) =1

(k − 1)!

dk−1

dzk−1

[(z − z0)kf(z)

]∣∣∣z=z0

.

We are now in the position to state the Residue Theorem.

Theorem 1.2 (Residue Theorem, cf. [BF09], Theorem III.6.3). Let D be a simply

connected domain, and take z0 ∈ D. Further let f : D \ z0 → C be holomorphic, and let

γ : [a, b]→ D \ z0 be a closed, piecewise differentiable curve. Then the following hold:

1

2πi

∫γf(ζ)dζ = Res(f, z0)χγ(z0). (1)

Note that Theorem 1.1 follows from Theorem 1.2 by applying (1) to the function

z 7→ f(z)z−z0 .

2. The complex logarithm

In this section we discuss the definition of the complex logarithm of a holomorphic

function on a subset of the complex plane or of the Riemann sphere C ∪ ∞. To under-

stand what can go wrong, let us start with an example.

Suppose we aim to define a branch of the complex logarithm of f(z) = z on a domain

D ⊂ C \ 0, call it F (z). In other words, we seek to define a function F holomorphic in

D with F ′(z) = 1/z for all z ∈ D. Let γ be any closed (i.e. starting and ending at the

same point) curve in D. Then ∫γ

1

zdz =

∫γF ′(z)dz = 0

by holomorphicity. On the other hand, by the Residue theorem the above integral is a

multiple of the winding number of γ around 0. This shows that in order for a branch of

log z to exist in D, D should not contain any curve γ winding around 0. In fact, we will

see that a branch F of the complex logarithm exists on any simply connected subset of Cnot containing 0.


What does it change when working with a subset of C∪∞ rather than of C? When

adding the point at infinity, we simply have to check whether we can allow curves to wind

around ∞. Again, we find∫γ

1

zdz = −2πiRes

(1

z,∞)

= 2πiRes(1

z, 0)

= 2πi,

where we have used that Res(f(z),∞

)= −Res

(1z2 f(

1z

), 0). This shows that the domain

of holomorphicity of F cannot contain any curve with non–trivial winding number around

∞. In order to avoid both curves winding around 0 and curves winding around ∞, one

often chooses to cut the Riemann sphere along the negative real axis.

Now let us look at the definition of a branch of the logarithm of a holomorphic function

f(z) on some simply connected subset D of the Riemann sphere containing the point at

infinity. Note that if f is holomorphic and non-zero in D∩C, then its logarithmic derivative

f ′(z)/f(z) is holomorphic on the same set, and so it has vanishing integral along any closed

curve contained in D ∩ C not winding around the point at infinity. Again, we only need

to check what the addition of the point at infinity entails. Take any closed curve γ in D

winding around ∞. Then∫γ

f ′(z)

f(z)dz = 2πiRes

(f ′(z)f(z)

,∞)

= −2πiRes( 1

z2

f ′(1/z)

f(1/z), 0). (2)

Suppose that our holomorphic function f is such that the far r.h.s. is equal to zero. Then∫γf ′(z)f(z) dz = 0 for any closed curve γ in D ⊂ C ∪ ∞, which implies that f ′(z)/f(z) has

a primitive in D (cf. [Car95], Chapter II). Thus we can set

F (z) = F (z0) +

∫z0→z

f ′(w)

f(w)dw

for arbitrary z0 ∈ D, where z0 → z denotes any curve from z0 to z contained in D, and

F (z0) is defined so that eF (z0) = f(z0) (this fixes the branch). The above definition is well

posed since the integral along any closed curve in D of f ′(z)/f(z) vanishes. We call F a

branch of the logarithm of f in D.

Remark 1.2. In the next chapters we will be interested in defining the logarithm of

the holomorphic functions Φn(z)/z, Γn(z)/z in a subset of the Riemann sphere. For these

functions we have a prescribed normalization at infinity, and the reader can check that

indeed the expressions in in (2) vanish for any curve winding around the point at infinity.

3. Conformal maps

Definition 1.7. A function f is a conformal map in D if it is holomorphic in D

and f ′(z) 6= 0 for all z ∈ D.

Definition 1.8. A function f is a conformal isomorphism it is a bijection in D

and both f and its inverse are conformal in D.

Note that, by definition, every conformal isomorphism is a conformal map. The vice

versa is not true: as a counterexample one can take the complex exponential (confor-

mal, but not injective). On the other hand, locally every conformal map is a conformal

isomorphism.

3. CONFORMAL MAPS 9

Definition 1.9 (Moebius transformations). A Moebius transformation is any function

f on C ∪ ∞ of the form

f(z) =az + b

cz + dwhere a, b, c, d ∈ C and ad− bc 6= 0.

It is important to point out that the set of Moebius transformations forms a group

under the operation of composition (very easy to check).

Let us look at different cases.

• If a, b, c, d are real, and ad − bc = 1, then f : H → H is an automorphism of

H. Indeed, f is simply the composition of translations (z 7→ z + c), dilatations

(z 7→ λz with λ > 0) and inversions (z 7→ 1/z).

• The map f(z) = i−zi+z : H → D := z ∈ C : |z| < 1 is a conformal isomorphism

from the upper half plane H to the unit disc D.

f (z) = i−zi+z

H

D∞

i

0

0 1−1f(∞) = −1f(i) = 0

f(0) = 1

• Define the map

φϑ,ω := eiϑz − ω1− ωz ϑ ∈ [0, 2π), ω ∈ D . (3)

Then φϑ,ω is a conformal automorphism of D which swaps 0 and ω.

φθ,ω D

1−1

D

0 φθ,ω(ω)

ωφθ,ω(0)

0 7→ ωω 7→ 0

Proposition 1.2. Let φ be a conformal automorphism of the unit disc D. Denote by

ω = φ−1(0) and with ϑ = arg φ′(ω). Then φ = φϑ,ω.

This says that Moebius transformations are the only automorphisms of the unit disc

D. The proof of this result uses the Schwarz Lemma, that we recall here.

Lemma 1.1 (Schwarz Lemma). For every holomorphic function f : D→ D with f(0) =

0, and every z ∈ D, it holds∣∣f(z)

∣∣ ≤ |z|. Moreover, if there exists a point z ∈ D,

z 6= 0, such that∣∣f(z)

∣∣ = |z|, then f is a rotation, i.e. there exists ϑ ∈ [0, 2π) such that

f(z) = eiϑz.

We will see the proof of Schwarz Lemma later on, as a corollary of the Maximum

Principle (cf. Theorem 1.6). For now, we use it to prove Proposition 1.2.


Proof of Proposition 1.2. To see that φ = φϑ,ω, we prove that f := φ φ0,ω is

a rotation by angle ϑ. Note that f(0) = 0 and f is a conformal automorphism on D.

Then by the Schwarz Lemma we have∣∣f(z)

∣∣ ≤ |z| for all z ∈ D \ 0. On the other

hand, f−1 has the same properties: f−1(0) = 0, f conformal automorphism of D and so∣∣f−1(f(z)

)∣∣ ≤ |f(z)| for all z 6= 0, i.e.∣∣f(z)

∣∣ ≥ |z|. We conclude that∣∣f(z)

∣∣ = |z| for all

z ∈ D, so f is a rotation. Check that ϑ = arg φ′(ω) as an exercise.

Exercise 1.1. Show that functions of the form f(z) = az+bcz+d with a, b, c, d real and

ad− bc = 1 are the only conformal automorphisms of H.

So at this point we have classified all automorphisms of D and H.

We conclude with one of the most important results about conformal isomorphisms.

Theorem 1.3 (Riemann mapping Theorem). For all simply connected domains D ( Cthere exists a conformal isomorphism φ : D → D.

Here is a useful corollary.

Corollary 1.1. Let D ( C be a simply connected domain, and ω ∈ D. Then there

exists a unique conformal isomorphism φ : D → D such that φ(ω) = 0 and Argφ′(ω) = 0.

D Dφ

ω 0ω 7→ 0

Above we have used the convention that Arg z := ϑ : z = reiϑ ∩ [−π, π).

Proof. By the Riemann theorem, there exists a conformal isomorphism φ0 from D to

D mapping ω to φ0(ω) ∈ D. Recall (3), and set φ := φ−ϑ,φ0(ω) φ0 where ϑ := Argφ′0(ω).

This proves the existence part. For the uniqueness, let φ1 and φ2 be two such maps. Set

f := φ2 φ−11 . Then f maps D to itself, f(0) = 0 and Arg f ′(0) = 0. f being a conformal

automorphism of D, it is a Moebius transformation of the form

z 7→ eiϑz − w1− wz .

But f(0) = 0 implies ω = 0, so f(z) = eiθz for some ϑ. Then f ′(0) = eiϑ, so the

requirement Arg f ′(0) = 0 forces ϑ = 0. This proves uniqueness.

4. Conformal invariance of Brownian motion

In this section we put all together the tools from stochastic calculus to prove the

conformal invariance of Brownian motion.

Theorem 1.4 (BM is conformal invariant). Consider two simply connected domains

D,D′ ( C. Let φ : D → D′ be a non constant holomorphic function. Pick z ∈ D and

4. CONFORMAL INVARIANCE OF BROWNIAN MOTION 11

set z′ = φ(z). Let B be a complex Brownian motion starting from z ∈ D, and define the

stopping time

T := inft ≥ 0 : Bt /∈ D .Moreover, introduce the quantity

T :=

∫ T

0

∣∣φ′(Bs)∣∣2ds ,

and define the new clock

T (t) := infs ≥ 0 :

∫ s

0

∣∣φ′(Bu)∣∣2du = t . (4)

Finally, let

Bt := φ(BT (t)

)be the time-changed image process of B under φ. Then the processes

BT = (Bt)t≤T and BT = (Bt)t≤T

have the same law.

Proof. First of all, we argue that we can assume D to be bounded, and that φ extends

to a C2 function in a neighbourhood of D. Indeed, if this is not the case one can simply

consider an increasing sequence of domains D1 ⊂ · · · ⊂ Dn ⊂ · ⊂ D such that Dn D as

n→∞, and show that all limiting procedures are justified.

Write φ = (f, g) with f = Reφ and g = Imφ. Also writeBt = (Xt, Yt) withXt = ReBtand Yt = ImBt. Since f is C2 (and harmonic) we can use Ito’s formula, that we write in

differential form for simplicity:

df(Xt, Yt) = ∂xf(Xt, Yt)dXt + ∂yf(Xt, Yt)dYt +1

2∂2xxf(Xt, Yt)d〈X〉t+

1

2∂2yyf(Xt, Yt)d〈Y 〉t + ∂2

xyf(Xt, Yt)d〈X,Y 〉t= ∂xf(Xt, Yt)dXt + ∂yf(Xt, Yt)dYt .

To get from the first to the second line above we have used the independence of X and Y ,

and the harmonicity of f . Hence we conclude that f(B) = f(X,Y ) is a local martingale (it

is the integral w.r.t. a local martingale). Similarly, f(B) = f(X,Y ) is a local martingale.

Our goal is to use Levy’s characterization of BM for the process φ(B) = (f(B), g(B)). For

now, we have shown that the components are local martingales. It remains to check that

φ(B) has the right quadratic variation. Check that

• d〈f(B)〉t =

[(∂xf(B)

)2+(∂yf(B)

)2]dt

• d〈g(B)〉t =

[(∂xg(B)

)2+(∂yg(B)

)2]dt

• d〈f(B), g(B)〉t = 0.

Then summing all terms, and using Cauchy-Riemann equations, we get

d〈φ(B)〉t =∣∣φ′(B)

∣∣2dt .


Now to get the conclusion we need a time-change, which makes d〈φ(B)〉t into dt. This is

clearly given by (4): indeed, the reader can check that 〈φ(BT (t))〉 = t. The theorem then

follows by Levy’s characterization of BM.

We list some useful corollaries.

4.1. Corollaries. As a first corollary of the conformal invariance, we can prove that

complex Brownian motion exits any simply connected domain in a finite time.

Corollary 1.2. Let D ( C be any simply connected domain. Pick any point z ∈ D,

and let B = (Bt)t≥0 be a complex Brownian motion started from z. Define

T := inft ≥ 0 : Bt /∈ D

to be the first exit time from D. Then T <∞ almost surely.

As an example, take D = C\z ∈ R : z < 0, the complex plane without half real axis.

Then the above results tells us that complex BM will hit the negative real axis in finite

time. Note that this does not follows from the neighbourhood recurrence of 2-dimensional

BM, since C \D has zero Lebesgue measure.

Proof. By the Riemann mapping theorem, there exists a conformal isomorphism φ

mapping D to the unit disc D with φ(z) = 0. The image process of B under φ is, by

conformal invariance, a time-changed BM starting from 0. We argue by contradiction:

assume that T = ∞, that is: φ(B) takes infinitely longer to reach the boundary of D.

Then there exists a finite time T1 < T such that |φ(Bt)| ≥ 1/2 ∀t ≥ T1. But φ(B) is a

time-changed complex BM, and therefore it is neighbourhood recurrent, so it must come

back inside the disc z : |z| ≤ 1/2 in finite time, which is a contradiction.

The next result states that the value of a harmonic function at any point inside a

bounded domain can be expressed as a suitable average of boundary values.

Theorem 1.5 (Kakutani’s formula). Let D ( C be any bounded domain. Let u be

a harmonic function on D which extends continuously to D. Pick any point z ∈ D, and

let B = (Bt)t≥0 be a complex Brownian motion started from z. Define the first exit time

T := inft ≥ 0 : Bt /∈ D from D as before. Then it holds

u(z) = Ez(u(BT )

).

z = B0

BT

D

4. CONFORMAL INVARIANCE OF BROWNIAN MOTION 13

Proof. The strategy of the proof is the following. If u is C2 in a neighbourhood of D,

then the thesis follows easily by Ito’s formula and the Optional Stopping Theorem (OST).

For the general case, we approximate D by an increasing sequence of domains Dn D in

which the previous point applies, and then use the continuity of BM to conclude. Let us

now implement this strategy.

Suppose first that u is C2 in a neighbourhood of D. Then we apply Ito’s formula for t ≤ Tto get

u(Bt) = u(B0) +

∫ t

0∇u(Bs)dBs +

1

2

∫ t

0∆u(Bs)︸︷︷︸

0

ds

= u(z) +

∫ t

0∇u(Bs)dBs .

We conclude that(u(Bt)

)t≤T is a local martingale. We stop it at time T to make it

bounded. But any bounded local martingale is a (bounded) true martingale, so we con-

clude that(u(Bt∧T )

)t≥0

is a (bounded) true martingale. By the OST, then, we have

u(z) = u(B0) = Ez(u(BT )

).

This concludes the proof for u ∈ C2(D). To deal with the general case, let Dn D be,

say, the increasing sequence of domains

Dn := z ∈ D : dist(z∂D) > 1/n .Then u ∈ C2(Dn), and for all z ∈ Dn it holds

u(z) = Ez(u(BTn)

)where Tn is the first exit time of B from Dn. It is straightforward to check that Tn T

almost surely which, together with the continuity of both B and the function u, gives

limn→∞

u(BTn) = u(BT ) almost surely.

It only remains to swap limit and expectation, but this is allowed since u is continuous

on the bounded closed domain D, and hence it is bounded. We conclude that by the

Dominated Convergence Theorem (DTC) it holds

u(z) = Ez(

limn→∞

u(BTn))

= Ez(u(BT )

).

As a first consequence of Kakutani’s formula, we see that for u and D as in the

Theorem, it holds

u(z) ≤ supw∈∂D

u(w) .

On a more interesting note, we obtain the mean value property of harmonic func-

tions, which goes as follows. Let u be harmonic in D. Then for all z ∈ D, and all R > 0

small enough so that the ball BR(z) of radius R centred at z is contained in D, it holds

u(z) = Ez(u(BT (BR(z)))

)=

1

2π

∫ 2π

0u(z +Reiϑ)dϑ


where T (BR(z)) denotes the first exit time from the ball BR(z). We remark that the mean

value property is also sufficient for a function to be harmonic in a given domain.

The following very useful result says that harmonic functions attain their maximum

on the boundary.

Theorem 1.6 (Maximum principle). Let u be harmonic in a domain D, and suppose

that there exists z ∈ D such that u(z) ≥ u(w) for all w ∈ D. Then u is constant in D.

Proof. Let m = u(z), and define the set D := w ∈ D : u(w) = m to be the set of

points at which the maximum is attained. Then:

• D is non empty (it contains at least z),

• D is closed inD, since it is the pre-image of the closed set m under a continuous

function,

• D is open in D, because if z0 ∈ D then we can apply the mean value property

to conclude that u must be identically m in a small neighbourhood of z0.

But the above implies that D is the whole D, which is what we wanted to show.

5. Density of harmonic measure

Let D be a simply connected domain, and denote by ∂D its boundary. Let z /∈ D be

a point strictly outside D, and let (Bt)t≥0 be a standard Brownian Motion on C starting

from z. Let T := inft ≥ 0 : BT ∈ D, and assume that ∂D is nice enough so that for any

arc γ ⊆ ∂D we have

Pz(BT ∈ γ) =

∫γhD(z, w)dw.

In this case hD(z, ·) is referred to as the density of harmonic measure on ∂D seen from z.

As an example, we note that by rotation invariance of Brownian Motion

hD(0, w) = hD(∞, w) =1

2π.

We use this, together with conformal invariance of Brownian Motion, to compute the

density of harmonic measure on the boundary of general domains. Given D domain in

C such that (C ∪ ∞) \ D is simply connected, and z /∈ D, let φ denote the unique

conformal isomorphism φ : (C ∪ ∞) \ D → (C ∪ ∞) \ D mapping ∞ to z and such

that φ′(∞) ∈ R+.

φ : ∞ 7→ zφ′(∞) > 0

∞ z

D

ϑ1

ϑ2

φ(ϑ1)

φ(ϑ2)

5. DENSITY OF HARMONIC MEASURE 15

Pick any 0 ≤ ϑ1 < ϑ2 < 2π. Then by conformal invariance of Brownian Motion we

haveϑ2 − ϑ1

2π= hD(∞, [eiϑ1 , eiϑ2 ]) = hD(z, [φ(eiϑ1), φ(eiϑ2)])

=

∫ φ(eiϑ2 )

φ(eiϑ1 )hD(z, w)dw = `([φ(eiϑ1), φ(eiϑ2)]) · hD(z, φ(eiϑ0)),

for some ϑ0 ∈ [ϑ1, ϑ2], where `(γ) denotes the arc length of a curve γ and the last equality

follows from the mean values theorem. We now divide both sides by ϑ2 − ϑ1 and let

ϑ2 ϑ1 to get 12π = |φ′(eiϑ1)| · hD(z, φ(eiϑ1)). This tells us that

hD(z, φ(eiϑ1)) =1

2π|φ′(eiϑ1)|−1.

The fact that the density of harmonic measure at φ(eiϑ) seen from z is inversely pro-

portional to |φ′(eiϑ1)| will be very useful to gain intuition on the effect of distortion in

Hastings–Levitov clusters.

CHAPTER 2

Hastings–Levitov models

The question of obtaining a rigorous description of spatial growth phenomena observed

in nature has received much attention in the last few decades in the mathematical com-

munity. One possible approach to this problem, on which we focus here, is to model the

growth as the result of subsequent aggregation of particles, which randomly move in the

surrounding space. We refer to such dynamics as random aggregation.

1. Discrete aggregation models

A random aggregation model consists of an underlying infinite graph together with

an aggregation rule. We focus here on deterministic, planar, undirected graphs, on whose

vertex set the particles live. Say that a vertex v of the graph is filled if it is occupied

by exactly one particle, and v is empty otherwise. Starting from one particle (i.e. filled

vertex), we grow increasing clusters of particles by aggregating one vertex at the time,

according to the prescribed aggregation rule. When a vertex is added to the cluster we

declare it filled, so that the cluster coincides with the set of filled vertices.

1.1. The Eden model. Arguably, the simplest of such models is the so called Eden

model [Ede61], introduced by Eden in 1961, according to which at each step one empty

site adjacent to the cluster is chosen uniformly at random, and added to the cluster. More

precisely, let the underlying graph be the d–dimensional lattice Zd, and take the initial

cluster to be the singleton E(0) = 0. Define a family of growing clusters (E(n))n≥0

recursively as follows. Write u ∼ v if u is adjacent to v, that is |u− v| = 1, and let

∂E(n− 1) = v /∈ E(n− 1) : v ∼ u for some u ∈ E(n− 1)denote the outer boundary of the set E(n− 1). Then, if ωn is a uniformly chosen element

of ∂E(n− 1), set

E(n) = E(n− 1) ∪ ωn.For such family of growing clusters it is natural to ask if there is a limiting shape as the

number of particles diverges, i.e. as n → ∞. This question was settled by Richardson in

[Ric73], showing that there exists indeed a limiting shape, and it is compact and convex

(see also [CD81, Kes93]). The problem of determining the limiting shape remains open.

1.2. Diffusion Limited Aggregation. A variant of the Eden model was introduced

a few years later by the physicists Witten and Sander [WJS81, WJS83], seeking to

explain the formation of arms, or dendrites, in metal–particles aggregation. This is the

celebrated Diffusion Limited Aggregation, in short DLA, and it is defined similarly to the

Eden model, with the crucial difference that the position of the next added particle is

17

18 2. HASTINGS–LEVITOV MODELS

Figure 1. A large Eden cluster (simulation by Jason Miller).

chosen according to the harmonic measure of the outer boundary of the cluster seen from

infinity, rather than according to the uniform measure on the same set. More precisely, at

each integer time n ≥ 1 we start a simple symmetric random walk ω(n) = (ω(n)(t))t≥0 on Zdfrom infinity, independent of everything else, and let τn := inft ≥ 0 : ω(n)(t) ∈ ∂D(n−1)denote the first time the walk reaches the outer boundary of D(n− 1). Then set

D(n) = D(n− 1) ∪ ω(n)(τn).Note that this aggregation rule promotes the formation of arms, or dendrites. Indeed,

once an arm starts forming, it gains a large harmonic measure, and it therefore keeps on

growing. This is also clear from simulations (see Figure 2).

The asymptotic behaviour of DLA clusters is very different from that of Eden clusters,

and it is far less understood. In 1987 Kesten [Kes87, Kes90] obtained an upper bound

for the length of arms in DLA, showing that in dimension d they can grow at most

at rate n2/(d+1) as the number n of particles in the cluster diverges. After that, the

only other result on DLA is due to Ebertz-Wagner [EW99], who showed that large 2–

dimensional DLA clusters contain almost surely infinitely many holes. We refer the reader

to [Hal00, MS13] for nice accounts on the topic, and to [SS16] for a multi–particle

version of DLA growth.

1.3. ...and many more. Aside from DLA and the Eden model, many discrete models

of random aggregation have been studied in the literature. In fact, to obtain a new model

from the ones described above, one could either change the aggregation rule (i.e. change

2. A CONTINUUM MODEL: HASTINGS–LEVITOV GROWTH 19

Figure 2. A large DLA cluster (simulation by Vincent Beffara).

the measure with respect to which the location of the new added particle is chosen), or

change the underlying graph on which the particles live, or change both. An example of

aggregation rule change is given by the Internal DLA (IDLA) [DF91, LBG92, JLS14a,

JLS14b, JLS13, JLS12, AG10, AG13a, AG13b, AG14], where the location of the

next added particle is sampled according to the harmonic measure of the outer boundary of

the cluster seen from the origin. Other important examples are the Dielectric Breakdown

Models of parameter γ > 0, in short DBMγ , according to which the location of the next

particle is sampled from the γth power of the harmonic measure of the outer boundary

of the cluster seen from infinity [NPW84]. Thus γ = 0 corresponds to the Eden model,

γ = 1 to DLA, and γ ∈ (0, 1) interpolates between the two. Finally, we would like to

mention DLA on cylinder graphs [BY08] and on regular trees [BPP97], Directed DLA

[Mar14], Stretched IDLA [BKP12], Diffusion Limited Deposition [ACSS16] and First

Passage Percolation [HW65, Kes86, Kes03, CD81, Mar04] (see also the recent survey

[AHD15] and references therein).

2. A continuum model: Hastings–Levitov growth

A common feature of some of the discrete aggregation models described above (e.g.

Eden model and DLA) is that their asymptotic properties, such as the limiting shape

and Hausdorff dimension, have been numerically found to depend on the underlying lat-

tice structure (see for example [BBRT85, MBRS87], where simulations of such models


are discussed). This led researchers to look, in the late 1980’s, for off–lattice analogues

[Roh11]. In 2000 Carleson and Makarov [CM01] introduced a deterministic growth

model on the complex plane, defined in terms of Loewner flows driven by time-dependent

measures on the unit circle. Following their ideas, Hastings and Levitov [HL98] proposed

a one parameter family of random growth models, now called HL(α) for α ∈ [0,∞). These

lecture notes are concerned with the study of the α = 0 case, that we now describe.

Figure 3. HL(0) cluster with 500 particles (simulation by Henry Jackson).

2.1. The HL(0) model. Let D denote the open unit disc in the complex plane, and

let K0 = D be the initial cluster. We grow an increasing family K0 ⊂ K1 ⊂ K2 . . . of

compact subsets of the complex plane, that we call clusters, as follows. Fix P ⊂ C \ Dto be a (non–empty) connected compact set having 1 as a limit point, and such that

the complement of K = D ∪ P in C ∪ ∞ is simply connected. We regard P as the

basic particle. For concreteness, note that the slit P = [1, 1 + δ] satisfies all the above

assumptions (this is indeed the particle shape used in all HL(0) simulations included in

these notes).

At each step, a new particle Pn, consisting of a distorted copy of P , attaches to the

cluster Kn−1 according to the following growth mechanism. Let D0 = (C∪∞)\K0 and

D = (C ∪ ∞) \K. Then by Theorem 1.3 there exists a unique conformal isomorphism

F : D0 → D such that F (∞) =∞ and F ′(∞) ∈ R>0. Set G = F−1, and let (Θn)n≥1 be a

sequence of i.i.d. random variables with Θn ∼Uniform[−π, π).


F P1 + δ1

Figure 4. The conformal map F attaching the particle P = [1, 1 + δ] at 1.

Define

Fn(z) := eiΘnF (e−iΘnz) , Gn(z) = F−1n (z) .

Then the map Fn attaches the particle P to the unit disc at the uniformly chosen point

eiΘn . Define Φn(z) := F1 · · · Fn(z), Dn = Φn(D0) and Kn = (C ∪ ∞) \Dn. We say

that the conformal map Φn grows an HL(0) cluster up to the n-th particle, while Γn = Φ−1n

maps it out. Note that, by conformal invariance, choosing the attachment angles to be

uniformly distributed corresponds to choosing the attachment point of the n-th particle

according to the harmonic measure of the boundary of the cluster Kn−1 seen from infinity.

With this notation, then, one has

Dn+1 = Φn Fn+1 Γn(Dn) .

This suggest the following interpretation for the attachment mechanism: given the cluster

Kn, first map it out via Γn, then attach a new copy of the particle P to the unit circle

at a uniformly chosen point eiΘn , and finally grow back the cluster Kn. It is then clear

that, although we are attaching identical copies of the particle P at each step, the particle

shape gets distorted each time by the application of the conformal map Φn, as shown in

Figure 5 below.

Φn = F1 · · · Fn

Figure 5. The conformal map Φn, which grows the cluster up to the nth particle.

Note that looking at the Laurent expansion of F around∞ we find that there exists a

unique constant c ∈ R such that F (z) = ecz +O(1) as |z| → ∞. The constant c is called

logarithmic capacity of P , and it will play a central role in the discussion of HL growth.

We will see that, although c could be a priori any real number, one can show that c > 0.

In fact, if (Bt)t≥0 is a standard complex Brownian Motion starting from a point z outside

the cluster K = D ∪ P , and T := inft ≥ 0 : Bt ∈ K denotes its first hitting time of K,

then

limz→∞

Ez(log |BT |) = c.


This is a consequence of Proposition 3.1 below, and it shows that the logarithmic capacity

is related to the size of the particle P . Indeed, the only positive contribution to the

expectation comes from Brownian trajectories hitting K on P , and the larger is P the

larger is such expectation, and hence c. We will use the logarithmic capacity as a measure

of the size of the particle throughout, and look at the small particle limit c→ 0.

2.2. Distortion in HL(0): a smoothing dynamics. To fix ideas, take the particle

P to be the slit [1, 1 + δ] for some small δ > 0, and let Pn+1 denote the (n+ 1)th copy of

P in the cluster, that is

Pn+1 = Φn

(eiΘn+1P

)=z : z = Φn(w) for some w such that e−iΘn+1w ∈ P

.

ΦnΘn+1

δ

`(Pn+1)

We measure the distortion of the conformal map Φn by comparing the arclength δ of

P with the one of Pn+1, denoted by `(·). By the mean value theorem, we have

`(Pn+1

)=

∫ 1+δ

1|Φ′n(reiΘn+1)|dr = δ |Φ′n(r0e

iΘn+1)|, (5)

for some r0 ∈ [1, 1 + δ]. For small δ, r0 ≈ 1, from which we deduce that

`(Pn+1

)≈ δ |Φ′n(eiΘn+1)|, δ 1.

On the other hand, we know from Section 5 that the density of harmonic measure at the

attachment point Φn(eiΘn+1), that is the probability density that the (n + 1)th particle

attaches to the cluster Kn at that point, is proportional to |Φ′n(eiΘn+1)|−1. This together

with (5) shows that

- particles attaching to the cluster at unlikely locations are large, and

- particles attaching to the cluster at likely locations are small.

Thus, although each new particle is added to the cluster at a location chosen accordingly

to the harmonic measure on the cluster boundary seen from∞, similarly to discrete DLA,

the distortion intrinsic in the model prevents the formation of arms. This suggests that

large HL(0) clusters should look much more regular than DLA clusters, as we will soon

see to be the case.

2.3. Overview of related models. The HL(0) model described above belongs to a

larger class of Hastings–Levitov models called HL(α), indexed by a real parameter α ≥ 0.

The idea behind the definition of HL(α) is that, while in HL(0) we let the size of the

particles in the actual cluster be determined by the distortion of the conformal map Φn

(see (5)), in HL(α) we insist on having some control on it, the strength of which depends

on α. Let, for the moment, P be the slit particle [1, 1 + δ]. Then we have seen in (5)


that for δ 1 the size (meaning arc length) of the particle Pn+1 in the actual cluster is

roughly δ|Φ′n(eiΘn+1)|. To cancel this distortion factor, one could therefore attach a slit of

(random) length δn+1 = δ/|Φ′n(eiΘn+1)|, which will result in a particle of size roughly δ in

the actual cluster. In general, one can vary the strength of such rescaling by introducing

a parameter α > 0 and setting

δn+1 =δ

|Φ′n(eiΘn+1)|α/2 .

Although the above considerations are specific to the slit particle, the same reasoning

can be carried out for more general particle shapes after replacing the arc length by

the logarithmic capacity. Indeed, Norris and Turner show in [NT12] that, in the small

particle limit, the logarithmic capacity of the particle equals (up to absolute multiplicative

constants) the square of its diameter (cf. Corollary 3.1). Thus for general particle shape P

we define the HL(α) model as obtained from HL(0) by attaching, at each step, a rescaled

copy of the basic particle P of logarithmic capacity

cn+1 =c

|Φ′n(eiΘn+1)|α . (6)

Note that α = 2 will result in particles having roughly the same size in the actual cluster.

Since the attachment point is chosen according to harmonic measure from∞, then, HL(2)

seems to be a good off-lattice analogous to discrete DLA. For α > 2, the size of the

particles in the actual cluster becomes proportional to a power of the harmonic measure

of the cluster boundary near the attachment point. This favours the growth of arms, since

to region of high harmonic measure we attach large particles (here by large we mean larger

than what they would have been without the α > 0 regularization). For α ∈ (0, 2), on the

other hand, the α regularization really contrasts the effect of distortion, since the size of

particles in the actual cluster becomes inversely proportional to a power of the density of

harmonic measure close to the attachment point. In fact, the regime α ∈ [1, 2] seems to be

the most interesting from the point of view of modelling real phenomena. Indeed, recall

the definition of Dielectric Breakdown Model (DBM in short) from Section 1.3. Then by

comparing local growth rates Hastings and Levitov argue that for α ∈ [1, 2] the HL(α)

model should be the right off–lattice analogue to DBMα−1, thus in particular recovering

the Eden model for α = 1 and DLA for α = 2. Let us explain this point in more detail.

Recall that in (discrete) DBMη at each step we choose the attachment location of the next

particle on the outer boundary of the cluster according to the ηth power of the harmonic

measure of the cluster boundary seen from infinity, which is roughly |Φ′n(z)|−η. At this

location we attach a particle of fixed size, say c. Thus the rate of growth of the cluster

boundary around a point z is roughly given by

Growth rate of DBMη at z ≈ c |Φ′n(z)|−η.

For HL(α), on the other hand, the rate of growth around z is given by the probability

of attaching a particle at z, roughly |Φ′n(z)|−1, times the size of the attached particle,

roughly c |Φ′n(z)|2−α (see (5)-(6)). Thus

Growth rate of HL(α) at z ≈ c |Φ′n(z)|1−α.


This suggests that for α ∈ [1, 2] the model HL(α) should be the right off–lattice analogue

of DBMα−1, which in particular gives the Eden model for α = 1 and DLA for α = 2.

Figure 6. HL(1) cluster (left) and HL(2) cluster (right) with 25000 par-

ticles. Different colours correspond to different arrival times (simulations

by Alan Sola).

Although taking α > 0 seems to be necessary for these models to be realistic, the

renormalization in (6) creates long range dependences which make them very difficult

to analyse. In fact, to the best of our knowledge there are no rigorous results on (the

non–regularised version of) HL(α) models for α > 0. A first regularised version of HL(α)

appears in [RZ05], in which Rohde and Zinsmeister obtained bounds for the Hausdorff

dimension of suitably regularised clusters for α ∈ [0, 2]. More recently, a different type

of regularization was considered by Sola, Turner and Viklund, in [JVST15], where they

showed that the limiting shape of regularised clusters is given by a disc for any α > 0,

provided that the regularization is strong enough (see Chapter 3 for more one this). In

agreement with [HL98], they conjecture that for α < 1 the regularization should not

matter, with non–regularised HL(α) clusters still converging to discs. Simulations, on the

other hand, suggest that the situation is very different for α > 1 (see [JVST15, HL98]).

In fact, by looking at the boundary fluctuations of large clusters, Hastings and Levitov

propose the existence of a sharp phase transition at α = 1, above which the fluctuations of

the cluster boundary do not seem to decrease as the particle size vanishes, thus suggesting

that a deterministic scaling limit might not exist.

The case α = 0 does not feature such long range dependences, and is much better

understood. In [RZ05] Rohde and Zinsmeister obtained a scaling limit for HL(0) clusters

as the particle size is kept fixed while n→∞. Moreover, they showed that the boundary of

these limiting clusters is almost surely one-dimensional. More recently, in [NT15, NT12]

Norris and Turner obtained a detailed description of HL(0) clusters in the small particle

limit, that we partially review in Chapter 3. They proved that in the small particle limit

large HL(0) clusters almost surely fill a disc, while the ancestral tree structure within the


cluster converges, always in the small particle limit, to the Brownian Web [FINR04].

This provides an interesting connection between these two a priori unrelated models.

A promising variant of Hastings–Levitov growth is the so called Angular Loewner

Evolution of parameter η, ALE(0, η) in short, which keeps the HL(0) distortion but adjusts

the rate at which particles are added to the cluster to compensate for it. Indeed, we saw

that in HL(0) growth large particles are attached to regions of low harmonic measure, and

small particles are attached to regions of high harmonic measure. To compensate this, we

could think of attaching particles to regions of low harmonic measure at a lower rate, and to

regions of high harmonic measure to a higher rate. More precisely, to balance the distortion

factor |Φ′n(eiΘn+1)| we attach particles around eiΘn+1 at rate |Φ′n(eiΘn+1)|−1, that is at a

rate which is directly proportional to the local density of harmonic measure. In general, in

ALE(0, η) at each step we attach a particle of constant logarithmic capacity c, at a random

point eiΘn+1 drawn from a probability density function proportional to |Φ′n(eiΘn+1)|1−η. A

regularised version of this model is studied by Sola, Turner and Viklund in a forthcoming

paper.

CHAPTER 3

Small particle limit

In this Chapter we review the scaling limit result of Norris and Turner [NT12], stating

that as the number of particles diverges and their size converges to 0, large HL(0) clusters

converge to disks. Recall that we take the basic particle P ⊂ C \ D to be a (non–

empty) connected compact set having 1 as a limit point, and such that the complement of

K = D∪P in C∪∞ is simply connected. We make the following additional assumptions

on P throughout.

Assumption 1. The basic particle P is such that the unique map F : D0 → D extends

continuously to D0 = z ∈ C : |z| ≥ 1. Moreover, there exists δ > 0 such that

P ⊆ z ∈ C : |z − 1| ≤ δ, 1 + δ ∈ P, P = z : z ∈ P. (7)

For concreteness the reader can think about the slit particle P = [1, 1 + δ].

Exercise 3.1. Write down the mapping out function Gs of the slit particle. Show

that c = − log |G′s(∞)| = − log(

1− δ2

(2+δ)2

). (Hint: reduce to the upper half plane.)

We regard δ as a measure of the diameter of the basic particle P , and think of it as

being small, vanishing as n → ∞. This chapter is devoted to the proof of the following

result.

Theorem 3.1 ([NT12], Proposition 5.1). For ε ∈ [2δ, 1] and any m ∈ N, denote by

Ω(m, ε) the event defined by the following conditions: for all n ≤ m, it holds

|e−cnΦn(z)− z| < εe6ε for all z : |z| ≥ e5ε,

and

|Γn(z)− e−cnz| < εe5ε for all z : |z| ≥ ecn+4ε.

Then there exists an absolute constant C ∈ (0,∞) such that

P(Ω(m, ε)c) ≤ (m+ ε−2)e−ε3/Cc.

Note that by taking m = bδ−6c and ε = δ2/3 log(1/δ) we get that P(Ω(m, ε)c) decays

faster than any polynomial in δ, as δ → 0. In particular, the event Ω(m, ε) holds almost

surely for δ small enough by Borel-Cantelli. This tells us that, for large enough n and

small enough δ, the conformal map Φn which grows the cluster up to the nth particle is

uniformly close to the map z 7→ ecnz, which simply grows a ball of radius ecn around the

origin. To keep the asymptotic size of clusters finite and macroscopic, then, we make the

following assumption.

Assumption 2 (Small particle limit). We study HL(0) clusters in the small particle

limit, that is in the limit as n→∞, c→ 0 and nc→ t, for arbitrary but fixed t ∈ (0,∞).

27

28 3. SMALL PARTICLE LIMIT

This assumption will be in force throughout these lecture notes, so that when we write

n→∞ or c→ 0 we in fact mean the above.

Remark 3.1. We will see in Corollary 3.1 that as c → 0 also δ → 0 with c δ2.

Thus when the logarithmic capacity of particles converges to 0 so does their Euclidean

size, measured by δ.

Figure 1. n = 100.

Figure 2. n = 5000.

Figure 3. n = 500.

Figure 4. n = 100000.

Figure 5. HL(0) clusters (simulations by Henry Jackson).

It follows that

Φn(z) ≈ etz as nc→ t,

3. SMALL PARTICLE LIMIT 29

which shows that the map that grows the n–particles cluster is close, in the small particle

limit, to the map that grows an Euclidean ball of radius et around the origin. In this sense

we say that the scaling limit of HL(0) clusters is an Euclidean ball.

Remark 3.2. Although one can let |z| → 1 as n → ∞ in Theorem 3.1 of Norris and

Turner, the limiting shape result breaks down (and it should) if |z| → 1 too fast with

respect to δ. On the other hand, we will show in Chapter 4 that it does not break too

much: it still holds in the sense of distributions, for a suitably small space of test functions.

The analysis carried out by Norris and Turner in [NT12] is rather sophisticated, and it

gives much more information than what stated in Theorem 3.1. In particular, it provides

very accurate estimates on the distortion of the conformal map Φn, that are crucial for

the analysis of fluctuations. On the other hand, the limiting shape of large clusters can

be guessed by very simple heuristic arguments.

Heuristics: a Loewner flow argument. The heuristics presented below is inspired

by the discussion of anisotropic Hastings–Levitov models of Sola, Turner and Viklund in

[JVST15], and it provides an exact guess for the limiting shape.

The main observation is that, if P is the slit particle P = [1, 1 + δ], then Hastings–

Levitov clusters are Loewner hulls associated to the Loewner flow driven by a specific

random measure, consisting of Dirac masses at the attachment points. More precisely,

recall that (Θk)k≥1 denotes the sequence of attachment angles, and define

ξn(t) :=n∑k=1

eiΘk1[c(k−1),ck)(t).

Let (ft)t≥0 be the Loewner flow driven by the random measure δξn(t), where δx denote the

Dirac delta measure centred at x, i.e.

∂tft(z) = zf ′t(z)

∫|z|=1

z + ζ

z − ζ dδξn(t)(ζ), f0(z) = z. (8)

Then Φn(z) = fnc(z) for the slit particle P = [1, 1 + δ]. Using the Loewner equation (8)

for ft, we find

∂tft(z) = zf ′t(z)

∫|z|=1

z + ζ

z − ζ

( n∑k=1

dδeiΘk (ζ)1[c(k−1),ck)(t)

)

= zf ′t(z)

n∑k=1

(∫|z|=1

z + ζ

z − ζ dδeiΘk (ζ)

)1[c(k−1),ck)(t)

= zf ′t(z)

n∑k=1

z + eiΘk

z − eiΘk 1[c(k−1),ck)(t).

Now, if we assume that there exists a deterministic scaling limit, then the random sum

appearing in the last term above should converge to a deterministic limit itself. To get an


idea of how this limit should look like, we compute its expectation:

E( n∑k=1

z + eiΘk

z − eiΘk 1[c(k−1),ck)(t)

)=

n∑k=1

E(z + eiΘk

z − eiΘk)

1[c(k−1),ck)(t)

= E(z + eiΘ1

z − eiΘ1

)=

∫|z|=1

z + ζ

z − ζdζ

2π= 1.

Thus if there is a deterministic limit for the random map Φn = fnc, then it must coincide

with the solution ft to ∂ft(z) = zf ′t(z)

f0(z) = z

evaluated at time t = nc. The above is easy to solve, to get ft(z) = etz. Thus we conclude

that if there exists a random conformal map ft such that Φn → ft as n→∞, nc→ t, then

it must be ft(z) = etz.

1. Preliminary estimates

In order to prove Theorem 3.1, and throughout the course, we will need to understand

how much the conformal maps Φn,Γn move points around. To this end, we look at

distortion estimates for the basic maps F,G. To start with, note that:

(P1) |F (z)| > |z| for all z ∈ D0, |G(z)| < |z| for all z ∈ D,

(P2) there exists a constant C > 0 such that |F (z)| ≤ C|z| for all z ∈ D0 and |G(z)| ≥|z|/C for all z ∈ D.

Indeed, (P1) follows from Schwarz lemma (cf. Lemma 1.1), while (P2) is a consequence

of the prescribed behaviour at infinity for F,G. In order to get more refined estimates, it

turns out to be useful to look at F,G in logarithmic coordinates. To this end, note that the

holomorphic functions F (z)/z and G(z)/z are non–zero on D0 and D respectively, with

a controlled behaviour at infinity. It follows that we can define their complex logarithm

on the same domains, to get the holomorphic functions logF (z)/z and logG(z)/z. We fix

the branch by requiring that logF (z)/z → c and logG(z)/z → c as |z| → ∞.

The following result appears in [NT12] (cf. Proposition 4.1 and Corollary 4.2 therein).

Proposition 3.1. There exists an absolute constant C <∞ such that, for all z ∈ D:

|z − 1| > 2δ, the following hold:∣∣∣∣ logG(z)

z+ c

∣∣∣∣ ≤ Cc

|z − 1| ,∣∣∣∣ d

dzlog

G(z)

z

∣∣∣∣ ≤ Cc

|z − 1|2 . (9)

Remark 3.3 (Geometric interpretation of c). Write log G(z)z = u(z) + iv(z). Then u, v

are bounded harmonic functions on D. It follows that if (Bt)t≥0 is a standard Brownian

Motion on D starting at z ∈ D, then (u(Bt))t≥0 and (v(Bt))t≥0 are continuous martingales.

Let T = inft ≥ 0 : Bt /∈ D denote the first hitting time of the boundary of the cluster.

Then T <∞ almost surely, and we can use the Optional Stopping theorem to see that

u(z) = E(u(B0)) = E(u(BT )) = E(

log|G(BT )||BT |

)= −E(log |BT |).

1. PRELIMINARY ESTIMATES 31

On the other hand, we know that lim|z|→∞ u(z) = lim|z|→∞ log G(z)z = log e−c = −c. We

therefore conclude that

c = lim|z|→∞

E(log |BT |). (10)

In particular, this tells us that c ≥ 0, which was not a priori clear. We will see in the

proof of Proposition 3.1 that in fact c > 0. Moreover, (10) shows that the logarithmic

capacity c of a particle P is closely related to the size of the particle. Indeed, the only

contribution to the expectation comes from Brownian trajectories hitting the cluster on

P , so that |BT | > 1, and the larger the particle P , the larger the expectation, and hence

c.

z

BT

The next corollary shows that, in fact, the relationship between c and the Euclidean

size of P can be made more precise.

Corollary 3.1. It holdsδ2

6≤ c ≤ 3δ2

4(11)

for δ small enough.

In light of (11) we use c and δ interchangeably, and all statements are intended to hold

for c, δ small enough (recall that we are looking at the small particle limit).

We can now proceed with the proof of Proposition 3.1.

Proof. Let u, v denote the real and imaginary part of log G(z)z respectively, so that

they are harmonic functions on D. Then by optional stopping u(z) = −E(log |BT |) < 0,

T being the first hitting time of K for a Brownian Motion B starting from z. Introduce

the particle P1 ⊃ P defined by P1 =z ∈ D0 :

∣∣ z−1z+1

∣∣ ≤ r

, for r = δ/(2 − δ), and set

D1 = (C ∪ ∞) \ (D ∪ P1). Then the unique conformal map G1 : D1 → D0 satisfying

G1(∞) =∞ and G′1(∞) > 0 is given by

G1(z) =z(γz − 1)

z − γ for γ =1− r2

1 + r2. (12)

Set F1 = G−11 , and A = z ∈ ∂P1 : |z| > 1. Then G1(A) = eiϑ : |ϑ| < ϑ0 with ϑ0 =

cos−1 γ. Moreover, u F1 is harmonic and bounded on D0 and, using that 12π

∫|ϑ|≤ϑ0

(u F1)(eiϑ)dϑ = −c, the optional stopping theorem yields

(u F1)(z) = −c+1

2π

∫|ϑ|≤ϑ0

(u F1)(eiϑ) Re( 2eiϑ

z − eiϑ)

dϑ .


AG1 ϑ0

G1(A)

Since (u F1)(eiϑ) < 0 for all ϑ in the integration range, we find∣∣(u F1)(z) + c∣∣ ≤ 2

dist(z,G1(A))

∣∣∣∣ 1

2π

∫|ϑ|≤ϑ0

(u F1)(eiϑ)dϑ

∣∣∣∣ =2c

dist(z,G1(A)).

Now take G1(z) in place of z, to get∣∣u(z) + c∣∣ ≤ 2c

dist(G1(z), G1(A)). (13)

Using the explicit expression for G1, one shows that there exists an absolute constant C1

such that dist(G1(z), G1(A)) ≥ |z − 1|/C1, to get∣∣u(z) + c∣∣ ≤ 2C1c

|z − 1| (14)

for all z ∈ D such that |z − 1| > 2δ. It remains to show that the imaginary part v(z) is

small. We adopt the following strategy. Use the integral representation of u in terms of

the Poisson kernel, to make (14) into a bound for |∇u(z)|. Then observe that by Cauchy–

Riemann equations |∇u(z)| = |∇v(z)|, so this gives us a bound on |∇v(z)|. Finally, from

this we recover a bound on |v(z)| by integration.

Fix z such that |z−1| > 2δ, and let BR(z) := w : |w−z| ≤ R, for R = |z−1|2 . Denote

by γ the boundary of BR(z). Then, for any w ∈ BR(z), writing w = z + reiθ we have

u(w) = Ew(u(BT )) =1

2π

∫γu(z +Reiφ)Pr/R(θ − φ)dφ,

where (Bt)t≥0 is a standard Brownian Motion starting from w, T is its first exit time from

BR(z) and Pr(φ) = Re(

1+reiφ

1−reiφ)

is the Poisson kernel for the unit disc. Then

|∇u(z)| ≤ 1

2π

∫γ|u(z + reiφ)||∇Pr/R(φ− ϑ)|r=0dφ. (15)

It is easier to compute |∇Pr/R(φ− θ)| directly in polar coordinates. By the chain rule, we

have

|∇Pr(φ)|2 =∣∣∣∂Pr(φ)

∂r

∣∣∣2 +1

r2

∣∣∣∂Pr(φ)

∂φ

∣∣∣2 =4

|reiφ − 1|4 .

It follows that ∣∣∣∇P rR

(ϑ− φ)∣∣∣r=0

=2

R2|rei(ϑ−φ) − 1|2∣∣∣r=0

=2

R2=

8

|z − 1|2 .

Plugging this into (15) and using (14) we conclude that

|∇u(z)| ≤ 8

2π|z − 1|2∫γ|u(ζ)|dζ ≤ 16C1c

|z − 1|2 ,

where in the last inequality we have used that if ζ ∈ γ then |ζ − 1| ≥ |z − 1|/2. By

Chauchy–Riemann, it follows that the same bound holds for |∇v(z)|. Finally, using the


Fundamental Theorem of Calculus (also known as Gradient Theorem in this setting) we

find

v(z) =

∫ ∞0∇v(z + s(z − 1)) · (z − 1)ds,

from which

|v(z)| ≤∫ ∞

0|∇v(z + s(z − 1))| · |z − 1|ds ≤ 16C1c

|z − 1|

∫ ∞0

1

(1 + s)2ds =

16C1c

|z − 1| .

This, together with (14), concludes the proof.

A similar strategy allows us to prove Corollary 3.1.

Proof of Corollary 3.1. We prove the upper bound by surrounding the particle

P by the larger particle P1, as in the previous proof, the advantage being that we have an

explicit expression for the mapping out function of P1. Recall that K0 = D, K = K0 ∪ P

G

K1

A

P

G(A)

G(K1 \K)K1 \K

G1 G†

P1

Figure 6. Proof of the upper bound. The functions G, G1 and G† map

out the particle P , P1 and G(K1 \K) respectively.

and K1 = K0 ∪ P1 ⊃ K. Then, if cap(K) denotes the logarithmic capacity of a compact

set K, we have

cap(K) ≤ cap(K) + cap(G(K1 \K)) = cap(K1) = − lim|z|→∞

logG1(z)

z

= − lim|z|→∞

logγz − 1

z − γ = − log γ = log1 + r2

1− r2≤ 3δ2

4,

where the second equality follows from (9), and the following ones are obtained by using

(12) and by recalling that r = δ/(2− δ).The proof of the lower bound is similar, but instead of surrounding the particle P by

a larger one we observe that P contains, by assumption, the slit particle Ps = [1, 1 + δ],

for which we are able to explicitly write down the mapping out function. Let Gs denote


Gs

K1

P

Gs(K \Ks)

G G††

Figure 7. Proof of the lower bound. The functions Gs, G and G†† map

out the particle Ps, P and Gs(K \Ks) respectively.

the mapping out function of the slit particle, and Ks = K0 ∪ Ps. Then we find

cap(K) = cap(Ks) + cap(Gs(K \Ks)) ≥ − log(

1− δ2

(2 + δ)2

)≥ δ2

6,

where the first inequality follows from the explicit computation in Exercise 3.1.

2. The scaling limit result

We can now set up things for the proof of Theorem 3.1. As already observed, it

is convenient to work in logarithmic coordinate. indeed, the underlying multiplicative

structure that comes from the iterated composition of conformal maps becomes an additive

structure in logarithmic coordinates. This is useful, for example, to build martingales, as

we will see in a second.

2.1. The logarithmic picture. Let

D := z : ez ∈ D, D0 := z : ez ∈ D0 = z : Re z > 0.There exist unique conformal maps F : D0 → D and G : D → D0 such that F (z)−(z+c)→0 and G(z)− (z− c)→ 0 as Re z → +∞. Note that G = F−1, F , G are 2π-periodic in the

imaginary coordinate, and G exp = exp G, F exp = exp F .

0

π

−π

P

D G

F0

π

−π

D0

2. THE SCALING LIMIT RESULT 35

Let (Θn)n≥1 denote the sequence of attachment angles. for each n ≥ 1, set Fn(z) =

F (z − iΘn) + iΘn, and define

Φn := F1 F2 . . . FnΓn := Gn Gn−1 . . . G1.

Then Φn grows the cluster Kn, while Γn maps it out.

Original picture Logarithmic pictureG(z)

z≈ e−c as |z| → ∞ G(z)− z ≈ −c as Re z →∞

F (z)

z≈ ec as |z| → ∞ F (z)− z ≈ c as Re z →∞∣∣∣ log

G(z)

z+ c∣∣∣ ≤ Cc

|z − 1| for |z − 1| > 2δ |G(z)− z + c| ≤ Cc

|ez − 1| for |ez − 1| > 2δ∣∣∣ d

dzlog

G(z)

z

∣∣∣ ≤ Cc

|z − 1|2 for |z − 1| > 2δ |G′(z)− 1| ≤ Cc|ez||z − 1|2 for |ez − 1| > 2δ.

Table 1. Recap of distortion estimates for F,G (left) and their analogues

in logarithmic coordinates (right).

Remark 3.4. Note that(Γn(z)

)n≤Nz is a Markov process with respect to σ(Θ1,Θ2, . . .)

for any fixed z, where Nz denotes the first time z is absorbed by the cluster.

For fixed m ∈ N and ε ∈ [2δ, 1], let the event Ω(m, ε) be defined by the following

conditions:

|Φn(z)− (z + cn)| < ε for all z : Re z ≥ 5ε,

and

|Γn(z)− (z − cn)| < ε for all z : Re z ≥ cn+ 4ε.

Exercise 3.2. Use the (complex version of) the mean value theorem to show that

Ω(m, ε) ⊆ Ω(m, ε).

By the above exercise, in order to prove Theorem 3.1 it suffices to show that

P(Ω(m, ε)c) ≤ (m+ ε−2)e−ε3/Cc (16)

for some absolute constant C ∈ (0,∞).

2.2. Proof of Theorem 3.1. Before proceeding with the proof, let us briefly sketch

the strategy. We say that Γn behaves well at z if |Γn(z)− (z − cn)| < ε (this is precisely

the condition appearing in Ω(m, ε)). For R ∈ 4ε, 6ε, 8ε, . . . , 2ε⌈cm2ε

⌉, proceed as follows:

(1) Define the event

ΩR := Γn behaves well on Re z ≥ R, for all n = 1, . . . , TR,where TR is a stopping time introduced so that Gn is well defined on Re z ≥ R

for all n ≤ TR.


(2) Argue that⋂R ΩR ⊆ Ω(m, ε), so it suffices to show that each ΩR holds with high

probability (then use the union bound).

(3) Focus on one R. We want to show that |Γn(z)− (z− cn)| < ε for all z : Re z ≥ Rand n ≤ TR.

(4) Since Γn(z)−(z−cn) is holomorphic, it suffices to prove that the bound holds for

z on `R := z : Re z = R. Note that we have 2 variables to control: n (discrete)

and Im z (continuous).

(5) For each z ∈ `R, Mn(z) := Γn(z)− (z − cn) is a C-valued martingale for n ≤ TR.

Use the distortion estimates of the previous section to control the increments,

and obtain a bound on supn≤TR |Mn(z)| thanks to Azuma–Hoeffding inequality.

(6) To control the continuous variable Im z, note that for all fixed z, z′ ∈ `R, Mn(z)−Mn(z′) is a martingale. Use standard martingale techniques and distortion esti-

mates to bound supn≤TR |Mn(z) −Mn(z′)| in terms of |z − z′|. Then conclude

a uniform control for supn≤TR |Mn(z) − Mn(z′)| in z thanks to Kolmogorov’s

continuity criterion.

We can now proceed with a detailed proof of (16), and hence of the scaling limit result.

Proof of Theorem 3.1. If ε3 < c then e−ε3/Cc ≥ e−1/C and so C(m+ε−2)e−ε

3/Cc ≥Ce−1/C which diverges as C diverges to infinity. It follows that by taking C large enough

the right hand side of (16) can be made larger than 1, and hence the statement holds

trivially. We can therefore assume ε3 ≥ c.Set M =

⌈cm2ε

⌉, k ∈ 1, 2, . . . ,M and1 R = 2(k+1)ε. Note that R ≤ cm+4ε. Recall

that `R = z : Re z = R, and let

TR := infn ≥ 0 : z /∈ Dn or Re Γn(z) ≤ R− cn− ε for some z ∈ `R ∧N,

where N is the largest integer such that cN ≤ R− 2ε. Note that, by definition,

Re Γn−1(z) > R− c(TR − 1)− ε ≥ c+ ε > ε > δ > log(1 + δ).

Hence at time TR − 1 the point z lies outside a ball of radius 1 + δ centred at the origin,

which implies that when the next particle is added the point z cannot be absorbed in the

cluster. Thus z ∈ DTR , and in fact

TR := infn ≥ 0 : Re Γn(z) ≤ R− cn− ε for some z ∈ `R ∧N.

For each R, define the event

ΩR :=

supn≤TR,z∈`R

|Γn(z)− z + cn| < ε,

and set Ω0(m, ε) :=M⋂R=1

ΩR. We claim that

P(ΩcR

)≤ Cε−6/5e−ε

3/Cc (17)

1Here we are abusing notation, since the value of R depends on k. We choose to omit the k dependence

to keep the notation simple.


(note that the upper bound does not depend on R). Assuming this, we find

P(Ω0(m, ε)c

)≤

M∑k=1

P(Ωc

2(k+1)ε

)≤MCε−6/5e−ε

3/Cc

≤(cm

2ε+ 1)Cε−6/5e−ε

3/Cc ≤ C(m+ ε−2)e−ε3/Cc.

The theorem will then follow provided we can show that Ω0(m, ε) ⊆ Ω(m, ε). To see

this, assume that Ω0(m, ε) holds. Then ΩR holds for all R = 1, . . . ,M . Take z such that

Re z ≥ 4ε+ cn. Then we can choose R so that Re z ≥ R, and so by the definition of ΩR

supn≤TR,z∈`R

|Γn(z)− z + cn| = supn≤TR,Re z≥R

|Γn(z)− z + cn| < ε,

where the first equality holds by the maximum principle (cf. Theorem 1.6). Moreover,

since on ΩR we have |ΓTR(z) − z + cTR| < ε for all z ∈ `R, it must be TR = N . It

follows that the second condition defining Ω(m, ε) holds. For the first one, simply map

everything back via Γn: we show that z : Re z ≥ 5ε gets mapped to z : Re z > cn+ εby Φn. Indeed, if z ∈ `R then Re Γn(z) < 5ε, so the curve Γn(`R) lies to the left of

z : Re z ≥ 5ε. Thus the image of z : Re z ≥ 5ε must lie to the right of `R. More

precisely, take R maximum so that R ≤ m + 4ε. Then if z ∈ `R, on the event ΩR we

have that |Γn(z) − z + cn| < ε, which implies Re Γn(z) < ε + Re(z) − cn ≤ 5ε. Thus if

Re z ≥ 5ε, Φn(z) > R, from which |Γn(Φn(z))− Φn(z) + cn| = |Φn(z)− z − cn| < ε. This

shows that Ω0(m, ε) ⊆ Ω(m, ε).

Φn

Γn

`5εΦn(`5ε)

`RΓn(`R)

In order to conclude the proof of the theorem now we only have to show that (17)

holds for each fixed R ∈ 1, . . . ,M. As mentioned, we do so via martingale techniques.

For z ∈ Dn, let

Mn(z) := Γn(z)− z + cn.

Then

Mn+1(z)−Mn(z) = Γn+1(z)− Γn(z) + c

= G(Γn(z)− iΘn+1)− (Γn(z)− iΘn+1) + c

= G0(Γn(z)− iΘn+1) + c,


where we have set G0(z) = G(z)− z. Recalling that Fn := σ(Θ1, . . . ,Θn), we have that

E(Mn+1(z)−Mn(z)|Fn) = E(G0(Γn(z)− iΘn+1)|Fn) + c

=1

2π

∫ 2π

0G0(w + iϑ)dϑ+ c = 0,

where in the last equality we have used that G0 is holomorphic and bounded in D, 2πi

periodic, with G0(z)→ −c as Re z → +∞, and so by Cauchy’s theorem the above integral

equals −c. It follows that (Mn(z))n≤T is a martingale for all z ∈ `R. Note that `R ⊂ D

at least up to time T , so Mn(z) is well defined for all n ≤ T .

By (9) in logarithmic coordinates (see Table 1), we have

|G0(z) + c| ≤ C1c

|ez − 1| ≤C1c

Re z,

and

|G′0(z)| ≤ 2C1c

Re z ∨ Re2 zfor Re z ≥ 2δ. Therefore we find

|Mn+1(z)−Mn(z)| = |G0(Γn(z)− iΘn+1) + c| ≤ C1c

Re(Γn(z)− iΘn+1)<

C1c

R− cn− ε,

since Re Γn(z) > R− cn− ε for n ≤ T . Thus, using that cN ≤ R− 2ε, we obtain

N−1∑n=0

|Mn+1(z)−Mn(z)|2 ≤N−1∑n=0

C21c

2

(R− cn− ε)2≤∫ cN

0

C21c

(R− s− ε)2ds ≤ C2

1c

ε.

We plug this into Azuma’s inequality to gather that

P(

supn≤N|Mn(z)| ≥ ε

2

)≤ 2e−ε

3/8C21c. (18)

This shows that, for each fixed z ∈ `R, the martingale Mn(z) is small up to time T ≤ N .

It remains to show that this holds uniformly in z ∈ `R. Take z, z′ ∈ `R, and define

Mn := Mn(z)−Mn(z′).

Note that (Mn)n≤T is a martingale (the reader should check this). We again have to show

that it is small. Let

f(n) = E(

supk≤T∧n

|Mk|2).

By Doob’s L2 inequality we have

f(n) ≤ 4E(|MT∧n|2) ≤ 4n−1∑k=0

E(|Mk+1 − Mk|21(k ≤ T )).

Now define q(r) := r ∨ r2 for brevity. Then

|Mk+1 − Mk| = |G0(Γn(z)− iΘk+1)− G0(Γn(z′)− iΘk+1)|

≤ maxw∈Γn(`R)

|G′0(w)| · |Γn(z)− Γn(z′)| ≤ 2C1c

q(R− cn− ε) |Γn(z)− Γn(z′)|

≤ 2C1c

q(R− cn− ε)(|z − z′|+ |Mn|).


It follows that

E(|Mk+1 − Mk|21(k ≤ T )) ≤ 4

[2C1c

q(R− cn− ε)

]2

E((|z − z′|+ |Mn|)2)

≤ 8

[2C1c

q(R− cn− ε)

]2

(|z − z′|2 + f(k)),

and hence

f(n) ≤ 32C21c

2n−1∑k=0

|z − z′|2 + f(k)

(q(R− ck − ε))2.

It is left to the reader as an exercise to show that this implies

f(n) = E(

supk≤T∧n

|Mk|2)≤ C2

c

ε3|z − z′|2

for some absolute constant C2 (hint: use the discrete version of Gronwall’s inequality).

Kolmogorov’s continuity theorem, then, yields the existence of a random variable M with

E(M 2) ≤ C2cε3

such that

supk≤T|Mk(z)−Mk(z

′)| ≤M |z − z′|1/3

for all z, z′ ∈ `R, almost surely. Note that by periodicity |z − z′| ≤ π. On the other hand,

the above estimate is good when z and z′ are close. To take this into account, we introduce

L ≥ 2 and observe that

P(


′)| ≥ ε

2for some z, z′ ∈ `R with |z − z′| ≤ π

L

)≤ P

(M |z − z′|1/3 ≥ ε

2for some z, z′ ∈ `R with |z − z′| ≤ π

L

)≤ 4

ε2

(πL

)2/3E(M 2) ≤ 4C2c

ε5

(πL

)2/3.

It follows that

P(ΩcR) = P

(sup

z∈`R,n≤T|Mn(z)| > ε

)= P

(∃z ∈ `R : sup

k≤T|Mk(z)| > ε

)≤ P

(∃z ∈ `R : sup

k≤T|Mk(z)−Mk(z

′)|+ supk≤T|Mk(z

′)| > ε for some |z − z′| ≤ π/L)

≤ P(


′)| > ε

2for some z, z′ ∈ `R, |z − z′| ≤ π/L

)+ P

(supk≤T|Mk(z

′)| > ε

2

)≤ 4C2c

ε5

(πL

)2/3+ Le−ε

3/8C21c,

where the last inequality follows from (18) using that L ≥ 2. It is now left to the reader

to optimise over L, to gather that

P(ΩcR) ≤ Cε−6/5e−ε

3/Cc

for some absolute constant C > 0. This proves Claim 17, thus concluding the proof of

Theorem 3.1.


3. Scaling limit of regularised HL(α) clusters

Recall from Section 2.3 the definition of HL(α) clusters, for arbitrary α > 0. These are

uniquely encoded by a sequence of random conformal maps that with abuse of notation

we still call (Φn)n≥1, built as follows. Let F be the (deterministic) unique conformal map

attaching the particle P at 1, with F (z) = ecz +O(1) as |z| → ∞, as in the definition of

HL(0). More generally, for any c > 0 denote by F (c) the unique conformal map attaching a

rescaled copy of P at 1 so to make the logarithmic capacity equal to c, i.e. F (z) = ecz+O(1)

as |z| → ∞. Recall that (Θk)k≥1 denotes a sequence of i.i.d. uniformly chosen angles. Set

Φ0(z) = z and c1 = c. Finally, for each n ≥ 1 define

Fn(z) := eiΘnF (cn)(e−iΘnz), Φn(z) = F1 F2 . . . Fn(z),

with

ck+1 =c

|Φ′k(eiΘk+1)|α, 0 ≤ k < n, (19)

assuming enough regularity of the map Φk close to the boundary of the unit disc for all k.

Then the sequence of random conformal maps (Φn)n≥0 uniquely encodes a growing family

of random clusters (Kn)n≥0 in the complex plane by setting

Kn := (C ∪ ∞) \ Φn(1 < |z| ≤ +∞).

See Section 2.3 for comments on the role of α, motivating the introduction of such models.

We are interested in describing the scaling limit, whenever it exists, of HL(α) clusters

in the small particle limit, that is as n → ∞ and c → 0. As stated, this is open, and

we can readily see why: the procedure of rescaling the original logarithmic capacity c by

the random factor |Φ′n(eiΘn+1)|−α introduces long range dependencies in the model, which

destroy the martingale structure exploited in the proof of Theorem 3.1. Note that, on the

other hand, while Φ′n(z) could be badly behaved at z on (or near) the boundary of the

unit disc, it is reasonable to expect to obtain some control on it for |z| large enough. This

is the main idea behind the introduction of a regularised version of HL(α) models due to

Sola, Turner and Viklund in [JVST15]. Indeed, following ideas of [CM02, RZ05], the

aforementioned authors define HL(α, σ) clusters as being built analogously to HL(α) ones,

but with

cn+1 =c

|Φ′n(eiΘn+1+σ)|α,

for σ > 0, possible converging to 0 as c → 0. The fact that the map Φ′n is evaluated at

distance eσ − 1 > σ from the boundary of the unit disc, rather than on the boundary

itself, makes it possible to use distortion estimates to show that, in fact, if σ c then the

random factor |Φ′n(eiΘn+1)|−α behaves roughly deterministically, so that the sequence of

random logarithmic capacities (cn)n≥1 is well approximated by a deterministic one:

cn+1 ≈ c∗n+1 :=c

1 + αcn, n ≥ 1. (20)

For the sequence (c∗n)n≥1 it is not hard to see that the proof of Theorem 3.1 goes through,

thus giving the scaling limit. This program is carried out in [JVST15], where the authors

specialise to the slit particle P = [1, 1 + δ] and prove the following result.

3. SCALING LIMIT OF REGULARISED HL(α) CLUSTERS 41

Theorem 3.2 ([JVST15]). Let α > 0 be fixed, and let (Φn)n≥0 denote the sequence

of random conformal maps growing HL(α, σ) clusters in the complex plane. Assume that

σ (log c−1)−1/2 as c → 0. Then, in the limit as n → ∞, c → 0 and nc → t ∈ (0,∞)

the map Φn converges in distribution to the deterministic map Ψt(z) := (1 +αt)1/αz, with

respect to uniform convergence on compact sets bounded away from the unit disc.

Note that as α 0 one recovers the scaling limit z 7→ etz of HL(0) clusters.

While we do not try to survey the proof of this result, we next briefly comment on

the form of the approximating sequence (c∗n)n≥1 defined in (20), following the discussion

in [JVST15]. To see how this sequence arises, let us look at very large values of σ. By

the chain rule we have

|Φ′n(z)| =n∏k=1

|F ′k(Φn−k(z))|, z = eiΘn+1+σ.

Now if σ 1, formally σ = +∞, we have

|F ′k(Φn−k(z))| = |F ′(∞)| = eck = exp

c

|Φ′k−1(eiΘk+1+σ)|α,

from which

|Φ′n(z)|−α =n∏k=1

exp

− α c

|Φ′k−1(z)|α. (21)

Setting q(k) = |Φ′k(z)|−α for brevity, we have

q(n) = exp− αc

n−1∑k=1

q(k).

It only remains to solve the recursion. We find

q(k)− q(k − 1)

q(k − 1)= eαcq(k−1) − 1 = −αcq(k − 1) +O(c2q(k − 1)2)

= −αcq(k − 1) +O(c2)

(22)

where we have used that c 1 to Taylor expand the exponential, and the fact that

q(k− 1) ≤ 1, which follows from (21). Dividing further both sides of (22) by q(k− 1) and

summing over k we findn∑k=1

q(k)− q(k − 1)

q(k − 1)2≈∫ n

0

q′(x)

q(x)2dx = 1− 1

q(n),

where q(x) is any differentiable interpolation of q(n). Thus we end up with 1 − 1q(n) =

−αcn+ nO(c2) which, rearranging and using that nc→ t, gives

q(n) =1

1 + αcn+O(c).

In conclusion, we have shown that for σ 1

cn+1 = cq(n) =c

1 + αcn+O(c2),

which justifies (20).

CHAPTER 4

Fluctuations

In the previous chapter we have shown that, in the small particle limit, the conformal

map Φn, which grows HL(0) clusters up to the nth particle, converges almost surely to

the deterministic map z 7→ etz, which simply grows a ball of radius et around the origin.

This convergence in particular holds on all compact subsets of D0 = (C ∪ ∞) \ D, with

respect to the supremum norm. This chapter is devoted to the analysis of the random

fluctuations around this deterministic behaviour, studied in [Sil15]. These can be divided

into local and global fluctuations.

Local fluctuations. Suppose we fix z ∈ C \ D, say z = eσ+ia for some σ > 0,

a ∈ [−π, π), and look at the limiting fluctuations of log Φn(z) around its mean as the

HL(0) cluster grows (n→∞) and the point z approaches the unit disc radially (σ → 0).

The next result shows that, provided σ → 0 slowly enough, these limiting fluctuations are

Gaussian. Moreover, approaching the unit disc radially from different angles results in

asymptotically independent fluctuations.

Theorem 4.1 (Local fluctuations, [Sil15]). Fix t > 0, and let z = eia+σ for some

a ∈ [−π, π), σ > 0. Then as n→∞, nc→ t and σ → 0 with σ δ1/3√

log(1/δ), it holds

log Φn(z)z − nc√

c log( 12σ )

−→ N (0, 1)

in distribution, where N (0, 1) denotes the law of a standard complex Gaussian random

variable. Moreover, if w = eσ and limσ→0a

2σ =: α ∈ [0,∞], then log Φn(z)z − nc√

c log( 12σ )

,log Φn(w)

w − nc√c log( 1

2σ )

−→ (N1,N2)

in distribution, where (N1,N2) is a random vector with centred complex Gaussian entries,

and covariance structure given by

E(N1N2) =

(1

1+α2 − α1+α2

α1+α2

11+α2

),

with the convention that 11+α2 = α

1+α2 = 0 when α =∞.

Note that this in particular shows that fluctuations at any two macroscopically sep-

arated points (i.e. a > 0 fixed) approaching the unit disc (i.e. σ → 0) are independent.

In other words, insisting on approaching the cluster boundary while the cluster grows has

the effect of killing correlations.

43

44 4. FLUCTUATIONS

Global fluctuations. At the price of keeping z away from the unit disc while the

cluster grows, we see that the fluctuations of log Φn become rather well behaved. Indeed,

by the same techniques that allow us to prove the local fluctuations result, we can show

(cf. Theorem 4.3) that, for any fixed z ∈ C \ D, these fluctuations are again centred

Gaussian, with variance now depending on |z| and t. Moreover, the correlation structure

is sufficiently well behaved to enable us to prove a functional central limit theorem for

log Φn when restricted to any circle of the form rT := z : |z| = r, r > 1 (cf. Theorem

4.5). Finally, we push our analysis forward to obtain limiting fluctuations of log Φn viewed

as a random variable in the space of holomorphic functions on C \ D. Our main result is

the following.

Theorem 4.2 (Global fluctuations, [Sil15]). Let H denote the space of holomorphic

functions on |z| > 1, and for n ≥ 1 set

Fn(z) =1√c

(log

Φn(z)

z− nc

).

Then there exists a random variable F in H such that Fn → F in distribution as n→∞,

with respect to the supremum norm on compacts of (C ∪ ∞) \ D. Moreover, F can be

obtained as the holomorphic extension of its boundary values on |z| = 1 to the outer

unit disc |z| > 1. These boundary values are given by a distribution–valued Gaussian

random variable W, formally defined in Fourier space by

W(ϑ) =∑

k∈Z\0

√1− e−2|k|t

|k|(Ak + iBk√

2

) eikϑ√2π

,

for (Ak)k, (Bk)k independent collections of i.i.d. N (0, 1) random variables.

This provides an explicit construction of the limiting Gaussian holomorphic field Fand, perhaps more interestingly, of its boundary values.

Remark 4.1. Note that, setting formally t =∞, we have

W∞(ϑ) =∑

k∈Z\0

|k|−1/2(Ak + iBk√

2

) eikϑ√2π

,

This is (the complex version of) a well-known Fractional Gaussian Field (FGF) on the

unit circle T, and it can be realised as W∞ = (−∆)−1/4W , where ∆ denotes the Laplace

operator1, and W is white noise on T. Fractional Gaussian Fields can be studied in greater

generality: we refer the interested reader to [DRSV14, LSSW14] and references therein.

Let ∆ and W denote the Laplace operator and White Noise on the d–dimensional torus

respectively, for any d ≥ 1. Then (−∆)−s/2W defines an FGF for any s ∈ R. This is

a (random) continuous function if s > d/2, and otherwise it takes values in the space

of distributions. Moreover, at the critical parameter s = d/2 it defines a Log–correlated

Gaussian Field (LGF), which coincides with the Gaussian Free Field when d = 2. By

taking d = 1, we see that W∞ coincides with the critical LGF on the 1–dimensional torus

T, which is also the restriction of a 2–dimensional Gaussian Free Field to T. It is possible

to show that the law of W for any finite t is absolutely continuous with respect to the one

1Recall that the eigenvalues of the operator −∆ on T are given by k2 : k ≥ 1.


of W∞, and in fact this can be made precise by constructing W as a stochastic process in

the space of distributions on T, which converges to W∞ as t→∞. We do not survey this

here, and refer the interested reader to [Sil15].

The rest of this chapter is devoted to the proof of Theorem 4.2, which is based on

martingale techniques. We proceed in steps. In Section 1 we collect some preliminary es-

timates for the basic conformal maps F,G, as in the previous chapter. We then introduce in

Section 2 our main tools, namely two sequences of backwards martingale difference arrays

(27), and prove Theorem 4.3 on pointwise fluctuations. This result is then generalised in

Sections 3-3.2 to obtain a central limit theorem for the sequence z 7→ log Φn(z)/z, viewed

as a random variable in the space of holomorphic functions on |z| > 1, thus completing

the proof.

1. Preliminary estimates

Recall that in Proposition 3.1 we showed that log G(z)z ≈ −c, and this approximation

gets better as z moves away from the cluster boundary. In order to study fluctuations,

on the other hand, a first order approximation turns out not to be enough. The following

result improves on Proposition 3.1.

Proposition 4.1. There exists an absolute constant C > 0 such that, for all z ∈ D:

|z − 1| > 2δ, it holds: ∣∣∣∣ log(G(z)

z

)+ c

z + 1

z − 1

∣∣∣∣ ≤ Cc3/2|z||z − 1|2 . (23)

The proof of the above result is spelled out in [Sil15], and it follows the one of Propo-

sition 3.1 with minor modifications. We combine Propositions 3.1 and 4.1 to obtain cor-

responding estimates for the function F , which are collected below.

Corollary 4.1. There exists a constant C > 0 such that, for all z ∈ D0 with |F (z)−1| > 2δ, it holds:

|F (z)− ecz| ≤ Cc|z||z| − 1

,

∣∣∣∣ log(F (z)

z

)− c z + 1

z − 1

∣∣∣∣ ≤ Cc3/2|z|2(|z| − 1)3

.

Proof. For the first inequality, note that (9) readily implies that

|G(z)− e−cz| ≤ Cc|z||z − 1| (24)

for all z ∈ D : |z − 1| > δ, and δ small enough. Therefore, by setting w = F (z) and using

(P1)-(P2), we obtain

|F (z)− ecz| = ec|G(w)− e−cw| ≤ Cc|w||w − 1| ≤

C ′c|z||z| − 1

, (25)

for all z ∈ D0 such that |F (z)− 1| > 2δ and δ small enough, as claimed.

46 4. FLUCTUATIONS

For the second inequality, note that, since |F (z)− 1| > 2δ by assumption, we can use

(23) to get∣∣∣∣ logF (z)

z− c z + 1

z − 1

∣∣∣∣ ≤ ∣∣∣∣ logG(w)

w+ c

w + 1

w − 1

∣∣∣∣+ 2c

∣∣∣∣ 1

w − 1− 1

G(w)− 1

∣∣∣∣≤ Cc3/2|w||w − 1|2 +

2c|G(w)− w||w − 1||G(w)− 1| ≤

Cc3/2|z|(|z| − 1)2

+2c|G(w)− w|

(|z| − 1)2.

Moreover, it follows from (25) that

|G(w)− w| ≤ |F (z)− ecz|+ (1− e−c)|w| ≤ Cc|z||z| − 1

+ Cc|z| ≤ Cc|z|2|z| − 1

for c small enough. Putting all together, we end up with∣∣∣∣ logF (z)

z− c z + 1

z − 1

∣∣∣∣ ≤ Cc3/2|z|(|z| − 1)2

(1 +

√c|z|

|z| − 1

)≤ 2Cc3/2|z|2

(|z| − 1)3

for c small enough, as claimed.

2. Pointwise fluctuations

In this section we prove that, fixed t ≥ 0 and z ∈ C \ D, the fluctuations of log Φn(z)

around its mean are given by a complex Gaussian random variable, whose variance is

independent of Arg(z).

Theorem 4.3. Pick a ∈ [−π, π), σ > 0, and let z = eia+σ. Define

v2t (σ) := log

1− e−2(σ+t)

1− e−2σ,

and let Fσ(t, eia) be a complex Gaussian random variable with i.i.d. real and imaginary

part, distributed according to N (0, v2t (σ)). Then it holds:

1√c

(log

Φn(z)

z− nc

)−→ Fσ(eia)

in distribution as n→∞, c→ 0 and nc→ t.

We prove Theorem 4.3 for σ ≤ 1, which we assume without further notice. This entails

no loss of generality, as it will become clear, and it has the advantage of slightly simplifying

the notation.

Our main tool for the proof of Theorem 4.3 consists of two sequences of backwards

martingale difference arrays, that we now define. Note that∣∣Φn(z)

z

∣∣ > 1 for all z ∈ D0, so

log Φn(z)z defines a holomorphic function on D0. For σ > 0 and a ∈ [−π, π) as in Theorem

4.3, then, we find:

logΦn(eia+σ)

eia+σ=

n∑k=1

logFk Fk+1 · · · Fn(eia+σ)

Fk+1 · · · Fn(eia+σ)=

n∑k=1

logF (e−iΘkZσk,n(a))

e−iΘkZσk,n(a),

2. POINTWISE FLUCTUATIONS 47

where Zσk,n(a) := Fk+1 · · · Fn(eia+σ). Moreover, if Fk,n := σ(Θk,Θk+1 . . .Θn), then

E(

logF (e−iΘkZσk,n(a))

e−iΘkZσk,n(a)

∣∣∣∣Fk+1,n

)=

1

2π

∫ π

−πlog

F (e−iϑZσk,n(a))

e−iϑZσk,n(a)dϑ = lim

|z|→∞log

F (z)

z= c .

(26)

We therefore set

Xσk,n(a) =

1√c

(log

∣∣∣∣F (e−iΘkZσk,n(a))

e−iΘkZσk,n(a)

∣∣∣∣− c),Y σk,n(a) =

1√c

Arg

(F (e−iΘkZσk,n(a))

e−iΘkZσk,n(a)

),

(27)

where Arg(z) ∈ [−π, π). The above computation then shows that (Xσk,n(a))k≤n and

(Y σk,n(a))k≤n form sequences of backwards martingale arrays with respect to the same

filtration (Fk,n)k≤n (see [Bil95], Section 35 for the definition of martingale arrays). More-

over,n∑k=1

(Xσk,n(a) + i Y σ

k,n(a))

=1√c

(log

Φn(eia+σ)

eia+σ− nc

).

Remark 4.2. We identify C with R2 throughout, and often refer to complex random

variables as random vectors and vice versa, depending on which point of view we seek to

emphasize.

The following result appears in [McL74], Corollary 2.8, as a Central Limit Theorem

for (forward) martingale difference arrays (see also [Bil95], Theorem 35.12). It is straight-

forward to adapt the proof to backwards martingale difference arrays, so that we have the

following.

Theorem 4.4. Let (Xk,n)1≤k≤n be a backwards martingale difference array with respect

to Fk,n = σ(Xk,n . . .Xn,n). Let Sk,n =∑n

j=k Xj,n. Assume that:

(I) for all η > 0,

n∑k=1

X 2k,n1(|Xk,n| > η)→ 0 in probability as n→∞,

(II)

n∑k=1

X 2k,n → s2 in probability as n→∞, for some s2 > 0.

Then S1,n converges in distribution to N (0, s2).

Note that the above theorem is concerned with scalar random variables, while we have

2–dimensional vectors. In order to reduce to the scalar case, recall that by the Cramer–

Wold Theorem (cf. [Dur10], Theorem 3.9.5) it suffices to prove convergence in distribution

of all linear combinations of the vector entries. To this end, pick any α, β ∈ R, and note

that by linearity(αXσ

k,n(a)+βY σk,n(a)

)k≤n is again a backwards martingale difference array

with respect to the filtration (Fk,n)k≤n. We are going to apply Theorem 4.4 to this linear

combination. To this end, we collect here some estimates for (Xσk,n(a)) and (Y σ

k,n(a)).

Since a and σ are fixed, we omit them from the notation throughout this section.

Lemma 4.1. There exists a constant C > 0 such that, for c small enough, it holds

|Xk,n| ≤ C/√c, |Yk,n| ≤ C/

√c for all n ≥ 1, k ≤ n.

48 4. FLUCTUATIONS

Proof. It follows from (P1) that log∣∣F (e−iΘkZk,n)

e−iΘkZk,n

∣∣ > 0, from which Xk,n > −√c.Moreover, (P2) gives |Xk,n| ≤ C/

√c+√c ≤ 2C/

√c for some constant C > 0 and c small

enough. Finally, since Arg(F (e−iΘkZk,n)

e−iΘkZk,n

)∈ [−π, π), we have |Yk,n| ≤ π/

√c.

Much better estimates can be obtained by restricting to the good event, Ω(m, ε) of

Theorem 3.1, on which for all n ≤ m it holds

|e−cnΦn(z)− z| < εe6ε for all z : |z| ≥ e5ε,

and

|Γn(z)− e−cnz| < εe5ε for all z : |z| ≥ ecn+4ε.

We take m = bδ−6c and ε = δ2/3 log(1/δ) throughout without further notice, so that2

ε δ, m n and P(Ω(m, ε)c) decays faster than any polynomial in δ.

Lemma 4.2. Assume that σ δ as δ → 0. Then there exists a constant C(t) > 0 such

that, for n large enough, on the good event Ω(m, ε) it holds

maxk≤n

|Xk,n| ∨ |Yk,n|

≤ C(t)

√c

σ. (28)

Proof. For any k ≤ n we have

max|Xk,n|; |Yk,n|

≤ 1√

c

(∣∣ logF (e−iΘkZk,n)− log(e−iΘkZk,n)∣∣+ c

)≤ 1√

c

[(sup|ξ|≥eσ

1

|ξ|)·∣∣F (e−iΘkZk,n)− e−iΘkZk,n

∣∣+ c

],

where the last inequality follows from the mean values theorem, and the fact that |F (e−iΘkZk,n)| >|e−iΘkZk,n| > eσ almost surely by (P1). Now note that |F (e−iΘkZk,n)− 1| > |Zk,n| − 1 ≥σ 2δ, so by Corollary 4.1 we have

|F (z)− z| ≤ |F (z)− ecz|+ (ec − 1)|z| ≤ Cc|z||z| − 1

+ 2c|z| ≤ 2Cc|z|2|z| − 1

(29)

for z = e−iΘkZk,n. Moreover, since we are on Ω(m, ε), there exists a constant C(t)

depending only on t such that eσ < |Zk,n| ≤ C(t). This, together with (29), yields

maxk≤n

|Xk,n| ∨ |Yk,n|

≤ 1√

c

(2Cc|Zk,n|2|Zk,n| − 1

+ c

)≤ C(t)

√c

eσ − 1+√c ≤ 2C(t) ·

√c

σ

as claimed.

We now make use of the above bounds to prove that the backwards martingale dif-

ference array (Xk,n)k≤n, with Xk,n := αXk,n + βYk,n, satisfies Assumptions (I)-(II) of

Theorem 4.4. In doing so, we provide an explicit formula for the limiting variance.

Lemma 4.3. Assume that σ δ as δ → 0. Then for all η > 0 it holdsn∑k=1

X 2k,n1(|Xk,n| > η)→ 0

in probability as n→∞.

2If a, b→ 0 (respectively a, b→∞) write a b to mean lima,b→0ba

= 0 (respectively lima,b→∞ba

= 0).


Proof. For any ε > 0 we have:

P( n∑k=1

X 2k,n1(|Xk,n| > η) > ε

)≤ P

(max

1≤k≤n|Xk,n| > η

)≤ 1

ηE(

max1≤k≤n

|Xk,n|)

=1

ηE(

max1≤k≤n

|Xk,n| ; Ω(m, ε)c)

+1

ηE(

max1≤k≤n

|Xk,n| ; Ω(m, ε)

).

The fact that the first term in the r.h.s. converges to zero as n→∞ follows from Lemma

4.1 and Theorem 3.1, while convergence to zero of the second term is a straightforward

consequence of Lemma 4.2.

We now concentrate on Assumption (II). The first step consists in replacing condition

(II) with a more convenient one, involving conditional second moments. The following

result shows that, provided σ is large enough with respect to c, this is allowed.

Lemma 4.4. Assume limδ→0

√δσ = 0, and that

n∑k=1

E(X 2k,n|Fk+1,n) → s2 in probability

as n→∞ for some s2 > 0. Then alson∑k=1

X 2k,n → s2 in probability as n→∞.

Proof. Let Mk,n := X 2k,n − E(X 2

k,n|Fk+1,n). It is readily checked that (Mk,n)k≤n is

a backwards martingale difference array with respect to the filtration (Fk,n)k≤n. We aim

to show that for any η > 0 it holds P(∣∣∣ n∑

k=1

Mk,n

∣∣∣ > η)→ 0 as n→∞. Indeed,

P(∣∣∣ n∑

k=1

Mk,n

∣∣∣ > η)≤ 1

η2E([ n∑

k=1

Mk,n

]2)=

1

η2

n∑k=1

E(M2k,n)

(∗)≤ 1

η2

n∑k=1

E(X 4k,n)

=1

η2

n∑k=1

E(X 4k,n;E(m, ε)c) +

1

η2

n∑k=1

E(X 4k,n;E(m, ε)) .

Above, (∗) follows from the general inequality E((X − E(X))2) ≤ E(X2), which we apply

to each term with respect to E( · |Fk,n). The fact that both terms in the r.h.s. converge

to zero as n→∞ is now a consequence of the bounds for |Xk,n| and |Yk,n|, and hence for

|Xk,n|, obtained in Lemmas 4.1–4.2.

In light of the above result, it remains to compute the limit in probability of

n∑k=1

E(X 2k,n|Fk+1,n) = α2

n∑k=1

E(X2k,n|Fk+1,n) + β2

n∑k=1

E(Y 2k,n|Fk+1,n)

+ 2αβn∑k=1

E(Xk,nYk,n|Fk+1,n) ,

and prove that it coincides with (α2 + β2)v2t (σ), where v2

t (σ) is the limiting variance

introduced in Theorem 4.3. The following result shows that, in fact, it suffices to compute

the limit of the first term in the r.h.s. above.

50 4. FLUCTUATIONS

Proposition 4.2. Almost surely, it holds

E(X2k,n|Fk+1,n) = E(Y 2

k,n|Fk+1,n)

E(Xk,nYk,n|Fk+1,n) = 0

for all k ≤ n.

Proof. All equalities in this proof are intended to hold almost surely. Introduce the

holomorphic function f(z) := 1√c

(log F (z)

z − c)

defined for |z| > 1, so that[f(e−iΘkZk,n)

]2= X2

k,n − Y 2k,n + 2iXk,nYk,n .

Taking conditional expectations both sides, we find

1

2π

∫ π

−π

[f(e−iϑZk,n)

]2dϑ =E(X2

k,n|Fk+1,n)− E(Y 2k,n|Fk+1,n)

+ 2iE(Xk,nYk,n|Fk+1,n).

(30)

Set z = eiϑ, and observe that the function z 7→[f(Zk,nz

)]2is holomorphic on D = |z| <

1, and extends continuously to D. Applying Cauchy’s integral formula to this function,

then, yields

1

2π

∫ π

−π

[f(e−iϑZk,n)

]2dϑ =

1

2πi

∫|z|=1

[f

(Zk,nz

)]2 dz

z= lim|z|→0

[f

(Zk,nz

)]2

= lim|z|→∞

f2(z) = 0.

Going back to (30), this implies that both real and imaginary part of the r.h.s. must vanish

almost surely, which is what we wanted to show.

Proposition 4.2 already shows that the limiting Gaussian vector Fσ(eia) must have

i.i.d. entries. It remains to compute the limiting variance, that is to show that for all

η > 0 it holds

P(∣∣∣∣ n∑

k=1

E(X2k,n|Fk+1,n)− v2

t (σ)

∣∣∣∣ > η

)→ 0 (31)

as n→∞. To this end it is clearly enough to work on Ω(m, ε), the advantage being that

on this event we have ∣∣Zσk,n(a)− eia+σ+(n−k)c∣∣ ≤ C(t)ε

for all k ≤ n, as it follows directly from the definition of Ω(m, ε) as long as σ ε.

Our strategy is then to replace each Zk,n by its deterministic approximation, and show

that, provided σ is large enough with respect to c, this does not affect the limiting variance.

Recall that the Poisson kernel for the unit disc D is given by Pr(ϑ) = Re(

1+reiϑ

1−reiϑ)

for r < 1, and that the function reiϑ 7→ Pr(ϑ) is harmonic in D. Moreover, given any

continuous function f on T = ∂D, its harmonic extension Hf inside D is given by Poisson’s

integral formula

(Hf)(reiϑ) =1

2π

∫ π

−πPr(ϑ− t)f(eit)dt = (Pr ∗ f)(ϑ) . (32)


We denote by Qr(ϑ) the harmonic conjugate of Pr(ϑ) in D which is 0 at 0, i.e. Qr(ϑ) =

Im(

1+reiϑ

1−reiϑ).

Before proceeding with the proof of (31), let us give a short sketch of how the limiting

variance is computed. Note that

E(X2k,n|Fk+1,n) =

1

2πc

∫ π

−π

[Re

(log

F (e−iϑZσk,n)

e−iϑZσk,n

)− c]2

dϑ

=1

2πc

∫ π

−π

[Re

(log

F (e−iϑZσk,n)

e−iϑZσk,n

)]2

dϑ− c,

where the second equality follows from (26). Now, using that on the event Ω(m, c) we

have3

Zσk,n ≈ eσ+(n−k)c ,

and that by Corollary 4.1

Re

(log

F (e−iϑZσk,n)

e−iϑZσk,n

)≈ cRe

(e−iϑZσk,n + 1

e−iϑZσk,n − 1

),

we obtain

1

2πc

∫ π

−π

[Re

(log

F (e−iϑZσk,n)

e−iϑZσk,n

)]2

dϑ ≈ c

2π

∫ π

−π

[Re

(e−iϑ+σ+(n−k)c + 1

e−iϑ+σ+(n−k)c − 1

)]2

dϑ

=c

2π

∫ π

−π

[Pe−σ−(n−k)c(ϑ)

]2dϑ = cPe−2σ−2(n−k)c(0) ,

where the last integral has been computed using Poisson’s integral formula (32). Finally,

this yields

n∑k=1

E(X2k,n|Fk+1,n) ≈ c

n∑k=1

Pe−2σ−2(n−k)c(0)− nc ≈∫ σ+t

σPe−2x(0)dx− t

= log1− e−2(σ+t)

1− e−2σ= v2

t (σ) .

Let us now implement this strategy by showing that all the errors in the above ap-

proximations vanish in the limit.

Lemma 4.5. Assume σ ε as δ → 0. Then there exists a constant C(t), depending

only on t, such that on the event Ω(m, ε) we have∣∣∣∣E(X2k,n|Fk+1,n) + c− c

2π

∫ π

−π

(Pe−σ−(n−k)c(ϑ)

)2dϑ

∣∣∣∣ ≤ C(t)cε

(σ + (n− k)c)3

for all k ≤ n, and n large enough.

3Write an ≈ bn if the sequences (an), (bn) converge to the same limit as n→∞.

52 4. FLUCTUATIONS

Proof. Fix any k ≤ n and set for simplicity Z = Zk,n(0), W = eσ+(n−k)c. Moreover,

introduce the functions

gϑ(z) = Re(

logF (e−iϑz)

e−iϑz

), hϑ(z) = cRe

(e−iϑz + 1

e−iϑz − 1

).

Then E(X2k,n|Fk+1,n) = 1

2πc

∫ π−π gϑ(Z)2dϑ− c, and we have to show that∣∣∣∣ 1

2πc

∫ π

−πgϑ(Z)2dϑ− 1

2πc

∫ π

−πhϑ(W )2dϑ

∣∣∣∣ ≤ C(t)cε

(σ + (n− k)c)3. (33)

Trivially, the l.h.s. is bounded above by

1

2πc

∫ π

−π

(|gϑ(Z)| · |gϑ(Z)− hϑ(W )|+ |hϑ(W )| · |gϑ(Z)− hϑ(W )|

)dϑ .

We bound each term separately. Recall the definition of Ω(m, ε), from which it follows

that

maxϑ∈[−π,π)

|Zk,n(ϑ)| ≤ eσ+(n−k)c(1 + 2ε) ≤ 2et+2 = C(t) ,

minϑ∈[−π,π)

|Zk,n(ϑ)| − 1 ≥ eσ+(n−k)c(1− 2ε) ≥ σ + (n− k)c

2

as long as n is large enough and σ ε. We combine the above estimates with the bounds

in Corollary 4.1, to get

|gϑ(Z)| ≤∣∣∣∣ log

F (e−iϑZ)

e−iϑZ− ce

−iϑZ + 1

e−iϑZ − 1

∣∣∣∣+ c

∣∣∣∣e−iϑZ + 1

e−iϑZ − 1

∣∣∣∣ ≤ Cc3/2|Z|2(|Z| − 1)3

+2c|Z||Z| − 1

≤ C(t)c3/2

(σ + (n− k)c)3+

C(t)c

σ + (n− k)c≤ 2C(t)c

σ + (n− k)c

(34)

for n large enough. Similarly, we find

|hϑ(W )| ≤ c(

1 +2

|W | − 1

)≤ c(

1 +2

σ + (n− k)c

)≤ C(t)c

σ + (n− k)c(35)

on the event Ω(m, ε) for, say, C(t) = t+ 3 and n large enough.

In order to bound the remaining terms, observe that, by definition, on Ω(m, ε) we have

maxϑ∈[−π,π)

∣∣Zk,n(ϑ)− eiϑ+σ+(n−k)c∣∣ ≤ C(t) ε ,

from which we get |Z −W | ≤ C(t)ε. Combining this with Corollary 4.1 we finally obtain

|g(Z)− h(W )| ≤ |g(Z)− h(Z)|+ |h(Z)− h(W )| ≤ Cc3/2|Z|2(σ + (n− k)c)3

+2c|Z −W |

(|Z| − 1)(|W | − 1)

≤ C(t)c3/2

(σ + (n− k)c)3+

C(t)cε

(σ + (n− k)c)2≤ 2C(t)cε

(σ + (n− k)c)2

for n large enough. Putting now this together with (34) and (35) gives (33).

3. GLOBAL FLUCTUATIONS 53

Assume now that σ √ε. Then we deduce from Lemma 4.5 that on Ω(m, ε) it holds:∣∣∣∣ n∑k=1

E(X2k,n|Fk,n)−

n∑k=1

c

2π

∫ π

−π


)2dϑ+ nc

∣∣∣∣ ≤ n∑k=1

C(t)cε

(σ + (n− k)c)3

≤ C(t)ε

∫ σ+nc

σ

dx

x3= C(t)ε

(1

2σ2− 1

2(σ + nc)2

)→ 0

as n → ∞. Note that√ε δ1/3 (apart from logarithmic corrections), so the assumption

σ √ε is stronger than the previous one σ √δ.

In conclusion, we have shown that that, provided σ √ε, the limiting variance is

given by the deterministic expression

limn→∞

(c

2π

n∑k=1

∫ π

−π


)2dϑ− nc

)=

∫ σ+t

σ

1

2π

∫ π

−π

(Pe−x(ϑ)

)2dϑdx− t

=

∫ σ+t

σ(Pe−x ∗ Pe−x)(0)dx− t =

∫ σ+t

σ

1 + e−2x

1− e−2xdx− t = log

1− e−2(σ+t)

1− e−2σ= v2

t (σ) .

Finally, if σ > 0 is kept fixed as n → ∞ the assumption σ √ε is trivially satisfied, so

this concludes the proof of Theorem 4.3.

3. Global fluctuations

Having a pointwise convergence result, it is natural to ask if this can be extended

to obtain convergence of random fields. We do this in two steps, first controlling the

fluctuations on any circle (cf. Theorem 4.5) and then extending this to the whole region

|z| > 1, thus proving Theorem 4.2.

3.1. Fluctuations on circles. Recall that T = |z| = 1 denotes the unit circle,

and let C(T) denote the space of continuous functions from T to C, equipped with the

supremum norm

‖x‖∞ = supϑ∈[−π,π)

|x(eiϑ)| . (36)

The following results controls the fluctuations on any circle outside the unit disc.

Theorem 4.5. Fix any σ > 0, and let Fσn denote the C(T)-valued random variable

defined by

Fσn (eia) =1√c

(log

Φn(eia+σ)

eia+σ− nc

)for eia ∈ T and t ≥ 0. Then there exists a zero mean Gaussian random variable Fσ on

C(T) whose covariance structure is given by

Cov(Fσ(eia)

)= v2

t (σ)

(1 0

0 1

), Cov

(Fσ(eia),Fσ(eib)

)=

(ct(σ, a− b) ct(σ, a− b)−ct(σ, a− b) ct(σ, a− b)

),

where v2t (σ) = ct(σ, 0), and

ct(σ, α) := Re

(log

1− e−2σ−2t+iα

1− e−2σ+iα

), ct(σ, α) = Im

(log

1− e−2σ−2t+iα

1− e−2σ+iα

),

such that Fσn → Fσ in distribution as n→∞, in the sense of weak convergence on C(T)

equipped with the supremum norm.

54 4. FLUCTUATIONS

Note that the limiting object is rotationally invariant in the spatial coordinate, as one

expects from the rotation invariance of the original model. The rest of this subsection is

devoted to the proof of the above result.

Fix any σ > 0. Clearly Fσn belongs to C(T) for all n ≥ 0. Since (C(T), ‖ · ‖∞)

is a complete separable metric space, it follows by Prohorov’s theorem that if the finite

dimensional distributions (FDDs) of Fσn converge to the ones of Fσ, and (Fσn )n≥0 is tight,

then Fσn → Fσ weakly.

3.1.1. Convergence of finite dimensional distributions. We have to show that for any

finite collection of angles −π ≤ a1 < a2 < . . . < aM < π, the random vector

(Fσn (eia1),Fσn (eia2), . . . ,Fσn (eiaM )) converges in distribution to the Gaussian vector

(Fσ(eia1),Fσ(eia2), . . . ,Fσ(eiaM )) as n→∞. To this end, let us again reduce to the scalar

case by considering linear combinations. Recall the definition of the random variables

Xσk,n( · ), Y σ

k,n( · ) given in (27). For (αj)1≤j≤M in RM , we look at the weak limit of∑j

αjFσn (eiaj ) =

n∑k=1

∑j

αj

[Xσk,n(aj) + iY σ

k,n(aj)]

=n∑k=1

X σk,n,

where we have set

X σk,n =

∑j

αj

[Xσk,n(aj) + iY σ

k,n(aj)]

(37)

for k ≤ n. Then(X σk,n

)k≤n is a backwards martingale difference array with respect to

the filtration (Fk,n)k≤n. We can therefore apply Theorem 4.4 to show weak convergence,

provided that Assumptions (I) and (II) are satisfied. Note that all terms in the r.h.s.

of (37) satisfy the estimates of Lemmas 4.1 and 4.2, from which one can easily show,

reasoning as in Lemma 4.3, that Assumption (I) holds. Furthermore, Lemma 4.4 is still

in force, from which we conclude that, provided σ √δ, the limiting variance is given

by the limit in probability ofn∑k=1

E((X σ

k,n)2|Fk+1,n

). We now focus on the computation

of this limit. Expand the square and use linearity to see that the above sum equals∑j

∑s

αjαs

n∑k=1

E[(Xσk,n(aj) + iY σ

k,n(aj))(Xσk,n(as) + iY σ

k,n(as))∣∣∣∣Fk,n

]It follows that it suffices to compute the limit in probability of

n∑k=1

E[(Xσk,n(a) + iY σ

k,n(a))(Xσk,n(b) + iY σ

k,n(b))∣∣∣Fk,n

]for arbitrary a, b ∈ [−π, π). Moreover, by rotational invariance we can set b = 0 without

loss of generality. The following result simplifies the computation.

Proposition 4.3. Almost surely, it holds

E(Xσk,n(a)Xσ

k,n(0)|Fk+1,n) = E(Y σk,n(a)Y σ

k,n(0)|Fk+1,n)

E(Xσk,n(a)Y σ

k,n(0)|Fk+1,n) = −E(Y σk,n(0)Xσ

k,n(a)|Fk+1,n)

for all k ≤ n.


Proof. The result follows by the same arguments used in the proof of Proposition

4.2, considering now[f(e−iΘkZσk,n(a))+f(e−iΘkZσk,n(0))

]2in place of

[f(e−iΘkZk,n)

]2.

It remains to compute the limit in probability of∑

k E(Xσk,n(a)Xσ

k,n(0)|Fk+1,n) and∑k E(Xσ

k,n(a)Y σk,n(0)|Fk+1,n). As in the previous section, we do this by approximating by

a deterministic quantity.

Lemma 4.6. Assume limε→0εσ = 0. Then there exists a constant C(t), depending only

on t, such that on the event Ω(m, ε) the following hold:∣∣∣∣E(Xσk,n(a)Xσ

k,n(0)|Fk+1,n) + c− c

2π

∫ π

−πPe−σ−(n−k)c(a− ϑ)Pe−σ−(n−k)c(ϑ)dϑ

∣∣∣∣ ≤ C(t)cε

(σ+(n− k)c)3∣∣∣∣E(Xσk,n(a)Y σ

k,n(0)|Fk+1,n) + c− c

2π

∫ π

−πPe−σ−(n−k)c(a− ϑ)Qe−σ−(n−k)c(ϑ)dϑ

∣∣∣∣ ≤ C(t)cε

(σ+(n− k)c)3

for all k ≤ n and n large enough. Above Qr(ϑ) = Im(

1+reiϑ

1−reiϑ), r < 1, denotes the conjugate

Poisson kernel.

The proof follows by the same arguments used in Lemma 4.5, and it is left to the

reader. As a consequence, after further assuming σ √ε, we conclude that on the event

Ω(m, ε) it holds:∣∣∣∣ n∑k=1

E(Xσk,n(a)Xσ

k,n(0)|Fk+1,n) + nc− c

2π

n∑k=1

∫ π

−πPe−σ−(n−k)c(a− ϑ)Pe−σ−(n−k)c(ϑ)dϑ

∣∣∣∣ ≤≤

n∑k=1

C(t)cε

(σ + (n− k)c)3≤ C(t)ε

∫ σ+nc

σ

dx

x3= C(t)ε

(1

2σ2− 1

2(σ + nc)2

)→ 0

as n→∞. This in turn implies that

limn→∞

n∑k=1

E(Xσk,n(a)Xσ

k,n(0)|Fk+1,n) = limn→∞

(c

2π

n∑k=1

∫ π

−πPe−σ−(n−k)c(a− ϑ)Pe−σ−(n−k)c(ϑ)dϑ− nc

)

=1

2π

∫ σ+t

σ

∫ π

−πPe−x(a− ϑ)Pe−x(ϑ)dϑ− s =

∫ σ+t

σPe−2x(a)dx− t = log

∣∣∣∣1− e−2σ−2t+ia

1− e−2σ+ia

∣∣∣∣ ,as claimed. Similarly,

limn→∞

n∑k=1

E(Xσk,n(a)Y σ

k,n(0)|Fk+1,n) =1

2π

∫ σ+t

σ

∫ π

−πPe−x(a− ϑ)Qe−x(ϑ)dϑ

=

∫ σ+t

σQe−2x(a)dx = Arg

(1− e−2σ−2t+ia

1− e−2σ+ia

).

This concludes the proof of convergence of Finite Dimensional Distributions.

3.1.2. Tightness. We now show that the sequence (Fσn )n≥1 of continuous random fields

on the circle T is tight. According to Theorem 7.5 in [Bil99], it suffices to show that for

all ν > 0 it holds

limη→0

lim supn→∞

P(

sup|a−b|<η

|Fσn (eia)−Fσn (eib)| ≥ ν)

= 0 . (38)

56 4. FLUCTUATIONS

It is clear that it suffices to restrict to the event Ω(m, ε). We switch to logrithmic coordi-

nates, following the notation introduced in Section 2.1. Then

Fσn (ϑ) =1√c

(Φn(iϑ+ σ)− iϑ− σ − nc

),

and we have:

P(

sup|a−b|<η

|Fσn (eia)−Fσn (eib)| ≥ ν ; Ω(m, ε)

)=

= P(∣∣(Φn(z)− z − nc

)−(Φn(z′)− z′ − nc

)∣∣ ≥ ν√c for

some z, z′ : Re(z) = Re(z′) = σ, |z − z′| < η ; Ω(m, ε)

)≤ P

(∣∣(Γn(z)− z + nc)−(Γn(z′)− z′ + nc

)∣∣ ≥ ν√c for some z, z′ :

|Re(z)− σ − nc| < ε, |Re(z′)− σ − nc| < ε, |z − z′| < η + 2ε ; Ω(m, ε)

).

(39)

Note that Dn := w ∈ C : ew ∈ Dn ⊇ w ∈ C : Re(w) ≥ nc+ 2ε on the event Ω(m, ε).

It follows that if Re(z),Re(z′) > σ + nc − ε, then z, z′ ∈ Dn, since σ − ε ≥ 3ε for n

large enough. We can therefore define Mk(w) = Γk(w) − w + nc for any k ≤ n and any

w in this range. The rightmost event in (39) is then a subset of Eν,η ∩ Ω(m, ε), where

Eη,ν := |Mn(z)−Mn(z′)| ≥ ν√c for some z, z′ : |Re(z)− σ− nc| < ε, |Re(z′)− σ− nc| <ε, |z − z′| < 2η.

Lemma 4.7. There exists a constant C(σ, t) depending only on σ, t such that for n

large enough

P(Eη,ν ∩ Ω(m, ε)

)≤ C(σ, t)

(2η)2/3

ν2.

This implies (38), and hence tightness, by letting first n→∞ and then η → 0.

We control the above probability by mean of a 2–dimensional version of Kolmogorov’s

continuity theorem, that we now recall.

Theorem 4.6 ([DPZ14], Theorem 3.5). Let (Xξ)ξ∈I be a collection of Rm–valued

random variables indexed by an open bounded set I ⊂ Rd of the form I = (i1, j1) ×(i2, j2)× . . .× (id, jd). Assume that for some constants C > 0, ε > 0 and δ > 1 it holds

E(|Xξ −Xη|δ) ≤ C|ξ − η|d+ε,

where, with a slight abuse of notation, | · | denotes the Euclidean norm in Rm on the l.h.s.

and in Rd on the r.h.s.. Then there exists a random variable M > 0 with E(M δ) ≤ C

such that for any α ∈ (0, ε/δ) it holds

supξ,η∈I

|Xξ −Xη| ≤M |ξ − η|α

almost surely.


We now show that for any w,w′ such that |Re(w)−σ−nc| < ε, |Re(w′)−σ−nc| < ε

and |w − w′| < 1, there exists a constant C = C(σ, t), depending only on σ and t, such

that

E(∣∣∣Mn(w)−Mn(w′)

∣∣∣3) ≤ C(σ, t)c3/2|w − w′|3,

where |w − w′| denotes the Euclidean norm.

Proof. For k ≤ n set Mk = Mk(w)−Mk(w′). Then

(Mk

)k≤n is a martingale. From

Table 1, if w is such that |ew − 1| > 2δ we have

|G′0(w)| ≤ Cc

(Re(w)− δ)2.

This together with the mean values theorem implies that∣∣Mk+1 − Mk

∣∣ =∣∣G0(Γk(w − iΘk+1)− G0(Γk(w

′ − iΘk+1)∣∣ ≤ Cc

σ2(|w − w′|+ |Mk|) .

Recall the following result, which appears in [DFJ68].

Theorem 4.7 ([DFJ68]). Let (Mn)n≥0 be a martingale with M0 = 0. Then, for all

ν ≥ 2 and n ≥ 1, it holds

E(|Mn|ν) ≤ Cν nν/2−1n∑k=1

E(|Mk −Mk−1|ν) ,

where Cν is an explicit constant, depending only on ν.

We apply the above result with ν = 3, to get

E(∣∣Mn

∣∣3) = E(∣∣∣ n−1∑

k=0

(Mk+1 − Mk

)∣∣∣3) ≤ C3n1/2

n−1∑k=0

E(∣∣Mk+1 − Mk

∣∣3)

≤ C3n1/2C

3c3

σ6

n−1∑k=0

(|w − w′|3 + E

(|Mk|3

)).

It then follows from Gronwall’s inequality that

E(∣∣Mn

∣∣3) ≤ n3/2Cc3

σ6|w − w′|3e

Cn3/2c3

σ6 ≤ C(σ, t)c3/2|w − w′|3 (40)

for n large enough and a constant C(σ, t) depending only on σ and t. Kolmogorov’s

continuity theorem now yields the existence of a random variable M > 0 such that

supw :|Re(w)−σ−nc|<εw′:|Re(w′)−σ−nc|<ε

|w−w′|<2η

∣∣∣Mn(w)−Mn(w′)∣∣∣ ≤M |w − w′|1/4 ,

with E(M 3) ≤ C(σ, t)c3/2. Therefore we find

P

(sup

w :|Re(w)−σ−nc|<εw′:|Re(w′)−σ−nc|<ε

|w−w′|<2η

∣∣∣Mn(w)−Mn(w′)∣∣∣ ≥ ν√c) ≤ P

(M 3 ≥ ν3c3/2

(2η)3/4

)≤ C(σ, t)

(2η)3/4

ν3,

and sending first n → ∞ and then η → 0 we conclude that (38) holds. This proves

tightness, and hence it concludes the proof of Theorem 4.5.

58 4. FLUCTUATIONS

3.2. Fluctuations on H . We can finally conclude the proof of Theorem 4.2. Recall

that T = z ∈ C : |z| = 1, and set αT = z ∈ C : |z| = α for any α ∈ R+. Let(C(αT), ‖·‖∞

)denote the space of continuous function on αT equipped with the supremum

norm. Moreover, for a subset D of the complex plane, denote by H (D) the space of

holomorphic functions on D. Whenever D = |z| > 1, denote H (D) simply by H .

We have shown in the previous section that, for any fixed σ > 0, Fσn → Fσ as n→∞in distribution with respect to the supremum norm on C(T). Note that, for any σ > 0,

the continuous function Fσn coincides with the restriction of the holomorphic function

Fn(z) =1√c

(log

Φn(z)

z− nc

), (41)

defined for all |z| > 1, to the circle eσT. We show below that also the limit Fσ can be

interpreted as the restriction of a holomorphic random function F , defined for all |z| > 1,

to eσT. Moreover, we provide an explicit construction of F , and prove that Fn → F in

distribution as n→∞. To this end, define, for all N ≥ 1, a distance dN on H by setting

dN (φ, ψ) = sup|z|≥e1/N

|φ(z)− ψ(z)| ∧ 1 , and let d(φ, ψ) =∑N≥1

dN (φ, ψ)

2N. (42)

Since (H , dN ) is a complete separable metric space for all N ≥ 1, this makes (H , d) into

a complete separable metric space (i.e. Polish space).

Let us now describe the explicit construction of F . Let D = R/2πZ, and recall that

if we set ek(ϑ) = eikϑ/√

2π for k ∈ Z, then (ek)k∈Z forms an orthonormal basis (in short

ONB) for L2(D) with respect to the inner product (f, g) =∫ π−π f(ϑ)g(ϑ)dϑ. On this basis,

that we refer to as Fourier basis, the Poisson kernel Re(

1+z1−z)

reads

P1/r(ϑ) = Re(1 + eiϑ/r

1− eiϑ/r)

=√

2π(e0 +

∑k∈Z\0

r−|k|ek(ϑ))

for any r > 1. Recall from Theorem 4.2 that W is formally defined by

W(ϑ) =∑

k∈Z\0

√1− e−2|k|t

|k|(Ak + iBk√

2

) eikϑ√2π

,

for (Ak)k, (Bk)k independent collections of i.i.d. N (0, 1) random variables. For r > 1 and

a ∈ [−π, π) then set

F(reia) =1√2π

(P1/r ∗W

)(a) =

1√2π

∫ π

−πRe(1 + ei(a−ϑ)/r

1− ei(a−ϑ)/r

)W(ϑ)dϑ .

We remark that, although W is only defined formally, its holomorphic extension F is a

well defined function. This can be seen, for example, by expanding F in Fourier basis,

and checking that VarF(z) <∞ for all z with |z| > 1.

Lemma 4.8. F is a random variable in H . Moreover, F is Gaussian, and its restric-

tion to eσT agrees in distribution with Fσ defined in Theorem 4.5, for any σ > 0.


Proof. The fact that F is Gaussian is true by construction. To check holomorphicity,

we expand F(·) in Fourier basis, to get

F(reia)=

(e0 +

∑k 6=0

r−|k|e−ikaek,∑k 6=0

√1− e−2|k|t

|k|(Ak + iBk√

2

)ek

)

=∑k 6=0

r−|k|

√1− e−2|k|t

|k|(Ak + iBk√

2

)eika

(d)=∑k≥1

r−k

√1− e−2|k|t

|k|[Ak cos ka−Bk sin ka

]+ i∑k≥1

r−k

√1− e−2|k|t

|k|[Bk cos ka+Ak sin ka

]

(43)

for any r > 1 and a ∈ [−π, π), where the last equality holds in law, and it follows from

the independence of the Ak’s and Bk’s. This provides an almost surely convergent power

series expansion for F at all z with |z| > 1, and hence it shows that F is a Gaussian

stochastic processes taking values in H .

We now show that F has the same covariance structure of Fσ on the circle eσT, for

arbitrary σ > 0. Indeed, it follows from (43) that real and imaginary parts of F(eσ+ia),

a ∈ [−π, π), are independent centred real Gaussian random variables, with

E[(ReF(eσ+ia))2

]= E

[(ImF(reσ+ia))2

]=∑k≥1

e−2kσ(1− e−2kt)

k= log

1− e−2(t+σ)

1− e−2σ.

Moreover, expanding F(eσ) as in (43), one further checks that

Cov(ReF(eσ+ia),ReF(eσ)

)= Cov

(ImF(eσ+ia), ImF(eσ)

)= Re

(log

1− e−2t−2σ+ia

1− e−2σ+ia

),

and

Cov(ReF(eσ+ia), ImF(eσ)

)= −Cov

(ImF(eσ+ia),ReF(eσ)

)= Im

(log

1− e−2t−2σ+ia

1− e−2σ+ia

).

By rotational invariance in the spatial coordinate, this is enough to conclude that F and

Fσ have the same covariance structure, and hence the same law, on every circle of the

form eσT, for arbitrary σ > 0, as claimed.

To conclude the proof of Theorem 4.2 it only remains to show that Fn → F in

distribution with respect to the metric d on H defined in (42). To this end, it clearly

suffices to show that for any N ≥ 1 the restriction of Fn to |z| ≥ e1/N converges to

the restriction of F with respect to the distance dN . But this follows from Theorem 4.5

together with the maximum principle, which applies since all Fn,F vanish as |z| → ∞.

This concludes the proof.

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