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Averaging principle and Shape Theoremfor growth with memory
Amir Dembo (Stanford)
Paris, June 19, 2018
Joint work with Ruojun Huang, Pablo Groisman,and Vladas Sidoravicius
Laplacian growth & motion in random media
Random growth processes arise in many physical and biologicalphenomena, going back to Eden (61’)/fpp (Hammersley-Welsh 65’).Math challenge to understand their evolution and pattern formation.
Laplacian growth models: growth at each portion of the boundarydetermined by the harmonic measure of the boundary from a source.
Ex.: Diffusion Limited Aggregation dla (Witten-Sander 81’),Dielectric Breakdown Model (dbl, Niemeyer et. al. 83’),idla (Meakin-Deuthch 86’; Diaconis-Fulton 91’),hl (Hastings-Levitov 98’); (see Miller-Sheffield 13’).
Also Abelian sandpiles (Bak et. al. 87’) & Rotor aggregation(see Levine-Peres 17’).
Related models of motion in random media:Once-Reinforced Random Walk orrw (Davis 90’),Origin-Excited Random Walk oerw (Kozma 06’).
Amir Dembo Random growth, Shape Theorem 1 / 16
Excited walk towards the center
Excited random walk: Benjamini-Wilson (04’), Kosygina-Zerner (12’).At first visit by rw to a vertex, it gets a (one-time) drift in a fixeddirection (e.g. ~e1), on subsequent visits to vertex perform srw.
Simplified model of orrw, where drift direction not fixed (one-timeedge/vertex conductance increase 1→ 1 + a).
Kozma (06’) studies oerw with normalized drift at −v direction:E[Xt+1 −Xt|Ft] = − δ
||v||2 v if Xt = v is first visit of v ∈ Zn.
Proving recurrence ∀n ≥ 1, δ > 0.
Kozma (07’, 13’): Conjectured Shape Theorem for “most” oerw-s.Sidoravicius: same for orrw if reinforcement strength a > 0 large.
Having a bulky limit shape connected to the conjectural recurrence ofsuch random walks (confined but not too trapped within a small region).By Shape Theorem get excursion probabilities and deduce recurrence.
Amir Dembo Random growth, Shape Theorem 2 / 16
Simulations of orrw
Vertex orrw on Z2.
Reinforcement strength a = 2 (left), a = 3 (middle), a = 100 (right)in a box of size 2000. Color proportional to
√· of vertex first visit time.
Amir Dembo Random growth, Shape Theorem 3 / 16
Simulations of oerw
oerw on Z2 with different excitation rules.
L: move one unit towards the origin in a coordinate chosenwith probability proportional to its absolute value.
M: move one unit towards the origin in the direction ofthe coordinate with largest absolute value.
R: move one unit towards the origin in each coordinate.
Sites colored according to the first visit time by the walk.
Amir Dembo Random growth, Shape Theorem 4 / 16
Shape Theorems: idla and generalizations
idla (Lawler-Bramson-Griffeath 92’): Each step runs srw till exitscurrent range R(t) ⊂ Zn. Then t← t+ 1 & particle back to the origin.
∀η > 0, a.s. B(o, (1− η)t) ⊂ R(cntn) ⊂ B(o, (1 + η)t) ∀t large.
Uniform idla (Benjamini-DuminilCopin-Kozma-Lucas 17’): Upon exitingR(t) jump to a uniformly chosen v ∈ R(t). Same conclusion as for idla.
idla with finitely fixed sources & particles moving simultaneously at rateε−1 (Gravner-Quastel 00’): particles empirical density on (εZ)n convergesweakly as ε→ 0 to solutions of a Stefan free boundary problem.
Amir Dembo Random growth, Shape Theorem 5 / 16
A random growth model in Rn
We construct a simplified growth model in Rn, n ≥ 2, to bypass sometechnical difficulties of the lattice models.
Family of domain processes (Dεt)t≥0 ⊂ Rn with scale & Poisson rate
parameter ε ∈ (0, 1]. Our rules keep domains star-shaped and compact,so represented by boundary Rεt : Sn−1 7→ R+, a pure jump (slow) processevolving by adding a small bump of approximate volume ε at randompoints on the boundary.
Growth location driven by a (fast) particle process (xεt)t≥0 via hittingprobability density F (Rεt , x
εt, ·) on Sn−1, specifying law of the angle ξt
where a bump is to be added. Mapping H(Rεt , ξt) says to where particleis transported upon hitting the boundary at angle ξt.
F (r, x, ·) : C(Sn−1)× Rn → L2(Sn−1), H(r, z) : C(Sn−1)× Sn−1 → Rn.
Example: F (r, x, ·) Poisson kernel of r, H(r, z) = αr(z)z, α ∈ [0, 1).
Amir Dembo Random growth, Shape Theorem 6 / 16
Spherical approximate identity & small bump on Sn−1
sai {gη(〈z, ·〉)}η>0 such that: gη ∈ C([−1, 1],R+), 1 ? gη = 1,
||f ? gη||2 ≤ ||f ||2 & ||f ? gη − f ||2 → 0 as η → 0
(where ? denotes spherical convolution).
E.g. gη(s) = cη−(n−1)φ(1− 1−s
η2
)for φ ∈ C([−1, 1],R+), φ(−1) = 0.
For a bump centered at angle ξ add damped sai ε1/nηn−1gη(〈ξ, ·〉)with η(ε, r, x) = ε1/ny
−1/(n−1)r,x . Such bump has height O(ε1/n)
& support on the spherical cap of Euclidean radius 2η around ξ.
-1 -0.5 0 0.5 1
L: gη(s) at different η. R: Adding gη(〈z, ·〉) to S2; z = (0, 0, 1).
Amir Dembo Random growth, Shape Theorem 7 / 16
Our random growth model
yr,x := ωn∫Sn−1 r(θ)
n−1F (r, x, θ)dσ(θ), ωn = σ(Sn−1)
=⇒ Each bump adds on average volume ε+ o(ε) to Dεt .
Hitting kernel: F (r, x, ·) : C(Sn−1)× Rn → L2(Sn−1).Transportation function: H(r, z) : C(Sn−1)× Sn−1 → Rn.
(Rεt , xεt)t≥0 updates at arrival times {T εi } of a rate ε−1 Poisson process.
At each t = T εi , conditional on Ft− := σ(Rεs, xεs, ξs : s ≤ t−), let
Rεt(θ) = Rεt−(θ) +
ε1/nη(ε,Rεt− ,x
εt− )n−1︷ ︸︸ ︷
ε
yRεt−,xεt−
gη(ε,Rεt−,xεt−
)(〈ξt, θ〉), θ ∈ Sn−1,
ξtd∼ F (Rεt− , x
εt− , ·) , xεt = H(Rεt− , ξt).
Amir Dembo Random growth, Shape Theorem 8 / 16
Simulations of the random growth model
L: H(r, ξ) = (r(ξ)− 1)+ξ. M: H(r, ξ) = (r(ξ)− |ξ|∞|ξ|2 )+ξ.
R: H(r, ξ) = (r(ξ)− |ξ|1|ξ|2 )+ξ. F (r, x, ·) harmonic measure on r (from x).
Snapshots t = 0.2 k2, 0 ≤ k ≤ 10, for ε = 10−4, φ = 1[0,1], c = 20,
Linear in k evolution ⇒ O(√t) asymptotic diameter growth.
Small t ≈ spherical shape (H = o, idla-like).
Large t⇒ Rεt “feels” the geometry ⇒ H-dependent limit shape(sphere, square, diamond; similar to different excitation for oerw).
Amir Dembo Random growth, Shape Theorem 9 / 16
Frozen domain & limiting ODE
Domain Rεt ≡ r frozen =⇒ (xε,rt )t≥0 Markov process.
Assume ∀r ∈ C(Sn−1), process {x1,rt } has a unique invariant law νr.
Then, limiting infinite-dimensional ode for Rεt is
r̄t(θ)= r̄0(θ) +
∫ t
0
b̄(r̄s)(θ)ds, θ ∈ Sn−1 (ode)
b̄(r)(θ) :=
∫Rnb(r, x)(θ)dνr(x) , b(r, x)(·) :=
ωnyr,x
F (r, x, ·)
mg decomposition of {Rεt} has bv-term (drift)
bε(r, x) := b(r, x) ? gη(ε,r,x) → b(r, x) as ε→ 0 (⇒ η → 0),
and mg term of O(ε) quadratic variation.
Leb(rt) = Leb(r0) + t in line with
E[Leb(Dεt)|Ft−]− Leb(Dε
t−) ≈ ε at each t = T εi
Amir Dembo Random growth, Shape Theorem 10 / 16
Assumptions on F & H
A1(a) :={r ∈ C(Sn−1) : inf
θr(θ) ≥ a, ||r||2 ≤ a−1
}A(a) :=
{(r, x) ∈ D(F ) : r ∈ A1(a), x ∈ Im(H(r, ·))
}Assumption (L)
∃K = K(a) <∞ so for all (r, x), (r′, x′) ∈ A(a), z, z′ ∈ Sn−1
||F (r, x, ·)− F (r′, x′, ·)||2 ≤ K(||r − r′||2 + |x− x′|
), (LF )
|H(r, z)−H(r′, z′)| ≤ K(||r − r′||2 + |z − z′|
), (LH)
||b̄(r)− b̄(r′)||2 ≤ K||r − r′||2 . (Lb̄)
Assumption (E)
∀r ∈ C(Sn−1), the invariant probability law νr of {x1,rt } exists & unique.
limt→∞
sup(r,x1,r
0 )∈A(a)
E[∣∣∣∣1
t
∫ t
0
[b(r, x1,rs )− b̄(r)]ds∣∣∣∣22
]= 0 .
For (E) suffices to have uniform minorization of jump kernel Pr of {x1,rTi }inf
(r,x)∈A(a){(Pr)
n0(x, ·)} ≥ m(·) .
Amir Dembo Random growth, Shape Theorem 11 / 16
Hydrodynamic limit by Averaging Principle
Theorem (DGHS,18’)
(a) Assume (L) & (E). Fix Rε0 = r0 ∈ C(Sn−1). Then,
limε→0
P(
sup0≤t≤T∧σε(δ)
||Rεt − r̄t||2 > ι)
= 0, ∀T, ι, δ > 0 (?)
with Ft-stopping time
σε(δ) := inf{t ≥ 0 : minθ{F (Rεt , x
εt, θ)} < δ}.
(b) No σε(δ) in (?) when inf{F (r, x, θ) : (r, x) ∈ A(a), θ ∈ Sn−1} > 0.
Proof: As ε→ 0, dynamic of {xεt} near equilibrium at a time scale where{Rεt} does not change macroscopically, namely we have here an
averaging principle.
Amir Dembo Random growth, Shape Theorem 12 / 16
Towards shape theorem: scale invariance
Assumption (I)
∀c > 0 if (r, x) ∈ D(F ) then (cr, cx) ∈ D(F ) and
F (r, x, ·) = F (cr, cx, ·) cH(r, ·) = H(cr, ·)
Example: (I) holds for F (r, x, ·) Poisson kernel at D (from x), r = ∂D;& H(r, z) = αr(z)z for z ∈ Sn−1, fixed α ∈ (0, 1).
Proposition (coupling)
Under Assumption (I), there exists coupling with(Rεt , x
εt) = (ε1/nR1
t/ε, ε1/n x1t/ε) for all t > 0, whenever holding for t = 0.
Amir Dembo Random growth, Shape Theorem 13 / 16
Candidate limiting shapes
ψ ∈ C(Sn−1) invariant (shape) for
r̄t(θ) = r̄0(θ) +
∫ t
0
b̄(r̄s)(θ)ds, θ ∈ Sn−1 (ode)
⇐⇒ r0 = ψ yields rt = ctψ in (ode); (ct)n = Leb(ψ) + t.
Recall: b̄(r)(θ) = ωn∫Rn y
−1r,xF (r, x, θ)dνr(x)
(I) ⇒ νcr(c ·) = νr(·), ycr,cx = cn−1yr,x ⇒ b̄(cr) = c−(n−1)b̄(r)
Under (I) shape ψ is invariant ⇐⇒ b̄(ψ)(θ) = 1nψ(θ) ∀θ ∈ Sn−1
Ex: F (r, x, ·) Poisson kernel & H(r, z) = α(z)r(z)z
ψ = c (Euclidean ball) invariant ⇐⇒ α(z) = α (constant).
Amir Dembo Random growth, Shape Theorem 14 / 16
Shape theorem
Theorem (DGHS 18’)
Assume (I) and hydrodynamic limit
limε→0
P(
sup0≤t≤T
||Rεt − r̄t||2 > ι)
= 0, ∀T, ι, δ > 0 (?)
If ψ invariant for ode (wlog Leb(ψ) = 1), then ∀T, c, ι > 0,
limN→∞
P(
sup1≤s≤T
∣∣∣∣(N(c+ s))−1/nRsN − ψ∣∣∣∣2> ι∣∣R0 = (Nc)1/nψ
)= 0
While ||F (r, x, ·)− F (r′, x, ·)||2 ≤ K(a)(||r − r′||2
)(LF )
fails for Poisson kernel F (r, x, ·), this is resolved upon regularizing F .
Amir Dembo Random growth, Shape Theorem 15 / 16
Thank you!
Generator & mg decomposition
The generator of (Rεt , xεt)t≥0 is(
Lεf)(r, x) =
1
ε
[ ∫Sn−1
f(r +
ε
yr,xgη(ε,r,x)(〈ξ, ·〉), H(r, ξ)
)F (r, x, ξ)dσ(ξ)− f(r, x)
].
Taking f(r, x) = r(θ) at fixed θ ∈ Sn−1 yields
Rεt(θ) = Rε0(θ) +
∫ t
0
bε(Rεs, xεs)(θ)ds+ Σεt(θ)
where Σεt(θ) a (small) mg.
Taking f(r, x) = x · ~ei, i = 1, . . . , n, yields
xεt = xε0 +
∫ t
0
ε−1h(Rεs, xεs)ds+M ε
t
h(r, x) :=
∫Sn−1
H(r, ξ)F (r, x, ξ)dσ(ξ)− x ,
for some Rn-valued mg M εt .
Amir Dembo Random growth, Shape Theorem 16 / 16
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