Gaussiancomplexzerosandeigenvalues ... · Gaussiancomplexzerosandeigenvalues: Rareeventsandtheemergenceof the‘forbidden’region LesDiablerets,13/02/2018 AlonNishry-TelAvivUniversity

Gaussian complex zeros and eigenvalues:Rare events and the emergence of

the ‘forbidden’ region

Les Diablerets, 13/02/2018

Alon Nishry - Tel Aviv University

Joint work with Subhroshekhar Ghosh (NUS)

Random point configurations

Poisson Point Process Ginibre ensemble Gaussian zeros







Point processes - Invariance

I X = {zj}j∈I - random point configurationI n (K ) is the number of points in a compact set KI T : C→ C a transformation (automorphism)I The distribution of the point process X is invariant with

respect to T ifX

d∼ T (X)

or

(n (K1) , . . . ,n (Kn))d∼ (n (T (K1)) , . . . ,n (T (Kn)))

Invariant point processes

All the examples we consider:I Homogeneous Poisson Point ProcessI Infinite Ginibre ensembleI Zeros of the Gaussian Entire Function

are invariant with respect to:I TranslationsI RotationsI Reflections

(Homogeneous) Poisson Point Process

I The distribution is given by

n (K )∼ Poisson(1π

Area(K )

).

I No “correlations”: K1,K2, . . . ,KN ⊂ C compact sets

K1,K2, . . . ,KN disjoint =⇒ n (K1) ,n (K2) , . . . ,n (KN) independent

I Can be thought of as “independent” points uniformlydistributed in the plane.

I “Gas” with no particle-particle interactions.

Ginibre ensemble (random eigenvalues)

Finite GinibreI Complex eigenvalues of non-Hermitian N×N matrixI Entries are independent standard complex GaussianI Standard complex Gaussian: density 1

πe−|w |

2, w ∈ C.

Infinite Ginibre - limit of finite Ginibre as N → ∞

I Determinantal point processI Probabilities are governed by eigenvalues of some integral

operators

I Gas with particle-particle interactions (repulsion) embedded inuniform background

Gaussian Entire Function (GEF)

I {ξn} - sequence of independent standard complex Gaussians.I The GEF is given by the Gaussian Taylor series:

F (z) =∞

∑n=0

ξnzn√n!, z ∈ C.

I Infinite radius of convergence (almost surely).I Zero set: Z (F ) = F−1 (0) is a discrete set in C.

Gaussian Entire Function - invariance

I F is a Gaussian function, distribution determined by covariancekernel:

K (z ,w) = E[F (z)F (w)

]=

∞

∑n=0

(zw)n

n!= ezw .

I Rotation: follows from rotation invariance of complexGaussians.

I Reflection: F (z) and F (z) have the same distribution.I Translation: For a ∈ C easy to check that

F (z +a) and F (z)eza−12 |a|

2have the same distribution

I eza−12 |a|

26= 0 so zeros are invariant with respect to translations.

Gaussian Entire Function - zero set

I Counting measure of zeros:

nF = ∑z∈Z(F )

δz

I Logarithm - fundamental solution to Laplacian:

δz =12π

∆ log |·− z | =⇒ dnF (w) =12π

∆ log |F (w)|

I Not difficult to check:

E [log |F (w)|] = log√

K (w ,w) +C =12|w |2 +C

I Using Fubini:

E [#{Z (F )∩D}] =1π

Area(D) .

Gaussian Entire Function - distribution of zeros

I GEF is unique (up to scaling) among Gaussian analyticfunctions in the plane with invariant zero set.

I Statistics of zero set are well understood. Some examples:I Edelman - Kostlan ’95I Forrester - Honner ’99 (Variance asymptotics)I Sodin - Tsirelson ’04 (Central limit theorem)

Point processes - rare events

I Write n (r) = n ({|z | ≤ r})I number of points in a (large) disk

I For all three models we normalize E [n (r)] = r2

I Consider rare events of the type:{n (r) =

⌊pr2⌋} , p 6= 1

I p = 0 - ‘hole’ (no points)I p < 1 - deficiencyI p > 1 - overcrowding

Rare events and conditional distribution

I We find the asymptotic rate of decay of

P(n (r) =

⌊pr2⌋) , as r → ∞.

I (very) rare events

I We also find the distribution of the points conditioned on thisrare event.

Poisson Point Process

I n (r) number of points in {|z | ≤ r} hasPoisson

(r2) distribution.

I In particular, not difficult to calculatethese probabilities:

I logP(n (r) =

⌊pr2⌋) is of order −Cpr

2

Poisson Point Process

I By the definition of the process:I For disjoint sets the distribution of the

points in each set is independent.

Infinite Ginibre ensemble (random ‘eigenvalues’)

I Determinantal point processI The number of points n (r) can be

written as the sum of independentBernoulli random variables.

I Moreover: Set of radii{|z1|2 , |z2|2 , . . .

}has same

distribution as the set ofindependent random variables{Γ(1,1) ,Γ(2,1) , . . .}.

Infinite Ginibre ensemble (random ‘eigenvalues’)

I It is possible to find the asymptoticdecay of rare events’ probabilities.

I Shirai (’06)I of order −Cpr

4

I Because the radii are independent nottoo difficult to find the conditionaldistribution.

I Jancovici, Lebowitz, Manificat (’93)

I Unlike Poisson, the number of points is‘conserved’.

Gaussian Entire Function (random zeros)

I Zeros of the GEF

F (z) =∞

∑n=0

ξnzn√n!

Z (F ) = F−1 (0)⊂ C


I Zeros of the GEF

F (z) =∞

∑n=0

ξnzn√n!

Z (F ) = F−1 (0)⊂ C

I Not a determinantal point process.



I Sodin and Tsirelson (’05): Founddecay rates are qualitatively likeGinibre ensemble.



I Sodin and Tsirelson (’05): Founddecay rates are qualitatively likeGinibre ensemble.

I F. Nazarov and M. Sodin asked whatis the conditional distribution for theGEF.

I In particular, is there a gap in thedistribution?

Ginibre vs. GEF zeros - deficiency n (r) = 14r

2

Ginibre ensemble: GEF zeros:


2



2


Ginibre vs. GEF zeros - overcrowding n (r) = 2r2






GEF zeros - hole event

Consider just the ‘hole’ event: Hole(r) = {Z (F )∩{|z | ≤ r}= /0}

Zeros on Hole(r): Limiting measure µH :







Limiting measure: dµH (w) = e ·δ{|w |=1}+1{|w |≥√e}dm(w)

π

Main results [Ghosh, N. ’16]

I ϕ - a smooth test function with compact support. Therandom variable (called linear statistics)

n (ϕ; r) = ∑z∈Z(F )

ϕ

(zr

)

TheoremE [n (ϕ; r) | Hole(r)] =

∫C ϕ (w) dµH (w) · r2 +o

(r2) , r → ∞.

I What is the actual number of zeros in the annulus?I Nε,r = #

{Z (F )∩

{r (1+ ε)≤ |w | ≤

√er (1− ε)

}}TheoremP(Nε,r > r1+ε | Hole(r)

)≤ exp

(−Cεr2(1+ε)

), r → ∞.



n (ϕ; r) = ∑z∈Z(F )

ϕ

(zr

)


∫C ϕ (w) dµH (w) · r2 +o

(r2) , r → ∞.


{Z (F )∩

{r (1+ ε)≤ |w | ≤

√er (1− ε)

}}

TheoremP(Nε,r > r1+ε | Hole(r)

)≤ exp

(−Cεr2(1+ε)

), r → ∞.



n (ϕ; r) = ∑z∈Z(F )

ϕ

(zr

)


∫C ϕ (w) dµH (w) · r2 +o

(r2) , r → ∞.


{Z (F )∩

{r (1+ ε)≤ |w | ≤

√er (1− ε)

}}TheoremP(Nε,r > r1+ε | Hole(r)

)≤ exp

(−Cεr2(1+ε)

), r → ∞.

Some ideas - Joint distribution

I Approximate the Taylor series with polynomials

P (z) =N

∑n=0

ξnzn√n!

=ξn√n!

N

∏j=1

(z− zj) =:ξn√n!QN (z) .

I Joint density of the zeros {z1, . . . ,zN} is known, butcomplicated:

1AN

∏j<k

|zj − zk |2(∫

C|QN (w)|2

[1πe−|w |

2]

dm (w)

)−(N+1)

I Main idea: approximate the joint density (at the exponentialscale), with a limiting functional acting on probabilitymeasures.

I Motivated by Zeitouni and Zelditch ’10



P (z) =N

∑n=0

ξnzn√n!

=ξn√n!

N

∏j=1

(z− zj) =:ξn√n!QN (z) .


1AN

∏j<k

|zj − zk |2

(∫C|QN (w)|2

[1πe−|w |

2]

dm (w)

)−(N+1)





P (z) =N

∑n=0

ξnzn√n!

=ξn√n!

N

∏j=1

(z− zj) =:ξn√n!QN (z) .


1AN

∏j<k

|zj − zk |2(∫

C|QN (w)|2

[1πe−|w |

2]

dm (w)

)−(N+1)



Joint distribution - Comparison with Ginibre

I Finite GinibreI Complex eigenvalues of non-Hermitian N×N matrix with i.i.d.

complex Gaussian entries.

I Joint density of Ginibre eigenvalues {w1, . . . ,wN} is:

1BN

∏j<k

|wj −wk |2 exp

(−

N

∑j=1|wj |2

)

I We can describe the limiting distribution of the eigenvalues,with appropriate scaling, in terms of probability measures.

Joint distribution - Comparison with Ginibre - cont.

I Joint density of Ginibre eigenvalues {w1, . . . ,wN} is:

1BN

∏j<k

|wj −wk |2 exp

(−

N

∑j=1|wj |2

)

I With a probability measure µ on C we can associate:I Log. potential: Uµ (z) =

∫C log |z−w | dµ (w).

I Log. energy:Σ(µ) =

∫∫C×C log |z−w | dµ (z)dµ (w) =

∫CUµ (z) dµ (z).

I Limiting functional is:

J (µ) = 2∫C

|w |2

2dµ (w)−Σ(µ) .

I Weighted logarithmic energy (external field |w |2

2 ).

Some ideas - Limiting functional

I In the same way, we can describe the limiting distribution ofthe zeros (with appropriate scaling) in terms of probabilitymeasures.

I Joint density of the zeros {z1, . . . ,zN} is:

1AN

∏j<k

|zj − zk |2(∫

C|QN (w)|2

[1πe−|w |

2]

dm (w)

)−(N+1)

I The (strictly convex) limiting functional is:

I (µ) = 2 supw∈C

{Uµ (w)− |w |

2

2

}−Σ(µ) .

I ‘Configurations’ which are more likely to occur correspond tosmaller values of the functional.

Some ideas - minimizing measures

I The (strictly convex) limiting functional is:

I (µ) = 2 supw∈C

{Uµ (w)− |w |

2

2

}−Σ(µ) .

I ‘Configurations’ which are more likely to occur correspond tosmaller values of the functional.

I Identify the limiting conditional distribution of the zeros:I find the (unique) measure µH = µH (t) minimizing the

functional I (µ), out of all measures µ satisfying the constraint

µ ({|z |< t}) = 0.

I The minimizing measure determine the limiting conditionaldistribution of the zeros.

Some ideas - minimizing measures - cont.

I By symmetrization and convexity of the funcational I (µ) theminimizing measures are radial, and the following version ofJensen’s formula is useful:∫ r

0

µ ({|z | ≤ t})t

dt = Uµ (r)−Uµ (0)

I In particular, the potential is constant inside the ‘hole’.

I For µeq the uniform probability measure on the disk {|z | ≤ 1}:

Uµeq (z) =|z |2

2− 1

2, |z | ≤ 1.

I Since, up to a constant, Uµeq (z) is the same as the externalfield, the measure µeq is the unconditional (limiting)distribution of the zeros/eigenvalues.

Ginibre - hole event potential

‘Eigenvalues’ (unconditional):

Potential:

External field:

r2

2

Distribution:

Ginibre - hole event potential

‘Eigenvalues’ on Hole(r):

Potential:

External field:

r2

2

Distribution:

GEF Zeros - hole event potential

Zeros (unconditional):

Potential:

‘External field’:

r2

2

Distribution:


Zeros on Hole(r):

Potential:


r2

2

Distribution:


Zeros on Hole(r):

Potential:


r2

2

Distribution:

Non-circular holes

I The same approach works in principle for other domains.I If someone will tell me how to solve the constrained

minimization problem.I Ginibre - Adhikari, Reddy (’16)

I ‘Perturbations’ of disks: can find asymptotic probability of thehole and conditional distribution.

I Have to define ‘center’ and ‘radius’ (“inner capacity”).

I It is still not clear what is the general picture, even forsimply-connected domains.

One-Component Plasma (OCP)

I OCP represents the simplest statistical mechanical model of aCoulomb system.

I Joint density for points {z1, . . . ,zN} is given by:

1BN;β

∏j<k

∣∣zj − zk∣∣β exp(−β

2

N

∑j=1

∣∣zj ∣∣2)

I Arbitrary (inverse) temperature parameter β > 0I (finite) Ginibre ensemble - a special ‘computable’ case (β = 2)

I Similar approach worksI Gives hope to study large fluctuations in the number of

particles.

The end

Proofs appear in:

I S. Ghosh, A. N. - Gaussian complex zeros on the hole event:the emergence of a forbidden region.

I arXiv:1609.00084.

Documents

Gaussiancomplexzerosandeigenvalues ... · Gaussiancomplexzerosandeigenvalues: Rareeventsandtheemergenceof the‘forbidden’region LesDiablerets,13/02/2018 AlonNishry-TelAvivUniversity