Gaussian complex zeros and eigenvalues: Rare events and the - - PowerPoint PPT Presentation

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

Random point configurations

Poisson Point Process Ginibre ensemble Gaussian zeros

Random point configurations

Poisson Point Process Ginibre ensemble Gaussian zeros

Random point configurations

Poisson Point Process Ginibre ensemble Gaussian zeros

Point processes - Invariance

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

respect to T if X d ∼ T (X)

• r

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

Invariant point processes

All the examples we consider:

◮ Homogeneous Poisson Point Process ◮ Infinite Ginibre ensemble ◮ Zeros of the Gaussian Entire Function

are invariant with respect to:

◮ Translations ◮ Rotations ◮ Reflections

(Homogeneous) Poisson Point Process

◮ The distribution is given by

n(K) ∼ Poisson 1 π Area(K)

• .

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

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

◮ Can be thought of as “independent” points uniformly

distributed in the plane.

◮ “Gas” with no particle-particle interactions.

Ginibre ensemble (random eigenvalues)

Finite Ginibre

◮ Complex eigenvalues of non-Hermitian N ×N matrix ◮ Entries are independent standard complex Gaussian ◮ Standard complex Gaussian: density 1 π e−|w|2, w ∈ C.

Infinite Ginibre - limit of finite Ginibre as N → ∞

◮ Determinantal point process

◮ Probabilities are governed by eigenvalues of some integral

• perators

◮ Gas with particle-particle interactions (repulsion) embedded in

uniform background

Gaussian Entire Function (GEF)

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

F (z) =

∞

∑

n=0

ξn zn √ n! , z ∈ C.

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

Gaussian Entire Function - invariance

◮ F is a Gaussian function, distribution determined by covariance

kernel: K (z,w) = E

• F (z)F (w)
• =

∞

∑

n=0

(zw)n n! = ezw.

◮ Rotation: follows from rotation invariance of complex

Gaussians.

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

F (z +a) and F (z)eza− 1

2|a|2 have the same distribution ◮ eza− 1 2 |a|2 = 0 so zeros are invariant with respect to translations.

Gaussian Entire Function - zero set

◮ Counting measure of zeros:

nF = ∑

z∈Z(F)

δz

◮ Logarithm - fundamental solution to Laplacian:

δz = 1 2π ∆log|·−z| = ⇒ dnF (w) = 1 2π ∆log|F (w)|

◮ Not difficult to check:

E[log|F (w)|] = log

• K (w,w)+C = 1

2 |w|2 +C

◮ Using Fubini:

E[#{Z (F)∩D}] = 1 π Area(D).

Gaussian Entire Function - distribution of zeros

◮ GEF is unique (up to scaling) among Gaussian analytic

functions in the plane with invariant zero set.

◮ Statistics of zero set are well understood. Some examples:

◮ Edelman - Kostlan ’95 ◮ Forrester - Honner ’99 (Variance asymptotics) ◮ Sodin - Tsirelson ’04 (Central limit theorem)

Point processes - rare events

◮ Write n(r) = n({|z| ≤ r})

◮ number of points in a (large) disk

◮ For all three models we normalize E[n(r)] = r2 ◮ Consider rare events of the type:

• n(r) =
• pr2

, p = 1

◮ p = 0 - ‘hole’ (no points) ◮ p < 1 - deficiency ◮ p > 1 - overcrowding

Rare events and conditional distribution

◮ We find the asymptotic rate of decay of

P

• n(r) =
• pr2

, as r → ∞.

◮ (very) rare events

◮ We also find the distribution of the points conditioned on this

rare event.

Poisson Point Process

◮ n(r) number of points in {|z| ≤ r} has

Poisson

• r2

distribution.

◮ In particular, not difficult to calculate

these probabilities:

◮ logP

• n(r) =
• pr2

is of order −Cpr2

Poisson Point Process

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

points in each set is independent.

Infinite Ginibre ensemble (random ‘eigenvalues’)

◮ Determinantal point process

◮ The number of points n(r) can be

written as the sum of independent Bernoulli random variables.

◮ Moreover: Set of radii

• |z1|2 ,|z2|2 ,...
• has same

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

Infinite Ginibre ensemble (random ‘eigenvalues’)

◮ It is possible to find the asymptotic

decay of rare events’ probabilities.

◮ Shirai (’06) ◮ of order −Cpr 4

◮ Because the radii are independent not

too difficult to find the conditional distribution.

◮ Jancovici, Lebowitz, Manificat (’93)

◮ Unlike Poisson, the number of points is

‘conserved’.

Gaussian Entire Function (random zeros)

◮ Zeros of the GEF

F (z) =

∞

∑

n=0

ξn zn √ n! Z (F) = F −1 (0) ⊂ C

Gaussian Entire Function (random zeros)

◮ Zeros of the GEF

F (z) =

∞

∑

n=0

ξn zn √ n! Z (F) = F −1 (0) ⊂ C

◮ Not a determinantal point process.

Gaussian Entire Function (random zeros)

◮ Not a determinantal point process. ◮ Sodin and Tsirelson (’05): Found

decay rates are qualitatively like Ginibre ensemble.

Gaussian Entire Function (random zeros)

◮ Not a determinantal point process. ◮ Sodin and Tsirelson (’05): Found

decay rates are qualitatively like Ginibre ensemble.

◮ F. Nazarov and M. Sodin asked what

is the conditional distribution for the GEF.

◮ In particular, is there a gap in the

distribution?

Ginibre vs. GEF zeros - deficiency n(r) = 1

4r2 Ginibre ensemble: GEF zeros:

Ginibre vs. GEF zeros - deficiency n(r) = 1

4r2 Ginibre ensemble: GEF zeros:

Ginibre vs. GEF zeros - deficiency n(r) = 1

4r2 Ginibre ensemble: GEF zeros:

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

Ginibre ensemble: GEF zeros:

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

Ginibre ensemble: GEF zeros:

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

Ginibre ensemble: GEF zeros:

GEF zeros - hole event

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

GEF zeros - hole event

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

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]

◮ ϕ - a smooth test function with compact support. The

random variable (called linear statistics) n(ϕ;r) = ∑

z∈Z(F)

ϕ z r

• Theorem

E[n(ϕ;r) | Hole(r)] =

• C ϕ (w) dµH (w)·r2 +o
• r2

, r → ∞.

Main results [Ghosh, N. ’16]

◮ ϕ - a smooth test function with compact support. The

random variable (called linear statistics) n(ϕ;r) = ∑

z∈Z(F)

ϕ z r

• Theorem

E[n(ϕ;r) | Hole(r)] =

• C ϕ (w) dµH (w)·r2 +o
• r2

, r → ∞.

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

• Z (F)∩
• r (1+ε) ≤ |w| ≤ √er (1−ε)

Main results [Ghosh, N. ’16]

◮ ϕ - a smooth test function with compact support. The

random variable (called linear statistics) n(ϕ;r) = ∑

z∈Z(F)

ϕ z r

• Theorem

E[n(ϕ;r) | Hole(r)] =

• C ϕ (w) dµH (w)·r2 +o
• r2

, r → ∞.

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

• Z (F)∩
• r (1+ε) ≤ |w| ≤ √er (1−ε)
• Theorem

P

• Nε,r > r1+ε | Hole(r)
• ≤ exp
• −Cεr2(1+ε)

, r → ∞.

Some ideas - Joint distribution

◮ Approximate the Taylor series with polynomials

P (z) =

N

∑

n=0

ξn zn √ n!

Some ideas - Joint distribution

◮ Approximate the Taylor series with polynomials

P (z) =

N

∑

n=0

ξn zn √ n! = ξn √ n!

N

∏

j=1

(z −zj) =: ξn √ n! QN (z).

◮ Joint density of the zeros {z1,...,zN} is known, but

complicated: 1 AN ∏

j<k

|zj −zk|2

Some ideas - Joint distribution

◮ Approximate the Taylor series with polynomials

P (z) =

N

∑

n=0

ξn zn √ n! = ξn √ n!

N

∏

j=1

(z −zj) =: ξn √ n! QN (z).

◮ Joint density of the zeros {z1,...,zN} is known, but

complicated: 1 AN ∏

j<k

|zj −zk|2

• C |QN (w)|2

1 π e−|w|2 dm(w) −(N+1)

◮ Main idea: approximate the joint density (at the exponential

scale), with a limiting functional acting on probability measures.

◮ Motivated by Zeitouni and Zelditch ’10

Joint distribution - Comparison with Ginibre

◮ Finite Ginibre

◮ Complex eigenvalues of non-Hermitian N ×N matrix with i.i.d.

complex Gaussian entries.

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

1 BN ∏

j<k

|wj −wk|2 exp

• −

N

∑

j=1

|wj|2

• ◮ We can describe the limiting distribution of the eigenvalues,

with appropriate scaling, in terms of probability measures.

Joint distribution - Comparison with Ginibre - cont.

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

1 BN ∏

j<k

|wj −wk|2 exp

• −

N

∑

j=1

|wj|2

• ◮ With a probability measure µ on C we can associate:

◮ Log. potential: Uµ (z) =

• C log|z −w| dµ (w).

◮ Log. energy:

Σ(µ) =

• C×C log|z −w| dµ (z)dµ (w) =
• C Uµ (z) dµ (z).

◮ Limiting functional is:

J (µ) = 2

• C

|w|2 2 dµ (w)−Σ(µ).

◮ Weighted logarithmic energy (external field |w|2

2 ).

Some ideas - Limiting functional

◮ In the same way, we can describe the limiting distribution of

the zeros (with appropriate scaling) in terms of probability measures.

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

1 AN ∏

j<k

|zj −zk|2

• C |QN (w)|2

1 π e−|w|2 dm(w) −(N+1)

◮ The (strictly convex) limiting functional is:

I (µ) = 2 sup

w∈C

• Uµ (w)− |w|2

2

• −Σ(µ).

◮ ‘Configurations’ which are more likely to occur correspond to

smaller values of the functional.

Some ideas - minimizing measures

◮ The (strictly convex) limiting functional is:

I (µ) = 2 sup

w∈C

• Uµ (w)− |w|2

2

• −Σ(µ).

◮ ‘Configurations’ which are more likely to occur correspond to

smaller values of the functional.

◮ Identify the limiting conditional distribution of the zeros:

◮ find the (unique) measure µH = µH (t) minimizing the

functional I (µ), out of all measures µ satisfying the constraint µ ({|z| < t}) = 0.

◮ The minimizing measure determine the limiting conditional

distribution of the zeros.

Some ideas - minimizing measures - cont.

◮ By symmetrization and convexity of the funcational I (µ) the

minimizing measures are radial, and the following version of Jensen’s formula is useful:

r

µ ({|z| ≤ t}) t dt = Uµ (r)−Uµ (0)

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

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

Uµeq (z) = |z|2 2 − 1 2, |z| ≤ 1.

◮ Since, up to a constant, Uµeq (z) is the same as the external

field, 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:

GEF Zeros - hole event potential

Zeros on Hole(r): Potential: ‘External field’: r2 2 Distribution:

GEF Zeros - hole event potential

Zeros on Hole(r): Potential: ‘External field’: r2 2 Distribution:

Non-circular holes

◮ The same approach works in principle for other domains.

◮ If someone will tell me how to solve the constrained

minimization problem.

◮ Ginibre - Adhikari, Reddy (’16)

◮ ‘Perturbations’ of disks: can find asymptotic probability of the

hole and conditional distribution.

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

◮ It is still not clear what is the general picture, even for

simply-connected domains.

One-Component Plasma (OCP)

◮ OCP represents the simplest statistical mechanical model of a

Coulomb system.

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

1 BN;β ∏

j<k

• zj −zk
• β exp
• −β

2

N

∑

j=1

• zj
• 2
• ◮ Arbitrary (inverse) temperature parameter β > 0

◮ (finite) Ginibre ensemble - a special ‘computable’ case (β = 2)

◮ Similar approach works

◮ Gives hope to study large fluctuations in the number of

particles.

The end

Proofs appear in:

◮ S. Ghosh, A. N. - Gaussian complex zeros on the hole event:

the emergence of a forbidden region.

◮ arXiv:1609.00084.