Powers of Ginibre Eigenvaluesdubach/Beamer_Ginibre_Powers.pdfPowers of Ginibre Eigenvalues...

Introduction Power-Ginibre decomposition Other ensembles

Powers of Ginibre EigenvaluesThe Laundry Machine Effect

Guillaume DubachCourant Institute, NYU

Northeast Probability SeminarNovember 15th, 2018

Guillaume Dubach Courant Institute, NYU Powers of Ginibre Eigenvalues 1 / 21

Complex Ginibre EnsembleGinibre ensemble: Gin(N), N ×N matrix G , with i.i.d. entries

Gi ,jd=NC (0, Id) .

Eigenvalues are almost surely distinct : G =P∆P−1.∆= Diag(λ1, . . . ,λN) with density proportional to∏

i<j|λi −λj |2e−

∑Ni=1 |λi |2

with respect to the Lebesgue measure on C.

Convergence of the empirical measure of the scaled eigenvalues tothe uniform measure on D=D(0,1).

N∑k=1

δ λkpN

d→ 1π1D.

Gi ,jd=NC (0, Id) .

Eigenvalues are almost surely distinct : G =P∆P−1.

∆= Diag(λ1, . . . ,λN) with density proportional to∏i<j

|λi −λj |2e−∑N

i=1 |λi |2

N∑k=1

δ λkpN

d→ 1π1D.

Gi ,jd=NC (0, Id) .

∑Ni=1 |λi |2

N∑k=1

δ λkpN

d→ 1π1D.

Gi ,jd=NC (0, Id) .

∑Ni=1 |λi |2

N∑k=1

δ λkpN

d→ 1π1D.

Circular Law : Ginibre points vs. uniform points

Kostlan’s Theorem

Definition (Gamma variables)

For any α> 0, γα is a real variable with density

1Γ(α)

tα−1e−t1R+ .

Theorem (Kostlan ’92){|λ1|2, . . . , |λN |2} d= {γ1, . . . ,γN

}, where the gamma variables are

independent, with parameters 1,2, . . . ,N.

One consequence : Gumbel fluctuations for max(|λi |2). (Rider ’03,Chafaï-Péché ’ 14)

Kostlan’s Theorem

1Γ(α)

tα−1e−t1R+ .

Kostlan’s Theorem

1Γ(α)

tα−1e−t1R+ .

Kostlan’s Theorem

Repulsion between eigenvalues

(Term∏

i<j |λi −λj |2 in the joint density.)

Kostlan’s Theorem

High powersPower map : πM(z)= zM . Image of Gin(N) under πM ?

Theorem (Hough, Krishnapur, Peres, Virág ’06)

For any integer M ≥N, the following equality in distribution holds:{λM1 , . . . ,λMN

}d= {Z1, . . . ,ZN }

where the variables Zk are independent.

}d= {Z1, . . . ,ZN }

Rains for the CUE

CUE(N) : unitary matrix chosen uniformly on UN(C).

Eigenvaluedensity proportional to: ∏

k<l|e iθk −e iθl |2

with respect to the Lebesgue measure on the unit circle.

Theorem (Rains ’03)

For any M ≥ 1,

CUE(N)Md=

N⋃k=1

(⌈N −k

where the CUE blocks in the right hand side are independent.

Rains for the CUE

CUE(N) : unitary matrix chosen uniformly on UN(C). Eigenvaluedensity proportional to: ∏

For any M ≥ 1,

CUE(N)Md=

N⋃k=1

(⌈N −k

Rains for the CUE

CUE(N) : unitary matrix chosen uniformly on UN(C). Eigenvaluedensity proportional to: ∏

For any M ≥ 1,

CUE(N)Md=

N⋃k=1

(⌈N −k

Power-Ginibre blocks

For fixed M ≤N and k ∈ [[1,M]], we consider the arithmeticprogressions

Ik :={i ∈ [[1,N]] | i ≡ k [M]

DefinitionThe Power-Ginibre distribution Gin(N ,M ,k) is the point processindexed by Ik with joint density proportional to∏

i<ji ,j∈Ik

|zi −zj |2∏i∈Ik

|zi |2(k−M)

M e−|zi |2/M

dm(zi ).

Example: Gin(5,2,1) consists of {z1,z3,z5} with the above density.

Ik :={i ∈ [[1,N]] | i ≡ k [M]

i<ji ,j∈Ik

|zi |2(k−M)

M e−|zi |2/M

dm(zi ).

Ik :={i ∈ [[1,N]] | i ≡ k [M]

i<ji ,j∈Ik

|zi |2(k−M)

M e−|zi |2/M

dm(zi ).

Power-Ginibre decomposition

Theorem (D. ’18)

The images of Ginibre eigenvalues under πM are distributed like asuperposition of M independent Power-Ginibre point processes.

Gin(N)Md=

M⋃k=1

Gin(N ,M ,k).

d= t t·· ·t︸︷︷︸M independent blocks

Theorem (D. ’18)

Gin(N)Md=

M⋃k=1

Gin(N ,M ,k).

Theorem (D. ’18)

Gin(N)Md=

M⋃k=1

Gin(N ,M ,k).

t t·· ·t︸︷︷︸M independent blocks

Theorem (D. ’18)

Gin(N)Md=

M⋃k=1

Gin(N ,M ,k).

t·· ·t︸︷︷︸M independent blocks

Theorem (D. ’18)

Gin(N)Md=

M⋃k=1

Gin(N ,M ,k).

Repulsion breaking

Gin(N)

πM : z 7→ zM

Gin(N)M

Gin(N,M,2)

Gin(N,M,1)Gin(N,M,1)

Gin(N,M,M)

(Superposition)

Gin(N)M

Twisted circular law

M fixed, N →∞. The pushforward of the uniform distribution onthe disk has density 1

πM |z |2/M−2.

• Convergence to the twisted circular law holds for every block.• Second order: i.i.d. Gaussian fluctuations coherent with GFF.

πM |z |2/M−2.

• Convergence to the twisted circular law holds for every block.

• Second order: i.i.d. Gaussian fluctuations coherent with GFF.

πM |z |2/M−2.

Proof : main tool

Lemma (Andréief 1883)

Let (E ,E ,ν) be a measure space.For any functions (φi ,ψi )

Ni=1 in L2(ν),

det(φi (λj)) det(ψi (λj)) dνN(λ)= det(fi ,j)

where fi ,j =∫Eφi (λ)ψj(λ)dν(λ).

With φi (z)= z i−1,ψi (z)= z i−1, this gives a formula for thestatistics E

(∏Ni=1P(λi )

)with any polynomial P .

Proof : main tool

Lemma (Andréief 1883)

Ni=1 in L2(ν),

det(φi (λj)) det(ψi (λj)) dνN(λ)= det(fi ,j)

where fi ,j =∫Eφi (λ)ψj(λ)dν(λ).

With φi (z)= z i−1,ψi (z)= z i−1, this gives a formula for thestatistics E

(∏Ni=1P(λi )

)with any polynomial P .

General 2D β-ensembles

These results do not depend on the Gaussian reference measure.

The same is true for the joint density∏i<j

|λi −λj |βe−∑N

i=1V (|λi |2)

with β= 2 and any suitable V .

For instance, Kostlan’s theorem holds with generalized Gammavariable (Density tα−1e−V (t) on R+).

Conditional results available for any β= 2p,p ∈N.

i=1V (|λi |2)

Quaternionic Ginibre Ensemble

Figure: Quaternionic ’eigenvalues’.

B Artistic view : actual product may vary.

Figure: Quaternionic ’eigenvalues’.

B Artistic view : actual product may vary.

QGE: eigenvalues (λ1,λ1, . . . ,λN ,λN) with joint density∏i<j

|λi −λj |2∏i≤j

|λi −λj |2e−∑ |λi |2dµN(λ).

Note that :∏i<j

|λi −λj |2 =∆2N(λ1, . . . ,λN ,λ1, . . . ,λN)N∏i=1

(λi −λi ).

∣∣∣∣∣∣∣∣∣∣1 . . . 1 1 . . . 1λ1 . . . λN λ1 . . . λN...

......

λ2N−11 . . . λ2N−1

N λ2N−11 . . . λ

2N−1N

∣∣∣∣∣∣∣∣∣∣

Note that :∏i<j

|λi −λj |2 =∆2N(λ1, . . . ,λN ,λ1, . . . ,λN)N∏i=1

(λi −λi ).

∣∣∣∣∣∣∣∣∣∣1 . . . 1 1 . . . 1λ1 . . . λN λ1 . . . λN...

......

λ2N−11 . . . λ2N−1

N λ2N−11 . . . λ

2N−1N

∣∣∣∣∣∣∣∣∣∣

Note that :∏i<j

|λi −λj |2 =∆2N(λ1, . . . ,λN ,λ1, . . . ,λN)N∏i=1

(λi −λi ).

∣∣∣∣∣∣∣∣∣∣1 . . . 1 1 . . . 1λ1 . . . λN λ1 . . . λN...

......

λ2N−11 . . . λ2N−1

N λ2N−11 . . . λ

2N−1N

∣∣∣∣∣∣∣∣∣∣

Proposition (De Bruijn)

2Ni=1 in L2(ν),∫

ENdet(φi (λj) | ψi (λj)) dνN(λ)=N!2NPf(fi ,j)

2Ni ,j=1

wherefi ,j =

∫Eφi (λ)ψj(λ)dν(λ).

where Pf is the pfaffian.

(PfM)2 = det(M−Mt

Proposition (De Bruijn)

2Ni=1 in L2(ν),∫

ENdet(φi (λj) | ψi (λj)) dνN(λ)=N!2NPf(fi ,j)

2Ni ,j=1

wherefi ,j =

∫Eφi (λ)ψj(λ)dν(λ).

where Pf is the pfaffian. (PfM)2 = det(M−Mt

Theorem (Rider ’03)

For QGE,{|λ1|2, . . . , |λN |2} d= {

γ2, . . . ,γ2N}

Theorem (D. ’18)

For any integer M ≥ 2N,{λM1 ,λ1

M, . . . ,λMN ,λN

{Z1,Z 1, . . . ,ZN ,ZN

}where the Zi ’s are independent variables.

Theorem (Rider ’03)

For QGE,{|λ1|2, . . . , |λN |2} d= {

γ2, . . . ,γ2N}

Theorem (D. ’18)

For any integer M ≥ 2N,{λM1 ,λ1

M, . . . ,λMN ,λN

{Z1,Z 1, . . . ,ZN ,ZN

}where the Zi ’s are independent variables.

Thank you for your attention !

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