Dynamics of a planar Coulomb gas Djalil C HAFA Universit - - PowerPoint PPT Presentation

Dynamics of a planar Coulomb gas

Dynamics of a planar Coulomb gas

Djalil CHAFAÏ

Université Paris-Dauphine

Workshop on Optimal and Random Point Configurations February 26, 2018 – ICERM, Brown University

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Dynamics of a planar Coulomb gas

Joint work with. . .

François BOLLEY and Joaquín FONTBONA

2/26

Dynamics of a planar Coulomb gas

Motivation: Ginibre Ensemble

−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0

♥❂✺✵✵❀♣❧♦t✭❡✐❣✭r❛♥❞♥✭♥✱♥✮✰✐✯r❛♥❞♥✭♥✱♥✮✴sqrt✭✷✯♥✮✮✮

3/26

Dynamics of a planar Coulomb gas

Motivation: Ginibre Ensemble

−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0

♥❂✺✵✵❀♣❧♦t✭❡✐❣✭r❛♥❞♥✭♥✱♥✮✰✐✯r❛♥❞♥✭♥✱♥✮✴sqrt✭✷✯♥✮✮✮

Stochastic process leaving invariant this random picture?

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Dynamics of a planar Coulomb gas Poincaré inequality

Outline

Poincaré inequality Dyson Process Ginibre process

4/26

Dynamics of a planar Coulomb gas Poincaré inequality

Markov diffusion processes

H : Rd Ñ R, Hpxq energy of state or configuration x P Rd

5/26

Dynamics of a planar Coulomb gas Poincaré inequality

Markov diffusion processes

H : Rd Ñ R, Hpxq energy of state or configuration x P Rd Gradient dynamical system with noise

xn`1 ´ xn “ ´∇Hpxnq` gn

5/26

Dynamics of a planar Coulomb gas Poincaré inequality

Markov diffusion processes

H : Rd Ñ R, Hpxq energy of state or configuration x P Rd Markov process pXtqtě0 stochastic differential equation in Rd dXt “ ´∇HpXtqdt `dBt

5/26

Dynamics of a planar Coulomb gas Poincaré inequality

Markov diffusion processes

H : Rd Ñ R, Hpxq energy of state or configuration x P Rd Markov process pXtqtě0 stochastic differential equation in Rd dXt “ ´∇HpXtqdt `dBt Well-posedness or non-explosion: if ∇2H ě cId, c P R

5/26

Dynamics of a planar Coulomb gas Poincaré inequality

Markov diffusion processes

H : Rd Ñ R, Hpxq energy of state or configuration x P Rd Markov process pXtqtě0 stochastic differential equation in Rd dXt “ ´∇HpXtqdt `dBt Well-posedness or non-explosion: if ∇2H ě cId, c P R Reversible invariant Boltzmann–Gibbs measure µpdxq “ e´Hpxq

Z

dx,

Z “

ż e´Hpxqdx

X0 „ µ

ñ pX0,Xtq

d

“ pXt,X0q @t ě 0

5/26

Dynamics of a planar Coulomb gas Poincaré inequality

Markov diffusion processes

H : Rd Ñ R, Hpxq energy of state or configuration x P Rd Markov process pXtqtě0 stochastic differential equation in Rd dXt “ ´α∇HpXtqdt `?αdBt Well-posedness or non-explosion: if ∇2H ě cId, c P R Reversible invariant Boltzmann–Gibbs measure µpdxq “ e´Hpxq

Z

dx,

Z “

ż e´Hpxqdx

X0 „ µ

ñ pX0,Xtq

d

“ pXt,X0q @t ě 0

5/26

Dynamics of a planar Coulomb gas Poincaré inequality

Markov diffusion processes

H : Rd Ñ R, Hpxq energy of state or configuration x P Rd Markov process pXtqtě0 stochastic differential equation in Rd dXt “ ´α∇HpXtqdt ` cα β dBt Well-posedness or non-explosion: if ∇2H ě cId, c P R Reversible invariant Boltzmann–Gibbs measure µβpdxq “ e´βHpxq

Zβ

dx

X0 „ µ

ñ pX0,Xtq

d

“ pXt,X0q @t ě 0

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Dynamics of a planar Coulomb gas Poincaré inequality

Fokker–Planck evolution equation and generator

Let pt : Rd Ñ R` be the density of LawpXtq with respect to µ

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Dynamics of a planar Coulomb gas Poincaré inequality

Fokker–Planck evolution equation and generator

Let pt : Rd Ñ R` be the density of LawpXtq with respect to µ Fokker–Planck evolution equation and generator Btpt “ Gpt

where Gf “ ∆f ´x∇H,∇fy.

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Dynamics of a planar Coulomb gas Poincaré inequality

Fokker–Planck evolution equation and generator

Let pt : Rd Ñ R` be the density of LawpXtq with respect to µ Fokker–Planck evolution equation and generator Btpt “ Gpt

where Gf “ ∆f ´x∇H,∇fy.

The operator ´G is symmetric and non-negative in L2pµq

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Dynamics of a planar Coulomb gas Poincaré inequality

Fokker–Planck evolution equation and generator

Let pt : Rd Ñ R` be the density of LawpXtq with respect to µ Fokker–Planck evolution equation and generator Btpt “ Gpt

where Gf “ ∆f ´x∇H,∇fy.

The operator ´G is symmetric and non-negative in L2pµq Decay to the equilibrium: for all p0, Bt}pt ´ 1}2

L2pµq “ BtVarµpptq “ 2EµpptGptq “ ´2Eµp|∇pt|2q ď 0.

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Dynamics of a planar Coulomb gas Poincaré inequality

Fokker–Planck evolution equation and generator

Let pt : Rd Ñ R` be the density of LawpXtq with respect to µ Fokker–Planck evolution equation and generator Btpt “ Gpt

where Gf “ ∆f ´x∇H,∇fy.

The operator ´G is symmetric and non-negative in L2pµq Decay to the equilibrium: for all p0, Bt}pt ´ 1}2

L2pµq “ BtVarµpptq “ 2EµpptGptq “ ´2Eµp|∇pt|2q ď 0.

Exponential decay equivalent to Poincaré inequality: for all ρ ą 0, @p0,t, Varµpptq ď e´2ρtVarµpp0q

iif

@f, Varµpfq ď ´Eµp|∇f|2q ρ .

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Dynamics of a planar Coulomb gas Poincaré inequality

Exactly solvable model: Ornstein–Uhlenbeck process

Gaussian model: Hpxq “ |x|2

2 , dXt “ ´Xtdt `dBt,

µ “ N p0,Idq,

Gfpxq “ ∆fpxq´xx,∇fpxqy

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Dynamics of a planar Coulomb gas Poincaré inequality

Exactly solvable model: Ornstein–Uhlenbeck process

Gaussian model: Hpxq “ |x|2

2 , dXt “ ´Xtdt `dBt,

µ “ N p0,Idq,

Gfpxq “ ∆fpxq´xx,∇fpxqy

Mehler formula: LawpXt | X0 “ xq “ N pxe´t,p1´e´2tqIdq.

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Dynamics of a planar Coulomb gas Poincaré inequality

Exactly solvable model: Ornstein–Uhlenbeck process

Gaussian model: Hpxq “ |x|2

2 , dXt “ ´Xtdt `dBt,

µ “ N p0,Idq,

Gfpxq “ ∆fpxq´xx,∇fpxqy

Mehler formula: LawpXt | X0 “ xq “ N pxe´t,p1´e´2tqIdq. Hermite orthonormal polynomials pPnqně0

GPn “ ´nPn and Gp¨q “ ´

8

ÿ

n“0

nx¨,PnyPn

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Dynamics of a planar Coulomb gas Poincaré inequality

Exactly solvable model: Ornstein–Uhlenbeck process

Gaussian model: Hpxq “ |x|2

2 , dXt “ ´Xtdt `dBt,

µ “ N p0,Idq,

Gfpxq “ ∆fpxq´xx,∇fpxqy

Mehler formula: LawpXt | X0 “ xq “ N pxe´t,p1´e´2tqIdq. Hermite orthonormal polynomials pPnqně0

GPn “ ´nPn and Gp¨q “ ´

8

ÿ

n“0

nx¨,PnyPn

Spectral decomposition and spectral gap

pt “

8

ÿ

n“0

e´ntxp0,PnyPn.

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Dynamics of a planar Coulomb gas Poincaré inequality

Exactly solvable model: Ornstein–Uhlenbeck process

Gaussian model: Hpxq “ |x|2

2 , dXt “ ´Xtdt `dBt,

µ “ N p0,Idq,

Gfpxq “ ∆fpxq´xx,∇fpxqy

Mehler formula: LawpXt | X0 “ xq “ N pxe´t,p1´e´2tqIdq. Hermite orthonormal polynomials pPnqně0

GPn “ ´nPn and Gp¨q “ ´

8

ÿ

n“0

nx¨,PnyPn

Spectral decomposition and spectral gap

pt “

8

ÿ

n“0

e´ntxp0,PnyPn. Optimal Poincaré inequality, equality achieved for f “ P1 Varµpfq ď Eµp|∇f|2q.

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Dynamics of a planar Coulomb gas Poincaré inequality

Comparison to Gaussianity via convexity

Theorem (Brascamp–Lieb 1976)

If µpdxq “ e´Hpxq

Z

dx, ∇2H ą 0 on Rd, then for any smooth f : Rd Ñ R, Varµpfq ď Eµxp∇2Hq´1∇f,∇fy

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Dynamics of a planar Coulomb gas Poincaré inequality

Comparison to Gaussianity via convexity

Theorem (Brascamp–Lieb 1976)

If µpdxq “ e´Hpxq

Z

dx, ∇2H ą 0 on Rd, then for any smooth f : Rd Ñ R, Varµpfq ď Eµxp∇2Hq´1∇f,∇fy Proof by induction on dimension d

8/26

Dynamics of a planar Coulomb gas Poincaré inequality

Comparison to Gaussianity via convexity

Theorem (Brascamp–Lieb 1976)

If µpdxq “ e´Hpxq

Z

dx, ∇2H ą 0 on Rd, then for any smooth f : Rd Ñ R, Varµpfq ď Eµxp∇2Hq´1∇f,∇fy Proof by induction on dimension d Ornstein–Uhlenbeck: ∇2H “ Id

8/26

Dynamics of a planar Coulomb gas Poincaré inequality

Comparison to Gaussianity via convexity

Theorem (Brascamp–Lieb 1976)

If µpdxq “ e´Hpxq

Z

dx, ∇2H ą 0 on Rd, then for any smooth f : Rd Ñ R, Varµpfq ď Eµxp∇2Hq´1∇f,∇fy Proof by induction on dimension d Ornstein–Uhlenbeck: ∇2H “ Id Convexity: ∇2H ě ρId ą 0 gives Poincaré with 1{ρ

8/26

Dynamics of a planar Coulomb gas Poincaré inequality

Comparison to Gaussianity via convexity

Theorem (Brascamp–Lieb 1976)

If µpdxq “ e´Hpxq

Z

dx, ∇2H ą 0 on Rd, then for any smooth f : Rd Ñ R, Varµpfq ď Eµxp∇2Hq´1∇f,∇fy Proof by induction on dimension d Ornstein–Uhlenbeck: ∇2H “ Id Convexity: ∇2H ě ρId ą 0 gives Poincaré with 1{ρ H convex means that µpdxq “ e´Hdx is log-concave

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Dynamics of a planar Coulomb gas Poincaré inequality

Comparison to Gaussianity via convexity

Theorem (Brascamp–Lieb 1976)

If µpdxq “ e´Hpxq

Z

dx, ∇2H ą 0 on Rd, then for any smooth f : Rd Ñ R, Varµpfq ď Eµxp∇2Hq´1∇f,∇fy Proof by induction on dimension d Ornstein–Uhlenbeck: ∇2H “ Id Convexity: ∇2H ě ρId ą 0 gives Poincaré with 1{ρ H convex means that µpdxq “ e´Hdx is log-concave Jensen divergence: Varµpfq “ EµΦpfq´ΦpEµfq, Φpuq “ u2

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Dynamics of a planar Coulomb gas Poincaré inequality

Comparison to Gaussianity via convexity

Theorem (Bakry–Émery 1984)

If µpdxq “ e´Hpxq

Z

dx, ∇2H ě ρId ą 0, then for any convex Φ : I Ñ R

with pu,vq ÞÑ Φ2puqv2 convex and any smooth f : Rd Ñ I

EµΦpfq´ΦpEµfq ď EµpΦ2pfq|∇f|2q ρ

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Dynamics of a planar Coulomb gas Poincaré inequality

Comparison to Gaussianity via convexity

Theorem (Bakry–Émery 1984)

If µpdxq “ e´Hpxq

Z

dx, ∇2H ě ρId ą 0, then for any convex Φ : I Ñ R

with pu,vq ÞÑ Φ2puqv2 convex and any smooth f : Rd Ñ I

EµΦpfq´ΦpEµfq ď EµpΦ2pfq|∇f|2q ρ Proof by semigroup interpolation ept´sqGpΦpesGfqq, etGf “ EfpXtq

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Dynamics of a planar Coulomb gas Poincaré inequality

Comparison to Gaussianity via convexity

Theorem (Bakry–Émery 1984)

If µpdxq “ e´Hpxq

Z

dx, ∇2H ě ρId ą 0, then for any convex Φ : I Ñ R

with pu,vq ÞÑ Φ2puqv2 convex and any smooth f : Rd Ñ I

EµΦpfq´ΦpEµfq ď EµpΦ2pfq|∇f|2q ρ Proof by semigroup interpolation ept´sqGpΦpesGfqq, etGf “ EfpXtq Ornstein–Uhlenbeck: ∇2H “ Id

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Dynamics of a planar Coulomb gas Poincaré inequality

Comparison to Gaussianity via convexity

Theorem (Bakry–Émery 1984)

If µpdxq “ e´Hpxq

Z

dx, ∇2H ě ρId ą 0, then for any convex Φ : I Ñ R

with pu,vq ÞÑ Φ2puqv2 convex and any smooth f : Rd Ñ I

EµΦpfq´ΦpEµfq ď EµpΦ2pfq|∇f|2q ρ Proof by semigroup interpolation ept´sqGpΦpesGfqq, etGf “ EfpXtq Ornstein–Uhlenbeck: ∇2H “ Id Poincaré: I “ R, Φpuq “ u2

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Dynamics of a planar Coulomb gas Poincaré inequality

Comparison to Gaussianity via convexity

Theorem (Bakry–Émery 1984)

If µpdxq “ e´Hpxq

Z

dx, ∇2H ě ρId ą 0, then for any convex Φ : I Ñ R

with pu,vq ÞÑ Φ2puqv2 convex and any smooth f : Rd Ñ I

EµΦpfq´ΦpEµfq ď EµpΦ2pfq|∇f|2q ρ Proof by semigroup interpolation ept´sqGpΦpesGfqq, etGf “ EfpXtq Ornstein–Uhlenbeck: ∇2H “ Id Poincaré: I “ R, Φpuq “ u2 Beckner: I “ R`, Φpuq “ up, 1 ă p ď 2

9/26

Dynamics of a planar Coulomb gas Poincaré inequality

Comparison to Gaussianity via convexity

Theorem (Bakry–Émery 1984)

If µpdxq “ e´Hpxq

Z

dx, ∇2H ě ρId ą 0, then for any convex Φ : I Ñ R

with pu,vq ÞÑ Φ2puqv2 convex and any smooth f : Rd Ñ I

EµΦpfq´ΦpEµfq ď EµpΦ2pfq|∇f|2q ρ Proof by semigroup interpolation ept´sqGpΦpesGfqq, etGf “ EfpXtq Ornstein–Uhlenbeck: ∇2H “ Id Poincaré: I “ R, Φpuq “ u2 Beckner: I “ R`, Φpuq “ up, 1 ă p ď 2 Logarithmic Sobolev: I “ R`, Φpuq “ u logpuq

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Dynamics of a planar Coulomb gas Poincaré inequality

Comparison to Gaussianity via convexity

Theorem (Caffarelli 2000)

If µpdxq “ e´Hpxq

Z

dx, ∇2H ě ρId ą 0, then µ is the image of N p0,Idq

by a Lipschitz function F : Rd Ñ Rd with }F}Lip ď

1

?ρ .

10/26

Dynamics of a planar Coulomb gas Poincaré inequality

Comparison to Gaussianity via convexity

Theorem (Caffarelli 2000)

If µpdxq “ e´Hpxq

Z

dx, ∇2H ě ρId ą 0, then µ is the image of N p0,Idq

by a Lipschitz function F : Rd Ñ Rd with }F}Lip ď

1

?ρ .

Proof by Monge–Ampère equation: f “ detpDFqg

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Dynamics of a planar Coulomb gas Poincaré inequality

Comparison to Gaussianity via convexity

Theorem (Caffarelli 2000)

If µpdxq “ e´Hpxq

Z

dx, ∇2H ě ρId ą 0, then µ is the image of N p0,Idq

by a Lipschitz function F : Rd Ñ Rd with }F}Lip ď

1

?ρ .

Proof by Monge–Ampère equation: f “ detpDFqg Gives Poincaré from the Gaussian by transportation: Varµpfq “ VarN p0,IdqpfpFqq ď EN p0,Idqp|∇fpFqq|2q ď Eµp|∇f|2q ρ

10/26

Dynamics of a planar Coulomb gas Poincaré inequality

Comparison to Gaussianity via convexity

Theorem (Caffarelli 2000)

If µpdxq “ e´Hpxq

Z

dx, ∇2H ě ρId ą 0, then µ is the image of N p0,Idq

by a Lipschitz function F : Rd Ñ Rd with }F}Lip ď

1

?ρ .

Proof by Monge–Ampère equation: f “ detpDFqg Gives Poincaré from the Gaussian by transportation: Varµpfq “ VarN p0,IdqpfpFqq ď EN p0,Idqp|∇fpFqq|2q ď Eµp|∇f|2q ρ Gives also any Φ-Sobolev inequality from the Gaussian!

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Dynamics of a planar Coulomb gas Poincaré inequality

KLS conjecture

Conjecture (Kannan–Lovász–Simonovits 1995)

There exists a universal constant C ą 0 such that for any dimension d ě 1 and any smooth H : Rd Ñ R with ∇2H ě 0 and Cov “ Id,

µpdxq “ e´Hpxq

Z

dx satisfies to a Poincaré inequality with constant C. . . .

11/26

Dynamics of a planar Coulomb gas Poincaré inequality

KLS conjecture

Conjecture (Kannan–Lovász–Simonovits 1995)

There exists a universal constant C ą 0 such that for any dimension d ě 1 and any smooth H : Rd Ñ R with ∇2H ě 0 and Cov “ Id,

µpdxq “ e´Hpxq

Z

dx satisfies to a Poincaré inequality with constant C. . . . KLS/Bobkov true with d1{2

11/26

Dynamics of a planar Coulomb gas Poincaré inequality

KLS conjecture

Conjecture (Kannan–Lovász–Simonovits 1995)

There exists a universal constant C ą 0 such that for any dimension d ě 1 and any smooth H : Rd Ñ R with ∇2H ě 0 and Cov “ Id,

µpdxq “ e´Hpxq

Z

dx satisfies to a Poincaré inequality with constant C. . . . KLS/Bobkov true with d1{2 . . . , Bourgain, . . . , Klartag, . . . , Eldan, . . .

11/26

Dynamics of a planar Coulomb gas Poincaré inequality

KLS conjecture

Conjecture (Kannan–Lovász–Simonovits 1995)

There exists a universal constant C ą 0 such that for any dimension d ě 1 and any smooth H : Rd Ñ R with ∇2H ě 0 and Cov “ Id,

µpdxq “ e´Hpxq

Z

dx satisfies to a Poincaré inequality with constant C. . . . KLS/Bobkov true with d1{2 . . . , Bourgain, . . . , Klartag, . . . , Eldan, . . . Lee–Vempala 2016: true with d1{4

11/26

Dynamics of a planar Coulomb gas Poincaré inequality

KLS conjecture

Conjecture (Kannan–Lovász–Simonovits 1995)

There exists a universal constant C ą 0 such that for any dimension d ě 1 and any smooth H : Rd Ñ R with ∇2H ě 0 and Cov “ Id,

µpdxq “ e´Hpxq

Z

dx satisfies to a Poincaré inequality with constant C. . . . KLS/Bobkov true with d1{2 . . . , Bourgain, . . . , Klartag, . . . , Eldan, . . . Lee–Vempala 2016: true with d1{4 . . .

11/26

Dynamics of a planar Coulomb gas Dyson Process

Outline

Poincaré inequality Dyson Process Ginibre process

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Dynamics of a planar Coulomb gas Dyson Process

Gaussian Hermitian Random Matrices

Hermnˆn ” Rn`2 n2´n

2

“ Rn2 “ Rd

13/26

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Hermitian Random Matrices

Hermnˆn ” Rn`2 n2´n

2

“ Rn2 “ Rd Boltzmann–Gibbs measure µpdMq “ e´ n

2 TrpM2q

Z

dM

13/26

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Hermitian Random Matrices

Hermnˆn ” Rn`2 n2´n

2

“ Rn2 “ Rd Boltzmann–Gibbs measure µpdMq “ e´ n

2 TrpM2q

Z

dM Stochastic Differential Equation à la Ornstein–Uhlenbeck dMt “ ´nMtdt `dBt.

13/26

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Hermitian Random Matrices

Hermnˆn ” Rn`2 n2´n

2

“ Rn2 “ Rd Boltzmann–Gibbs measure µpdMq “ e´ n

2 TrpM2q

Z

dM Stochastic Differential Equation à la Ornstein–Uhlenbeck dMt “ ´αnnMtdt `?αndBt.

13/26

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Hermitian Random Matrices

Hermnˆn ” Rn`2 n2´n

2

“ Rn2 “ Rd Boltzmann–Gibbs measure µpdMq “ e´ n

2 TrpM2q

Z

dM Stochastic Differential Equation à la Ornstein–Uhlenbeck dMt “ ´Mtdt ` dBt ?

n.

13/26

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Hermitian Random Matrices

Hermnˆn ” Rn`2 n2´n

2

“ Rn2 “ Rd Boltzmann–Gibbs measure µpdMq “ e´ n

2 TrpM2q

Z

dM Stochastic Differential Equation à la Ornstein–Uhlenbeck dMt “ ´Mtdt ` dBt ?

n.

Change of variable: if specpMq “ tx1,...,xnu,

M “ UDU˚ with D “ diagpx1,...,xnq

13/26

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Hermitian Random Matrices

Hermnˆn ” Rn`2 n2´n

2

“ Rn2 “ Rd Boltzmann–Gibbs measure µpdMq “ e´ n

2 TrpM2q

Z

dM Stochastic Differential Equation à la Ornstein–Uhlenbeck dMt “ ´Mtdt ` dBt ?

n.

Change of variable: if specpMq “ tx1,...,xnu,

M “ UDU˚ with D “ diagpx1,...,xnq

Stochastic process of spectrum?

13/26

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Unitary Ensemble and Dyson Process

State space

D “ tpx1,...,xnq P Rn : x1 ă ¨¨¨ ă xnu

14/26

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Unitary Ensemble and Dyson Process

State space

D “ tpx1,...,xnq P Rn : x1 ă ¨¨¨ ă xnu

Boltzmann–Gibbs measure via change of variable µpdxq “ e´ n

2

řn

i“1 x2 i ś

iăjpxj ´ xiq2

Z

dx

14/26

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Unitary Ensemble and Dyson Process

State space

D “ tpx1,...,xnq P Rn : x1 ă ¨¨¨ ă xnu

Boltzmann–Gibbs measure via change of variable µpdxq “ e

´ n

2

řn

i“1 x2 i ´2ř iăj log 1 xj ´xi

Z

dx

14/26

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Unitary Ensemble and Dyson Process

State space

D “ tpx1,...,xnq P Rn : x1 ă ¨¨¨ ă xnu

Boltzmann–Gibbs measure via change of variable µpdxq “ e

´ n

2

řn

i“1 x2 i ´2ř iăj log 1 xj ´xi

Z

dx Dyson Ornstein–Uhlenbeck process via Itô formula dX i

t “ ´

ˆ

X i

t ` 2

n

ÿ

iăj

1 X j

t ´ X i t

˙ dt` dBi

t

?

n

14/26

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Unitary Ensemble and Dyson Process

State space

D “ tpx1,...,xnq P Rn : x1 ă ¨¨¨ ă xnu

Boltzmann–Gibbs measure via change of variable µpdxq “ e

´ n

2

řn

i“1 x2 i ´2ř iăj log 1 xj ´xi

Z

dx Dyson Ornstein–Uhlenbeck process via Itô formula dX i

t “ ´

ˆ

X i

t ` 2

n

ÿ

iăj

1 X j

t ´ X i t

˙ dt` dBi

t

?

n

Well-posedness: . . . , Rogers–Shi, . . .

14/26

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Unitary Ensemble and Dyson Process

State space

D “ tpx1,...,xnq P Rn : x1 ă ¨¨¨ ă xnu

Boltzmann–Gibbs measure via change of variable µpdxq “ e

´ n

2

řn

i“1 x2 i ´2ř iăj log 1 xj ´xi

Z

dx Dyson Ornstein–Uhlenbeck process via Itô formula dX i

t “ ´

ˆ

X i

t ` 2

n

ÿ

iăj

1 X j

t ´ X i t

˙ dt` dBi

t

?

n

Well-posedness: . . . , Rogers–Shi, . . . Poincaré and log-Sobolev: . . . , Erd˝

• s–Yau et al, . . .

14/26

Dynamics of a planar Coulomb gas Dyson Process

James Dyson (1947 –)

15/26

Dynamics of a planar Coulomb gas Dyson Process

Freeman Dyson (1923 –)

15/26

Dynamics of a planar Coulomb gas Dyson Process

Freeman Dyson (1923 –)

Freeman Dyson

A Brownian-motion model for the eigenvalues of a random matrix Journal of Mathematical Physics 3 (1962) 1191–1198.

15/26

Dynamics of a planar Coulomb gas Dyson Process

Freeman Dyson (1923 –)

Freeman Dyson

A Brownian-motion model for the eigenvalues of a random matrix Journal of Mathematical Physics 3 (1962) 1191–1198.

Greg Anderson & Alice Guionnet & Ofer Zeitouni

An introduction to random matrices (CUP 2009)

15/26

Dynamics of a planar Coulomb gas Dyson Process

Freeman Dyson (1923 –)

Freeman Dyson

A Brownian-motion model for the eigenvalues of a random matrix Journal of Mathematical Physics 3 (1962) 1191–1198.

Greg Anderson & Alice Guionnet & Ofer Zeitouni

An introduction to random matrices (CUP 2009)

László Erd˝

• s & Horng-Tzer Yau

Dynamical Approach To Random Matrix Theory (AMS 2017)

15/26

Dynamics of a planar Coulomb gas Dyson Process

Optimal Poincaré constant (mind the gap!)

Boltzmann–Gibbs measure µpdxq “ e´Hpxq

Z

dx

with Hpxq “ n 2

n

ÿ

i“1

x2

i ` 2

ÿ

iăj

log

1 xj ´ xi

16/26

Dynamics of a planar Coulomb gas Dyson Process

Optimal Poincaré constant (mind the gap!)

Boltzmann–Gibbs measure µpdxq “ e´Hpxq

Z

dx

with Hpxq “ n 2

n

ÿ

i“1

x2

i ` 2

ÿ

iăj

log

1 xj ´ xi

Log-concavity ∇2Hpxq ě n.

16/26

Dynamics of a planar Coulomb gas Dyson Process

Optimal Poincaré constant (mind the gap!)

Boltzmann–Gibbs measure µpdxq “ e´Hpxq

Z

dx

with Hpxq “ n 2

n

ÿ

i“1

x2

i ` 2

ÿ

iăj

log

1 xj ´ xi

Log-concavity ∇2Hpxq ě n. Brascamp–Lieb or Bakry–Émery or Caffarelli Varµpfq ď Eµp|∇f|2q

n

.

16/26

Dynamics of a planar Coulomb gas Dyson Process

Optimal Poincaré constant (mind the gap!)

Boltzmann–Gibbs measure µpdxq “ e´Hpxq

Z

dx

with Hpxq “ n 2

n

ÿ

i“1

x2

i ` 2

ÿ

iăj

log

1 xj ´ xi

Log-concavity ∇2Hpxq ě n. Brascamp–Lieb or Bakry–Émery or Caffarelli Varµpfq ď Eµp|∇f|2q

n

. Equality achieved for fpxq “ x1 `¨¨¨` xn (compute traces)

16/26

Dynamics of a planar Coulomb gas Dyson Process

Optimal Poincaré constant (mind the gap!)

Boltzmann–Gibbs measure µpdxq “ e´Hpxq

Z

dx

with Hpxq “ n 2

n

ÿ

i“1

x2

i ` 2

ÿ

iăj

log

1 xj ´ xi

Log-concavity ∇2Hpxq ě n. Brascamp–Lieb or Bakry–Émery or Caffarelli Varµpfq ď Eµp|∇f|2q

n

. Equality achieved for fpxq “ x1 `¨¨¨` xn (compute traces) Lipschitz deformation of Gaussian (Hoffman–Wielandt)

16/26

Dynamics of a planar Coulomb gas Ginibre process

Outline

Poincaré inequality Dyson Process Ginibre process

17/26

Dynamics of a planar Coulomb gas Ginibre process

Ginibre process

Boltzmann–Gibbs measure on MatnˆnpCq µpMq “ e´nTrpMM˚q

Z

dM

18/26

Dynamics of a planar Coulomb gas Ginibre process

Ginibre process

Boltzmann–Gibbs measure on MatnˆnpCq µpMq “ e´nTrpMM˚q

Z

dM Schur unitary decomposition: if tx1,...,xnu “ specpMq,

M “ UTU˚ with T “ D ` N and D “ diagpx1,...,xnq.

18/26

Dynamics of a planar Coulomb gas Ginibre process

Ginibre process

Boltzmann–Gibbs measure on MatnˆnpCq µpMq “ e´nTrpMM˚q

Z

dM Schur unitary decomposition: if tx1,...,xnu “ specpMq,

M “ UTU˚ with T “ D ` N and D “ diagpx1,...,xnq.

Lack of normality is generic: µptN “ 0uq “ 0

18/26

Dynamics of a planar Coulomb gas Ginibre process

Ginibre process

Boltzmann–Gibbs measure on MatnˆnpCq µpMq “ e´nTrpMM˚q

Z

dM Schur unitary decomposition: if tx1,...,xnu “ specpMq,

M “ UTU˚ with T “ D ` N and D “ diagpx1,...,xnq.

Lack of normality is generic: µptN “ 0uq “ 0 Process on spectrum melts N and D (Ñ Bourgade–Dubach)

18/26

Dynamics of a planar Coulomb gas Ginibre process

Ginibre process

Boltzmann–Gibbs measure on MatnˆnpCq µpMq “ e´nTrpMM˚q

Z

dM Schur unitary decomposition: if tx1,...,xnu “ specpMq,

M “ UTU˚ with T “ D ` N and D “ diagpx1,...,xnq.

Lack of normality is generic: µptN “ 0uq “ 0 Process on spectrum melts N and D (Ñ Bourgade–Dubach) How about an O.-U. like diffusion leaving invariant µ?

18/26

Dynamics of a planar Coulomb gas Ginibre process

State space

D “ CnzYi‰j tpx1,...,xnq P Cn : xi “ xju

19/26

Dynamics of a planar Coulomb gas Ginibre process

State space

D “ CnzYi‰j tpx1,...,xnq P Cn : xi “ xju

Boltzmann–Gibbs measure via change of variable µpdxq “ e´nřn

i“1 |xi|2

Z

ź

iăj

|xi ´ xj|2dx

19/26

Dynamics of a planar Coulomb gas Ginibre process

State space

D “ CnzYi‰j tpx1,...,xnq P Cn : xi “ xju

Boltzmann–Gibbs measure via change of variable µpdxq “ e

´nřn

i“1 |xi|2´2ř iăj log 1

|xi ´xj |

Z

dx

19/26

Dynamics of a planar Coulomb gas Ginibre process

State space

D “ CnzYi‰j tpx1,...,xnq P Cn : xi “ xju

Boltzmann–Gibbs measure via change of variable µpdxq “ e

´nřn

i“1 |xi|2´2ř iăj log 1

|xi ´xj |

Z

dx Ginibre process on Cn “ pR2qn dX i

t “ ´2X i t dt ´ 2

n

ÿ

i‰j

X j

t ´ X i t

|X i

t ´ X j t |2 dt ` dBi t

?

n.

19/26

Dynamics of a planar Coulomb gas Ginibre process

State space

D “ CnzYi‰j tpx1,...,xnq P Cn : xi “ xju

Boltzmann–Gibbs measure via change of variable µpdxq “ e

´βn ´

1 n

řn

i“1 |xi|2` 1 n2

ř

iăj log 1

|xi ´xj |

¯

Z

dx Ginibre process on Cn “ pR2qn dX i

t “ ´2αn

n X i

t dt ´ 2αn

n

ÿ

j‰i

X j

t ´ X i t

|X i

t ´ X j t |2 dt `

cαn βn dBi

t.

19/26

Dynamics of a planar Coulomb gas Ginibre process

State space

D “ CnzYi‰j tpx1,...,xnq P Cn : xi “ xju

Boltzmann–Gibbs measure via change of variable µpdxq “ e

´βn ´

1 n

řn

i“1 |xi|2` 1 n2

ř

iăj log 1

|xi ´xj |

¯

Z

dx Ginibre process on Cn “ pR2qn dX i

t “ ´2αn

n X i

t dt ´ 2αn

n

ÿ

j‰i

X j

t ´ X i t

|X i

t ´ X j t |2 dt `

cαn βn dBi

t.

RMT: βn “ n2

19/26

Dynamics of a planar Coulomb gas Ginibre process

State space

D “ CnzYi‰j tpx1,...,xnq P Cn : xi “ xju

Boltzmann–Gibbs measure via change of variable µpdxq “ e

´βn ´

1 n

řn

i“1 |xi|2` 1 n2

ř

iăj log 1

|xi ´xj |

¯

Z

dx Ginibre process on Cn “ pR2qn dX i

t “ ´2αn

n X i

t dt ´ 2αn

n

ÿ

j‰i

X j

t ´ X i t

|X i

t ´ X j t |2 dt `

cαn βn dBi

t.

RMT: βn “ n2 No convexity / Brascamp–Lieb / Bakry–Émery / Caffarelli

19/26

Dynamics of a planar Coulomb gas Ginibre process

State space

D “ CnzYi‰j tpx1,...,xnq P Cn : xi “ xju

Boltzmann–Gibbs measure via change of variable µpdxq “ e

´βn ´

1 n

řn

i“1 |xi|2` 1 n2

ř

iăj log 1

|xi ´xj |

¯

Z

dx Ginibre process on Cn “ pR2qn dX i

t “ ´2αn

n X i

t dt ´ 2αn

n

ÿ

j‰i

X j

t ´ X i t

|X i

t ´ X j t |2 dt `

cαn βn dBi

t.

RMT: βn “ n2 No convexity / Brascamp–Lieb / Bakry–Émery / Caffarelli No Hoffman–Wielandt for non-normal matrices

19/26

Dynamics of a planar Coulomb gas Ginibre process

Well posedness

Explosion time

TBD “ lim

RÑ8TR

where TR “ inf tt ě 0 : HpXtq ą Ru

20/26

Dynamics of a planar Coulomb gas Ginibre process

Well posedness

Explosion time

TBD “ lim

RÑ8TR

where TR “ inftt ě 0 : distpXt,BDq ď 1{Ru

20/26

Dynamics of a planar Coulomb gas Ginibre process

Well posedness

Explosion time

TBD “ lim

RÑ8TR

where TR “ inf

"

t ě 0 : max

i

|X i

t | ě R or min i‰j |X i t ´ X j t | ď 1{R

*

20/26

Dynamics of a planar Coulomb gas Ginibre process

Well posedness

Theorem (Well-posedness)

For all X0 “ x P D, n ě 2, βn ą 0, we have PpTBD “ `8q “ 1.

20/26

Dynamics of a planar Coulomb gas Ginibre process

Well posedness

Theorem (Well-posedness)

For all X0 “ x P D, n ě 2, βn ą 0, we have PpTBD “ `8q “ 1.

No constraint on β in contrast with Rogers–Shi for Dyson O.–U. !

20/26

Dynamics of a planar Coulomb gas Ginibre process

Well posedness

Theorem (Well-posedness)

For all X0 “ x P D, n ě 2, βn ą 0, we have PpTBD “ `8q “ 1.

No constraint on β in contrast with Rogers–Shi for Dyson O.–U. ! Positivity and coercivity inf

xPD Hpxq ą 0

and

lim

xÑBD Hpxq “ `8

20/26

Dynamics of a planar Coulomb gas Ginibre process

Well posedness

Theorem (Well-posedness)

For all X0 “ x P D, n ě 2, βn ą 0, we have PpTBD “ `8q “ 1.

No constraint on β in contrast with Rogers–Shi for Dyson O.–U. ! Positivity and coercivity inf

xPD Hpxq ą 0

and

lim

xÑBD Hpxq “ `8

Cutoff

Wpxq “ r Wpxq on |x| ă R with r W smooth

20/26

Dynamics of a planar Coulomb gas Ginibre process

Well posedness

Theorem (Well-posedness)

For all X0 “ x P D, n ě 2, βn ą 0, we have PpTBD “ `8q “ 1.

No constraint on β in contrast with Rogers–Shi for Dyson O.–U. ! Positivity and coercivity inf

xPD Hpxq ą 0

and

lim

xÑBD Hpxq “ `8

Cutoff

Wpxq “ r Wpxq on |x| ă R with r W smooth

Itô formula ExpHpXt^Tqq´ Hpxq “ Ex ˆż t^T

GHpXsqds

˙ .

20/26

Dynamics of a planar Coulomb gas Ginibre process

Well posedness

Theorem (Well-posedness)

For all X0 “ x P D, n ě 2, βn ą 0, we have PpTBD “ `8q “ 1.

No constraint on β in contrast with Rogers–Shi for Dyson O.–U. ! Positivity and coercivity inf

xPD Hpxq ą 0

and

lim

xÑBD Hpxq “ `8

Cutoff

Wpxq “ r Wpxq on |x| ă R with r W smooth

Itô formula ExpHpXt^Tqq´ Hpxq “ Ex ˆż t^T

GHpXsqds

˙ . R1TRďt ď HpXt^TRq and GH ď cn on D

20/26

Dynamics of a planar Coulomb gas Ginibre process

Poincaré inequality

Theorem (Poincaré inequality)

For any n, the law µn satisfies a Poincaré inequality.

21/26

Dynamics of a planar Coulomb gas Ginibre process

Poincaré inequality

Theorem (Poincaré inequality)

For any n, the law µn satisfies a Poincaré inequality.

Proof using Lyapunov criterion

21/26

Dynamics of a planar Coulomb gas Ginibre process

Poincaré inequality

Theorem (Poincaré inequality)

For any n, the law µn satisfies a Poincaré inequality.

Proof using Lyapunov criterion Bakry–Barthe–Cattiaux–Guillin Lyapunov approach

Hpxq “ 1 n

n

ÿ

i“1

|xi|2 ` 1

n2

ÿ

j‰i

log

1

|xi ´ xj|

Gf “ αn

βn ∆f ´αn∇H ¨∇f

21/26

Dynamics of a planar Coulomb gas Ginibre process

Poincaré inequality

Theorem (Poincaré inequality)

For any n, the law µn satisfies a Poincaré inequality.

Proof using Lyapunov criterion Bakry–Barthe–Cattiaux–Guillin Lyapunov approach

Hpxq “ 1 n

n

ÿ

i“1

|xi|2 ` 1

n2

ÿ

j‰i

log

1

|xi ´ xj|

Gf “ αn

βn ∆f ´αn∇H ¨∇f

GΨ ď ´cΨ` c11K

21/26

Dynamics of a planar Coulomb gas Ginibre process

Poincaré inequality

Theorem (Poincaré inequality)

For any n, the law µn satisfies a Poincaré inequality.

Proof using Lyapunov criterion Bakry–Barthe–Cattiaux–Guillin Lyapunov approach

Hpxq “ 1 n

n

ÿ

i“1

|xi|2 ` 1

n2

ÿ

j‰i

log

1

|xi ´ xj|

Gf “ αn

βn ∆f ´αn∇H ¨∇f

GΨ ď ´cΨ` c11K

Ψ “ eγH

21/26

Dynamics of a planar Coulomb gas Ginibre process

Poincaré inequality

Theorem (Poincaré inequality)

For any n, the law µn satisfies a Poincaré inequality.

Proof using Lyapunov criterion Bakry–Barthe–Cattiaux–Guillin Lyapunov approach

Hpxq “ 1 n

n

ÿ

i“1

|xi|2 ` 1

n2

ÿ

j‰i

log

1

|xi ´ xj|

Gf “ αn

βn ∆f ´αn∇H ¨∇f

GΨ ď ´cΨ` c11K

Ψ “ eγH

21/26

Dynamics of a planar Coulomb gas Ginibre process

Uniform Poincaré for the one particle marginal

Theorem (Uniform Poincaré for one-particle)

If βn “ n2 then the one-particle marginal of µ is log-concave and satisfies a Poincaré inequality with a constant uniform in n.

22/26

Dynamics of a planar Coulomb gas Ginibre process

Uniform Poincaré for the one particle marginal

Theorem (Uniform Poincaré for one-particle)

If βn “ n2 then the one-particle marginal of µ is log-concave and satisfies a Poincaré inequality with a constant uniform in n.

If βn “ n2 then one particle marginal of µ has density

z P C ÞÑ ϕpzq “ e´n|z|2

π

n´1

ÿ

ℓ“0

nℓ|z|2ℓ

ℓ! .

22/26

Dynamics of a planar Coulomb gas Ginibre process

Uniform Poincaré for the one particle marginal

Theorem (Uniform Poincaré for one-particle)

If βn “ n2 then the one-particle marginal of µ is log-concave and satisfies a Poincaré inequality with a constant uniform in n.

If βn “ n2 then one particle marginal of µ has density

z P C ÞÑ ϕpzq “ e´n|z|2

π

n´1

ÿ

ℓ“0

nℓ|z|2ℓ

ℓ! . Circular law lim

nÑ8sup zPK

ˇ ˇ ˇ ˇϕpzq´

1t|z|ď1u

π ˇ ˇ ˇ ˇ “ 0.

22/26

Dynamics of a planar Coulomb gas Ginibre process

Uniform Poincaré for the one particle marginal

Theorem (Uniform Poincaré for one-particle)

If βn “ n2 then the one-particle marginal of µ is log-concave and satisfies a Poincaré inequality with a constant uniform in n.

If βn “ n2 then one particle marginal of µ has density

z P C ÞÑ ϕpzq “ e´n|z|2

π

n´1

ÿ

ℓ“0

nℓ|z|2ℓ

ℓ! . Circular law lim

nÑ8sup zPK

ˇ ˇ ˇ ˇϕpzq´

1t|z|ď1u

π ˇ ˇ ˇ ˇ “ 0. The function z ÞÑ logřn´1

ℓ“0 |z|2ℓ ℓ!

is concave!

22/26

Dynamics of a planar Coulomb gas Ginibre process

Uniform Poincaré for the one particle marginal

Theorem (Uniform Poincaré for one-particle)

If βn “ n2 then the one-particle marginal of µ is log-concave and satisfies a Poincaré inequality with a constant uniform in n.

If βn “ n2 then one particle marginal of µ has density

z P C ÞÑ ϕpzq “ e´n|z|2

π

n´1

ÿ

ℓ“0

nℓ|z|2ℓ

ℓ! . Circular law lim

nÑ8sup zPK

ˇ ˇ ˇ ˇϕpzq´

1t|z|ď1u

π ˇ ˇ ˇ ˇ “ 0. The function z ÞÑ logřn´1

ℓ“0 |z|2ℓ ℓ!

is concave!

Second moment of ϕ bounded in n then KLS/Bobkov theorem

22/26

Dynamics of a planar Coulomb gas Ginibre process

Second moment dynamics

Theorem (Second moment dynamics) pRtqtě0 “ p |Xt|2

n qtě0 is an ergodic Cox–Ingersoll–Ross process:

dRt “ 4αn

n

„

n

βn ` n ´ 1

2n ´ Rt

 dt ` d

4αn nβn Rt dBt.

23/26

Dynamics of a planar Coulomb gas Ginibre process

Second moment dynamics

Theorem (Second moment dynamics) pRtqtě0 “ p |Xt|2

n qtě0 is an ergodic Cox–Ingersoll–Ross process:

dRt “ 4αn

n

„

n

βn ` n ´ 1

2n ´ Rt

 dt ` d

4αn nβn Rt dBt. In particular, with Γn “ Gammapn ` n´1

2n βn,βnq, for any t ě 0

W1pLawpRtq,Γnq ď e´4 αn

n t W1pLawpR0q,Γnq. 23/26

Dynamics of a planar Coulomb gas Ginibre process

Second moment dynamics

Theorem (Second moment dynamics) pRtqtě0 “ p |Xt|2

n qtě0 is an ergodic Cox–Ingersoll–Ross process:

dRt “ 4αn

n

„

n

βn ` n ´ 1

2n ´ Rt

 dt ` d

4αn nβn Rt dBt. Moreover for any x P D and t ě 0, we have

EpRt | R0 “ rq “ re´ 4αn

n t `

ˆ

1 2 ` n

βn ´ 1

2n

˙´

1´e´ 4αn

n t¯

.

23/26

Dynamics of a planar Coulomb gas Ginibre process

McKean–Vlasov mean-field limit

νn,t “ 1

n

n

ÿ

i“1

δX i

t 24/26

Dynamics of a planar Coulomb gas Ginibre process

McKean–Vlasov mean-field limit

νn,t “ 1

n

n

ÿ

i“1

δX i

t

dX i

t “ ´2αn

n X i

t dt ´ 2αn

n

ÿ

j‰i

X i

t ´ X j t

|X i

t ´ X j t |2 dt `

cαn βn dBi

t.

24/26

Dynamics of a planar Coulomb gas Ginibre process

McKean–Vlasov mean-field limit

Theorem? (Limiting McKean–Vlasov equation)

If σ “ limnÑ8 αn

βn P r0,8q then limnÑ8 νn,t “ νt with . . .

24/26

Dynamics of a planar Coulomb gas Ginibre process

McKean–Vlasov mean-field limit

Theorem? (Limiting McKean–Vlasov equation)

If σ “ limnÑ8 αn

βn P r0,8q then limnÑ8 νn,t “ νt with

d dt ż

R2fpxqνtpdxq “ σ

ż ∆fpxqνtpdxq´ 2 ż

R2x ¨∇fpxqνtpdxq

` ż

R4

px ´ yq¨p∇fpxq´∇fpyqq |x ´ y|2 νtpdxqνtpdyq

24/26

Dynamics of a planar Coulomb gas Ginibre process

McKean–Vlasov mean-field limit

Theorem? (Limiting McKean–Vlasov equation)

If σ “ limnÑ8 αn

βn P r0,8q then limnÑ8 νn,t “ νt with

Btνt “ σ∆νt `∇¨pp∇V `∇W ˚νtqνtq.

24/26

Dynamics of a planar Coulomb gas Ginibre process

McKean–Vlasov mean-field limit

Theorem? (Limiting McKean–Vlasov equation)

If σ “ limnÑ8 αn

βn P r0,8q then limnÑ8 νn,t “ νt with

Btνt “ σ∆νt `∇¨pp∇V `∇W ˚νtqνtq. Semi-linear Partial Differential Equation (Sznitman)

24/26

Dynamics of a planar Coulomb gas Ginibre process

McKean–Vlasov mean-field limit

Theorem? (Limiting McKean–Vlasov equation)

If σ “ limnÑ8 αn

βn P r0,8q then limnÑ8 νn,t “ νt with

Btνt “ σ∆νt `∇¨pp∇V `∇W ˚νtqνtq. Semi-linear Partial Differential Equation (Sznitman) Regimes: pαn,βnq “ pn,n2q and pαn,βnq “ pn,nq equation

24/26

Dynamics of a planar Coulomb gas Ginibre process

McKean–Vlasov mean-field limit

Theorem? (Limiting McKean–Vlasov equation)

If σ “ limnÑ8 αn

βn P r0,8q then limnÑ8 νn,t “ νt with

Btνt “ σ∆νt `∇¨pp∇V `∇W ˚νtqνtq. Semi-linear Partial Differential Equation (Sznitman) Regimes: pαn,βnq “ pn,n2q and pαn,βnq “ pn,nq equation Dyson O.-U. Cauchy–Stieltjes transform Ñ Burger’s equation

24/26

Dynamics of a planar Coulomb gas Ginibre process

McKean–Vlasov mean-field limit

Theorem? (Limiting McKean–Vlasov equation)

If σ “ limnÑ8 αn

βn P r0,8q then limnÑ8 νn,t “ νt with

Btνt “ σ∆νt `∇¨pp∇V `∇W ˚νtqνtq. Semi-linear Partial Differential Equation (Sznitman) Regimes: pαn,βnq “ pn,n2q and pαn,βnq “ pn,nq equation Dyson O.-U. Cauchy–Stieltjes transform Ñ Burger’s equation Dyson O.-U. . . . , Cépa–Lépingle, . . . , Cabanal–Guionnet,. . .

24/26

Dynamics of a planar Coulomb gas Ginibre process

McKean–Vlasov mean-field limit

Theorem? (Limiting McKean–Vlasov equation)

If σ “ limnÑ8 αn

βn P r0,8q then limnÑ8 νn,t “ νt with

Btνt “ σ∆νt `∇¨pp∇V `∇W ˚νtqνtq. Semi-linear Partial Differential Equation (Sznitman) Regimes: pαn,βnq “ pn,n2q and pαn,βnq “ pn,nq equation Dyson O.-U. Cauchy–Stieltjes transform Ñ Burger’s equation Dyson O.-U. . . . , Cépa–Lépingle, . . . , Cabanal–Guionnet,. . . Smooth interactions: . . . , Carrillo–McCann–Villani (2003), . . .

24/26

Dynamics of a planar Coulomb gas Ginibre process

McKean–Vlasov mean-field limit

Theorem? (Limiting McKean–Vlasov equation)

If σ “ limnÑ8 αn

βn P r0,8q then limnÑ8 νn,t “ νt with

Btνt “ σ∆νt `∇¨pp∇V `∇W ˚νtqνtq. Semi-linear Partial Differential Equation (Sznitman) Regimes: pαn,βnq “ pn,n2q and pαn,βnq “ pn,nq equation Dyson O.-U. Cauchy–Stieltjes transform Ñ Burger’s equation Dyson O.-U. . . . , Cépa–Lépingle, . . . , Cabanal–Guionnet,. . . Smooth interactions: . . . , Carrillo–McCann–Villani (2003), . . . Vortex model: . . . , Fournier–Hauray–Mischler (2014), . . .

24/26

Dynamics of a planar Coulomb gas Ginibre process

McKean–Vlasov mean-field limit

Theorem? (Limiting McKean–Vlasov equation)

If σ “ limnÑ8 αn

βn P r0,8q then limnÑ8 νn,t “ νt with

Btνt “ σ∆νt `∇¨pp∇V `∇W ˚νtqνtq. Semi-linear Partial Differential Equation (Sznitman) Regimes: pαn,βnq “ pn,n2q and pαn,βnq “ pn,nq equation Dyson O.-U. Cauchy–Stieltjes transform Ñ Burger’s equation Dyson O.-U. . . . , Cépa–Lépingle, . . . , Cabanal–Guionnet,. . . Smooth interactions: . . . , Carrillo–McCann–Villani (2003), . . . Vortex model: . . . , Fournier–Hauray–Mischler (2014), . . . Without noise and confinement: . . . , Duerinckx (2016),. . .

24/26

Dynamics of a planar Coulomb gas Ginibre process

Numerical analysis and stochastic simulation

Overdamped Langevin dynamics dXt “ ´α∇HpXtqdt ` cα β dBt

25/26

Dynamics of a planar Coulomb gas Ginibre process

Numerical analysis and stochastic simulation

Overdamped Langevin dynamics dXt “ ´α∇HpXtqdt ` cα β dBt Ergodic theorem limtÑ8

1 t

şt

0 δXsds “ e´βH

25/26

Dynamics of a planar Coulomb gas Ginibre process

Numerical analysis and stochastic simulation

Overdamped Langevin dynamics dXt “ ´α∇HpXtqdt ` cα β dBt Ergodic theorem limtÑ8

1 t

şt

0 δXsds “ e´βH

Ñ Metropolis Adjusted Langevin Algorithm (MALA)

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Dynamics of a planar Coulomb gas Ginibre process

Numerical analysis and stochastic simulation

Overdamped Langevin dynamics dXt “ ´α∇HpXtqdt ` cα β dBt Ergodic theorem limtÑ8

1 t

şt

0 δXsds “ e´βH

Ñ Metropolis Adjusted Langevin Algorithm (MALA) Underdamped Langevin dynamics # dXt “ ∇UpYtqdt dYt “ ´∇HpXtq´γ∇UpYtqdt ` b

γ β dBt.

25/26

Dynamics of a planar Coulomb gas Ginibre process

Numerical analysis and stochastic simulation

Overdamped Langevin dynamics dXt “ ´α∇HpXtqdt ` cα β dBt Ergodic theorem limtÑ8

1 t

şt

0 δXsds “ e´βH

Ñ Metropolis Adjusted Langevin Algorithm (MALA) Underdamped Langevin dynamics # dXt “ ∇UpYtqdt dYt “ ´∇HpXtq´γ∇UpYtqdt ` b

γ β dBt.

Ergodic theorem limtÑ8

1 t

şt

0 δpXs,Ysqds “ e´βH be´γU

25/26

Dynamics of a planar Coulomb gas Ginibre process

Numerical analysis and stochastic simulation

Overdamped Langevin dynamics dXt “ ´α∇HpXtqdt ` cα β dBt Ergodic theorem limtÑ8

1 t

şt

0 δXsds “ e´βH

Ñ Metropolis Adjusted Langevin Algorithm (MALA) Underdamped Langevin dynamics # dXt “ ∇UpYtqdt dYt “ ´∇HpXtq´γ∇UpYtqdt ` b

γ β dBt.

Ergodic theorem limtÑ8

1 t

şt

0 δpXs,Ysqds “ e´βH be´γU

Ñ Hamiltonian or Hybrid Monte Carlo (HMC, C.–Ferré–Stoltz)

25/26

Dynamics of a planar Coulomb gas Ginibre process

That’s all folks! Thank you for your attention.

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