Dynamics of gases of particles with singular repulsion Djalil CHAFA - - PowerPoint PPT Presentation

Dynamics of gases of particles

Dynamics of gases of particles

with singular repulsion Djalil CHAFAÏ

Paris-Dauphine / PSL

Random Matrices and Related Topics KIAS, Seoul, Korea May 6-10, 2019

1/28

Dynamics of gases of particles Introduction

Outline

Introduction Coulomb gases Concentration of measure Dynamics for planar case

2/28

Dynamics of gases of particles Introduction

High dimensional phenomenon : random matrix spectrum

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

plot(eig(randn(n,n)+i*randn(n,n))/sqrt(2*n)))

3/28

Dynamics of gases of particles Introduction

High dimensional phenomenon : random matrix spectrum

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

plot(eig(randn(n,n)+i*randn(n,n))/sqrt(2*n)))

Law of large numbers: orthonormal rows/columns for large n

3/28

Dynamics of gases of particles Introduction

Ginibre ensemble in nature

Jean Ginibre (1938 – )

Statistical Ensembles of Complex, Quaternion, and Real Matrices Journal of Mathematical Physics (1965)

4/28

Dynamics of gases of particles Introduction

Ginibre ensemble in nature

Jean Ginibre (1938 – )

Statistical Ensembles of Complex, Quaternion, and Real Matrices Journal of Mathematical Physics (1965)

Robert May (1938 – )

Will a large complex system be stable? Nature (1972) Stability and Complexity in Model Ecosystems. Princeton Press (1973)

4/28

Dynamics of gases of particles Introduction

Ginibre ensemble in nature

Jean Ginibre (1938 – )

Statistical Ensembles of Complex, Quaternion, and Real Matrices Journal of Mathematical Physics (1965)

Robert May (1938 – )

Will a large complex system be stable? Nature (1972) Stability and Complexity in Model Ecosystems. Princeton Press (1973)

Robert B. Laughlin (1950 – )

Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations Physical Review Letters (1983)

4/28

Dynamics of gases of particles Introduction

Ginibre ensemble in nature

Jean Ginibre (1938 – )

Statistical Ensembles of Complex, Quaternion, and Real Matrices Journal of Mathematical Physics (1965)

Robert May (1938 – )

Will a large complex system be stable? Nature (1972) Stability and Complexity in Model Ecosystems. Princeton Press (1973)

Robert B. Laughlin (1950 – )

Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations Physical Review Letters (1983)

Dan-Virgil Voiculescu (1949 – )

Limit laws for random matrices and free products Inventiones Mathematicae (1991)

4/28

Dynamics of gases of particles Introduction

Ginibre ensemble in nature

Jean Ginibre (1938 – )

Statistical Ensembles of Complex, Quaternion, and Real Matrices Journal of Mathematical Physics (1965)

Robert May (1938 – )

Will a large complex system be stable? Nature (1972) Stability and Complexity in Model Ecosystems. Princeton Press (1973)

Robert B. Laughlin (1950 – )

Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations Physical Review Letters (1983)

Dan-Virgil Voiculescu (1949 – )

Limit laws for random matrices and free products Inventiones Mathematicae (1991)

Terence Tao (1975 – ), Sylvia Serfaty (1975 – ), Robert Berman (1976 – ), . . .

4/28

Dynamics of gases of particles Introduction

Gibbs measures: Ginibre versus Dyson

Non-Hermitian (Ginibre) 2D e−nTrace(MM∗) =

n

∏

j,k=1

e−n|Mjk|2

5/28

Dynamics of gases of particles Introduction

Gibbs measures: Ginibre versus Dyson

Non-Hermitian (Ginibre) 2D M = U(D + N)U∗ {λ ∈ Cn} e−nTrace(MM∗) =

n

∏

j,k=1

e−n|Mjk|2

5/28

Dynamics of gases of particles Introduction

Gibbs measures: Ginibre versus Dyson

Non-Hermitian (Ginibre) 2D M = U(D + N)U∗ {λ ∈ Cn} e−nTrace(MM∗) =

n

∏

j,k=1

e−n|Mjk|2

spectrum

e−n∑n

i=1 |λi|2∏

j<k

|λj −λk|2

5/28

Dynamics of gases of particles Introduction

Gibbs measures: Ginibre versus Dyson

Non-Hermitian (Ginibre) 2D M = U(D + N)U∗ {λ ∈ Cn} e−nTrace(MM∗) =

n

∏

j,k=1

e−n|Mjk|2

spectrum

e−n∑n

i=1 |λi|2∏

j<k

|λj −λk|2 Hermitian (Dyson) 1D e−nTrace(H2) =

n

∏

j,k=1

e−|Hjk|2

5/28

Dynamics of gases of particles Introduction

Gibbs measures: Ginibre versus Dyson

Non-Hermitian (Ginibre) 2D M = U(D + N)U∗ {λ ∈ Cn} e−nTrace(MM∗) =

n

∏

j,k=1

e−n|Mjk|2

spectrum

e−n∑n

i=1 |λi|2∏

j<k

|λj −λk|2 Hermitian (Dyson) 1D H = UDU∗ {λ ∈ Rn} e−nTrace(H2) =

n

∏

j,k=1

e−|Hjk|2

5/28

Dynamics of gases of particles Introduction

Gibbs measures: Ginibre versus Dyson

Non-Hermitian (Ginibre) 2D M = U(D + N)U∗ {λ ∈ Cn} e−nTrace(MM∗) =

n

∏

j,k=1

e−n|Mjk|2

spectrum

e−n∑n

i=1 |λi|2∏

j<k

|λj −λk|2 Hermitian (Dyson) 1D H = UDU∗ {λ ∈ Rn} e−nTrace(H2) =

n

∏

j,k=1

e−|Hjk|2

spectrum

e−n∑n

i=1 λ 2 i ∏

j<k

(λj −λk)2 Both are determinantal and exactly solvable

5/28

Dynamics of gases of particles Introduction

Gibbs measures: Ginibre versus Dyson

Non-Hermitian (Ginibre) 2D M = U(D + N)U∗ {λ ∈ Cn} e−nTrace(MM∗) =

n

∏

j,k=1

e−n|Mjk|2

spectrum

e

−

• n∑n

i=1 |λi|2+2∑j<k log 1

|λj −λk |

• Hermitian (Dyson) 1D H = UDU∗ {λ ∈ Rn : λ1 < ··· < λn}

e−nTrace(H2) =

n

∏

j,k=1

e−|Hjk|2

spectrum

e

−

• n∑n

i=1 λ 2 i +2∑j<k log 1

λk −λj

• Both are determinantal and exactly solvable

Both are Gibbs measures and Coulomb gases

5/28

Dynamics of gases of particles Introduction

Gibbs measures: Ginibre versus Dyson

Non-Hermitian (Ginibre) 2D M = U(D + N)U∗ {λ ∈ Cn} e−nTrace(MM∗) =

n

∏

j,k=1

e−n|Mjk|2

spectrum

e

−

• n∑n

i=1 |λi|2+2∑j<k log 1

|λj −λk |

• Hermitian (Dyson) 1D H = UDU∗ {λ ∈ Rn : λ1 < ··· < λn}

e−nTrace(H2) =

n

∏

j,k=1

e−|Hjk|2

spectrum

e

−

• n∑n

i=1 λ 2 i +2∑j<k log 1

λk −λj

• Both are determinantal and exactly solvable

Both are Gibbs measures and Coulomb gases Ginibre is not log-concave (contrary to Dyson)

5/28

Dynamics of gases of particles Coulomb gases

Outline

Introduction Coulomb gases Concentration of measure Dynamics for planar case

6/28

Dynamics of gases of particles Coulomb gases

Coulomb kernel in mathematical physics

Coulomb kernel in Rd, d ≥ 2,

x ∈ Rd → g(x) =

       log 1 |x|

if d = 2, 1

|x|d−2

if d ≥ 3.

7/28

Dynamics of gases of particles Coulomb gases

Coulomb kernel in mathematical physics

Coulomb kernel in Rd, d ≥ 2,

x ∈ Rd → g(x) =

       log 1 |x|

if d = 2, 1

|x|d−2

if d ≥ 3.

Fundamental solution of Poisson’s equation ∆g = −cd δ0

where cd =

• 2π

if d = 2,

(d − 2)|Sd−1|

if d ≥ 3.

7/28

Dynamics of gases of particles Coulomb gases

Coulomb kernel in mathematical physics

Coulomb kernel in Rd, d ≥ 2,

x ∈ Rd → g(x) =

       log 1 |x|

if d = 2, 1

|x|d−2

if d ≥ 3.

Fundamental solution of Poisson’s equation ∆g = −cd δ0

where cd =

• 2π

if d = 2,

(d − 2)|Sd−1|

if d ≥ 3.

For both Dyson and Ginibre: two dimensional repulsion

7/28

Dynamics of gases of particles Coulomb gases

Coulomb energy and equilibrium measure

Coulomb energy of probability measure µ on Rd: E (µ) =

• g(x − y)µ(dx)µ(dy) ∈ R∪{+∞}.

8/28

Dynamics of gases of particles Coulomb gases

Coulomb energy and equilibrium measure

Coulomb energy of probability measure µ on Rd: E (µ) =

• g(x − y)µ(dx)µ(dy) ∈ R∪{+∞}.

Coulomb energy with confining potential (external field) EV(µ) = E (µ)+

• V(x)µ(dx).

8/28

Dynamics of gases of particles Coulomb gases

Coulomb energy and equilibrium measure

Coulomb energy of probability measure µ on Rd: E (µ) =

• g(x − y)µ(dx)µ(dy) ∈ R∪{+∞}.

Coulomb energy with confining potential (external field) EV(µ) = E (µ)+

• V(x)µ(dx).

Equilibrium probability measure (electrostatics) µ∗ = arginf EV

8/28

Dynamics of gases of particles Coulomb gases

Coulomb energy and equilibrium measure

Coulomb energy of probability measure µ on Rd: E (µ) =

• g(x − y)µ(dx)µ(dy) ∈ R∪{+∞}.

Coulomb energy with confining potential (external field) EV(µ) = E (µ)+

• V(x)µ(dx).

Equilibrium probability measure (electrostatics) µ∗ = arginf EV If V is stronger than g at infinity then µ∗ is compactly supported

8/28

Dynamics of gases of particles Coulomb gases

Coulomb energy and equilibrium measure

Coulomb energy of probability measure µ on Rd: E (µ) =

• g(x − y)µ(dx)µ(dy) ∈ R∪{+∞}.

Coulomb energy with confining potential (external field) EV(µ) = E (µ)+

• V(x)µ(dx).

Equilibrium probability measure (electrostatics) µ∗ = arginf EV If V is stronger than g at infinity then µ∗ is compactly supported If V is smooth then µ∗ has density ∆V

2cd

8/28

Dynamics of gases of particles Coulomb gases

Examples of equilibrium measures

d Interaction g Confinement V Equilibrium µ∗ 1 2

∞1intervalc(x)

arcsine 1 2 x2 semicircle (Dyson) 2 2

|x|2

uniform on a disc (Ginibre)

≥ 3

d

x2

uniform on a ball

≥ 2

d radial radial in a ring

9/28

Dynamics of gases of particles Coulomb gases

Coulomb gas or one component plasma

Energy of n Coulomb charges 1

n at positions x1,...,xn in Rd:

H(x1,...,xn) = 1 n

n

∑

i=1

V(xi)+ 1 n2 ∑

i=j

g(xi − xj)

10/28

Dynamics of gases of particles Coulomb gases

Coulomb gas or one component plasma

Energy of n Coulomb charges 1

n at positions x1,...,xn in Rd:

H(x1,...,xn) =

• V(x)µn(dx)+
• x=y

g(x − y)µn(dx)µn(dy)

Empirical measure: µn = 1

n ∑n i=1 δxi

10/28

Dynamics of gases of particles Coulomb gases

Coulomb gas or one component plasma

Energy of n Coulomb charges 1

n at positions x1,...,xn in Rd:

H(x1,...,xn) =

• V(x)µn(dx)+
• x=y

g(x − y)µn(dx)µn(dy)

• E =

V (µn)

Empirical measure: µn = 1

n ∑n i=1 δxi

10/28

Dynamics of gases of particles Coulomb gases

Coulomb gas or one component plasma

Energy of n Coulomb charges 1

n at positions x1,...,xn in Rd:

H(x1,...,xn) =

• V(x)µn(dx)+
• x=y

g(x − y)µn(dx)µn(dy)

• E =

V (µn)

Empirical measure: µn = 1

n ∑n i=1 δxi

Gibbs measure on (Rd)n: P(dx) = exp

• −β n2

2 H(x1,...,xn)

• Z

dx.

10/28

Dynamics of gases of particles Coulomb gases

Coulomb gas or one component plasma

Energy of n Coulomb charges 1

n at positions x1,...,xn in Rd:

H(x1,...,xn) =

• V(x)µn(dx)+
• x=y

g(x − y)µn(dx)µn(dy)

• E =

V (µn)

Empirical measure: µn = 1

n ∑n i=1 δxi

Gibbs measure on (Rd)n: P(dx) = exp

• − β

2 n2E = V (µn)

• Z

dx.

10/28

Dynamics of gases of particles Coulomb gases

Coulomb gas or one component plasma

Energy of n Coulomb charges 1

n at positions x1,...,xn in Rd:

H(x1,...,xn) =

• V(x)µn(dx)+
• x=y

g(x − y)µn(dx)µn(dy)

• E =

V (µn)

Empirical measure: µn = 1

n ∑n i=1 δxi

Gibbs measure on (Rd)n: P(dx) = exp

• −βnE =

V (µn)

• Z

dx, βn = β

2 n2.

10/28

Dynamics of gases of particles Coulomb gases

Coulomb gas or one component plasma

Energy of n Coulomb charges 1

n at positions x1,...,xn in Rd:

H(x1,...,xn) =

• V(x)µn(dx)+
• x=y

g(x − y)µn(dx)µn(dy)

• E =

V (µn)

Empirical measure: µn = 1

n ∑n i=1 δxi

Gibbs measure on (Rd)n: P(dx) = exp

• −βnE =

V (µn)

• Z

dx, βn = β

2 n2.

∝ e−βn(V,µn+∆−1µn,µn) CLT with Gaussian Free Field

10/28

Dynamics of gases of particles Coulomb gases

Empirical measure and equilibrium measure

Random empirical measure under P: µn = 1

n

n

∑

i=1

δxi.

11/28

Dynamics of gases of particles Coulomb gases

Empirical measure and equilibrium measure

Random empirical measure under P: µn = 1

n

n

∑

i=1

δxi. If V is smooth and if βn = β

2 n2 ≫ n then with probability one

µn − →

n→∞ µ∗.

11/28

Dynamics of gases of particles Coulomb gases

Empirical measure and equilibrium measure

Random empirical measure under P: µn = 1

n

n

∑

i=1

δxi. If V is smooth and if βn = β

2 n2 ≫ n then with probability one

µn − →

n→∞ µ∗.

Laplace method Large Deviation Principle (Gozlan-C.-Zitt) logP

• dist(µn,µ∗) ≥ r
• n2

− →

n→∞ −β

2

inf

dist(µ,µ∗)≥r

• EV(µ)−EV(µ∗)
• .

11/28

Dynamics of gases of particles Coulomb gases

Empirical measure and equilibrium measure

Random empirical measure under P: µn = 1

n

n

∑

i=1

δxi. If V is smooth and if βn = β

2 n2 ≫ n then with probability one

µn − →

n→∞ µ∗.

Laplace method Large Deviation Principle (Gozlan-C.-Zitt) logP

• dist(µn,µ∗) ≥ r
• n2

− →

n→∞ −β

2

inf

dist(µ,µ∗)≥r

• EV(µ)−EV(µ∗)
• .

Quantitative estimates? How to relate dist and EV(·)−EV(µ∗)?

11/28

Dynamics of gases of particles Concentration of measure

Outline

Introduction Coulomb gases Concentration of measure Dynamics for planar case

12/28

Dynamics of gases of particles Concentration of measure

Probability metrics and topologies

Coulomb divergence (Large Deviations rate function) EV(µ)−EV(µ∗)

13/28

Dynamics of gases of particles Concentration of measure

Probability metrics and topologies

Coulomb divergence (Large Deviations rate function) EV(µ)−EV(µ∗) Coulomb metric

• E (µ −ν)

13/28

Dynamics of gases of particles Concentration of measure

Probability metrics and topologies

Coulomb divergence (Large Deviations rate function) EV(µ)−EV(µ∗) Coulomb metric

• E (µ −ν)

Bounded-Lipschitz or Fortet–Mourier distance dBL(µ,ν) = sup

fLip≤1 f∞≤1

• f(x)(µ −ν)(dx),

13/28

Dynamics of gases of particles Concentration of measure

Probability metrics and topologies

Coulomb divergence (Large Deviations rate function) EV(µ)−EV(µ∗) Coulomb metric

• E (µ −ν)

Bounded-Lipschitz or Fortet–Mourier distance dBL(µ,ν) = sup

fLip≤1 f∞≤1

• f(x)(µ −ν)(dx),

(Monge-Kantorovich-)Wasserstein distance Wp(µ,ν) = inf

(X,Y)

X∼µ,Y∼ν

E(|X − Y|p)1/p.

13/28

Dynamics of gases of particles Concentration of measure

Probability metrics and topologies

Coulomb divergence (Large Deviations rate function) EV(µ)−EV(µ∗) Coulomb metric

• E (µ −ν)

Bounded-Lipschitz or Fortet–Mourier distance dBL(µ,ν) = sup

fLip≤1 f∞≤1

• f(x)(µ −ν)(dx),

(Monge-Kantorovich-)Wasserstein distance Wp(µ,ν) =

• inf

π∈Π(µ,ν)

• |x − y|pπ(dx,dy)

1/p .

13/28

Dynamics of gases of particles Concentration of measure

Probability metrics and topologies

Coulomb divergence (Large Deviations rate function) EV(µ)−EV(µ∗) Coulomb metric

• E (µ −ν)

Bounded-Lipschitz or Fortet–Mourier distance dBL(µ,ν) = sup

fLip≤1 f∞≤1

• f(x)(µ −ν)(dx),

Kantorovich-Rubinstein duality W1(µ,ν) = sup

fLip≤1

• f(x)(µ −ν)(dx).

13/28

Dynamics of gases of particles Concentration of measure

Probability metrics and topologies

Coulomb divergence (Large Deviations rate function) EV(µ)−EV(µ∗) Coulomb metric

• E (µ −ν)

Bounded-Lipschitz or Fortet–Mourier distance dBL(µ,ν) = sup

fLip≤1 f∞≤1

• f(x)(µ −ν)(dx),

Kantorovich-Rubinstein duality dBL(µ,ν) ≤ W1(µ,ν) = sup

fLip≤1

• f(x)(µ −ν)(dx).

13/28

Dynamics of gases of particles Concentration of measure

Local Coulomb transport inequality

Theorem (Transport type inequality – C.-Hardy-Maïda) W1(µ,ν)2 ≤ CD E (µ −ν).

14/28

Dynamics of gases of particles Concentration of measure

Local Coulomb transport inequality

Theorem (Transport type inequality – C.-Hardy-Maïda)

D ⊂ Rd compact, supp(µ +ν) ⊂ D, E (µ) < ∞ and E (ν) < ∞,

W1(µ,ν)2 ≤ CD E (µ −ν).

14/28

Dynamics of gases of particles Concentration of measure

Local Coulomb transport inequality

Theorem (Transport type inequality – C.-Hardy-Maïda)

D ⊂ Rd compact, supp(µ +ν) ⊂ D, E (µ) < ∞ and E (ν) < ∞,

W1(µ,ν)2 ≤ CD E (µ −ν). Constant CD is ≈ Vol(B4Vol(D))

14/28

Dynamics of gases of particles Concentration of measure

Local Coulomb transport inequality

Theorem (Transport type inequality – C.-Hardy-Maïda)

D ⊂ Rd compact, supp(µ +ν) ⊂ D, E (µ) < ∞ and E (ν) < ∞,

W1(µ,ν)2 ≤ CD E (µ −ν). Constant CD is ≈ Vol(B4Vol(D)) Extends free transport inequality to any d

14/28

Dynamics of gases of particles Concentration of measure

Idea of proof of Coulomb transport inequality

Potential: if Uµ(x) = g ∗ µ(x) then ∆Uµ(x) = −cd µ

15/28

Dynamics of gases of particles Concentration of measure

Idea of proof of Coulomb transport inequality

Potential: if Uµ(x) = g ∗ µ(x) then ∆Uµ(x) = −cd µ Electric field: ∇Uµ(x). “Carré du champ”: |∇Uµ|2

15/28

Dynamics of gases of particles Concentration of measure

Idea of proof of Coulomb transport inequality

Potential: if Uµ(x) = g ∗ µ(x) then ∆Uµ(x) = −cd µ Integration by parts & Schwarz’s inequality in Rd and L2

cd

• f(x)(µ −ν)(dx) = −
• f(x)∆Uµ−ν(x)dx

≤ fLip

• |D|
• |∇Uµ−ν(x)|2dx

1/2

15/28

Dynamics of gases of particles Concentration of measure

Idea of proof of Coulomb transport inequality

Potential: if Uµ(x) = g ∗ µ(x) then ∆Uµ(x) = −cd µ Integration by parts & Schwarz’s inequality in Rd and L2

cd

• f(x)(µ −ν)(dx) = −
• f(x)∆Uµ−ν(x)dx

≤ fLip

• |D|
• |∇Uµ−ν(x)|2dx

1/2 Integration by parts again

• |∇Uµ−ν(x)|2dx = −
• Uµ−ν(x)∆Uµ−ν(x)dx

= cd E (µ −ν).

15/28

Dynamics of gases of particles Concentration of measure

Idea of proof of Coulomb transport inequality

Potential: if Uµ(x) = g ∗ µ(x) then ∆Uµ(x) = −cd µ Integration by parts & Schwarz’s inequality in Rd and L2

cd

• f(x)(µ −ν)(dx) = −
• f(x)∆Uµ−ν(x)dx

≤ fLip

• |D|
• |∇Uµ−ν(x)|2dx

1/2 Integration by parts again

• |∇Uµ−ν(x)|2dx = −
• Uµ−ν(x)∆Uµ−ν(x)dx

= cd E (µ −ν). Finally W1(µ,ν)2 ≤ |D|cdE (µ −ν).

15/28

Dynamics of gases of particles Concentration of measure

Coulomb transport inequality for equilibrium measures

Corollary (Transport type inequality – C.-Hardy-Maïda) dBL(µ,µ∗)2 ≤ CBL

• EV(µ)−EV(µ∗)
• .

16/28

Dynamics of gases of particles Concentration of measure

Coulomb transport inequality for equilibrium measures

Corollary (Transport type inequality – C.-Hardy-Maïda)

For any probability measure µ on Rd with E (µ) < ∞

dBL(µ,µ∗)2 ≤ CBL

• EV(µ)−EV(µ∗)
• .

Moreover if V has at least quadratic growth then

W1(µ,µ∗)2 ≤ CW1

• EV(µ)−EV(µ∗)
• .

16/28

Dynamics of gases of particles Concentration of measure

Coulomb transport inequality for equilibrium measures

Corollary (Transport type inequality – C.-Hardy-Maïda)

For any probability measure µ on Rd with E (µ) < ∞

dBL(µ,µ∗)2 ≤ CBL

• EV(µ)−EV(µ∗)
• .

Moreover if V has at least quadratic growth then

W1(µ,µ∗)2 ≤ CW1

• EV(µ)−EV(µ∗)
• .

Growth condition is optimal for W1

16/28

Dynamics of gases of particles Concentration of measure

Concentration of measure for Coulomb gases

Theorem (Concentration inequality – C.-Hardy-Maïda)

If V has reasonable growth then for every β,n,r

P

• dBL(µn,µ∗) ≥ r
• ≤ e−aβn2r2

.

Moreover if V has at least quadratic growth then W1 instead of dBL.

Optimal order in n

17/28

Dynamics of gases of particles Concentration of measure

Concentration of measure for Coulomb gases

Theorem (Concentration inequality – C.-Hardy-Maïda)

If V has reasonable growth then for every β,n,r

P

• dBL(µn,µ∗) ≥ r
• ≤ e−aβn2r2+1d=2( β

4 nlogn)+bβn2−2/d+c(β)n.

Moreover if V has at least quadratic growth then W1 instead of dBL.

Optimal order in n Explicit constants a,b,c if V is quadratic

17/28

Dynamics of gases of particles Concentration of measure

Concentration of measure for Coulomb gases

Theorem (Concentration inequality – C.-Hardy-Maïda)

If V has reasonable growth then for every β,n,r

P

• dBL(µn,µ∗) ≥ r
• ≤ e−aβn2r2+1d=2( β

4 nlogn)+bβn2−2/d+c(β)n.

Moreover if V has at least quadratic growth then W1 instead of dBL.

Optimal order in n Explicit constants a,b,c if V is quadratic Implies speed of convergence: P

• W1(µn,µ∗) ≥ r
• ≤ e−cn2r2,

r ≥

• logn

n

if d = 2, n−1/d if d ≥ 3.

17/28

Dynamics of gases of particles Concentration of measure

Concentration of measure for Coulomb gases

Theorem (Concentration inequality – C.-Hardy-Maïda)

If V has reasonable growth then for every β,n,r

P

• dBL(µn,µ∗) ≥ r
• ≤ e−aβn2r2+1d=2( β

4 nlogn)+bβn2−2/d+c(β)n.

Moreover if V has at least quadratic growth then W1 instead of dBL.

Optimal order in n Explicit constants a,b,c if V is quadratic Implies speed of convergence: P

• W1(µn,µ∗) ≥ r
• ≤ e−cn2r2,

r ≥

• logn

n

if d = 2, n−1/d if d ≥ 3.

See also Rougerie & Serfaty

17/28

Dynamics of gases of particles Concentration of measure

Idea of proof of concentration

Starting point P(W1(µn,µ∗) ≥ r) = 1

Z

• W1(µn,µ∗)≥r e− β

2 n2E = V (µn)dx. 18/28

Dynamics of gases of particles Concentration of measure

Idea of proof of concentration

Starting point P(W1(µn,µ∗) ≥ r) = 1

Z

• W1(µn,µ∗)≥r e− β

2 n2E = V (µn)dx.

Normalizing constant

1 Z ≤ exp

• n2 β

2 EV(µ∗)− n

β

2 E (µ∗)−S(µ∗)

• .

18/28

Dynamics of gases of particles Concentration of measure

Idea of proof of concentration

Starting point P(W1(µn,µ∗) ≥ r) = 1

Z

• W1(µn,µ∗)≥r e− β

2 n2E = V (µn)dx.

Normalizing constant

1 Z ≤ exp

• n2 β

2 EV(µ∗)− n

β

2 E (µ∗)−S(µ∗)

• .

Regularization: g superharmonic, µ(ε)

n

= µn ∗λε, −n2E =

V (µn) ≤ −n2EV(µ(ε) n )+ nE (λε)+ n n

∑

i=1

(V ∗λε − V)(xi).

18/28

Dynamics of gases of particles Concentration of measure

Idea of proof of concentration

Starting point P(W1(µn,µ∗) ≥ r) = 1

Z

• W1(µn,µ∗)≥r e− β

2 n2E = V (µn)dx.

Normalizing constant

1 Z ≤ exp

• n2 β

2 EV(µ∗)− n

β

2 E (µ∗)−S(µ∗)

• .

Regularization: g superharmonic, µ(ε)

n

= µn ∗λε, −n2E =

V (µn) ≤ −n2EV(µ(ε) n )+ nE (λε)+ n n

∑

i=1

(V ∗λε − V)(xi). Coulomb transport −EV(µ(ε)

n )+EV(µ∗) ≤ − 1 CW2 1(µ(ε) n ,µ∗).

18/28

Dynamics of gases of particles Concentration of measure

Concentration for spectrum of Ginibre matrices

Corollary (Ginibre Random Matrices)

If M is n × n with iid Gaussian entries of variance 1

n in C

Eigenvalues of M ∝ exp(−n∑n

i=1 |xi|2)∏i<j |xi − xj|2

19/28

Dynamics of gases of particles Concentration of measure

Concentration for spectrum of Ginibre matrices

Corollary (Ginibre Random Matrices)

If M is n × n with iid Gaussian entries of variance 1

n in C

Eigenvalues of M ∝ exp(−n∑n

i=1 |xi|2 −∑i=j g(xi − xj))

Here d = 2, β = 2, V = |·|2

19/28

Dynamics of gases of particles Concentration of measure

Concentration for spectrum of Ginibre matrices

Corollary (Ginibre Random Matrices)

If M is n × n with iid Gaussian entries of variance 1

n in C then

P

• W1(µn,µ•) ≥ r
• ≤ e− 1

4C n2r2+ 1 2 nlogn+n[ 1 C + 3 2 −logπ].

Eigenvalues of M ∝ exp(−n∑n

i=1 |xi|2 −∑i=j g(xi − xj))

Here d = 2, β = 2, V = |·|2 Provides a.s. W1(µn,µ•) = O

• log(n)

n

• .

19/28

Dynamics of gases of particles Concentration of measure

Concentration for spectrum of Ginibre matrices

Corollary (Ginibre Random Matrices)

If M is n × n with iid Gaussian entries of variance 1

n in C then

P

• W1(µn,µ•) ≥ r
• ≤ e− 1

4C n2r2+ 1 2 nlogn+n[ 1 C + 3 2 −logπ].

Eigenvalues of M ∝ exp(−n∑n

i=1 |xi|2 −∑i=j g(xi − xj))

Here d = 2, β = 2, V = |·|2 Provides a.s. W1(µn,µ•) = O

• log(n)

n

• .

Bernoulli ±1 random matrices (universality) µn → µ• (Tao-Vu 2010) but P(W1(µn,µ•) ≥ r) is not known

19/28

Dynamics of gases of particles Concentration of measure

Concentration for spectrum of Ginibre random matrices

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

plot(eig(rand(n,n)+i*randn(n,n))/sqrt(2*n)))

Dynamics leaving invariant in law this picture

20/28

Dynamics of gases of particles Dynamics for planar case

Outline

Introduction Coulomb gases Concentration of measure Dynamics for planar case

21/28

Dynamics of gases of particles Dynamics for planar case

Ginibre process

Ginibre energy H(x1,...,xn) = 1

n ∑n i=1 |xi|2 + 1 n2 ∑i=j log 1

|xi−xj|

22/28

Dynamics of gases of particles Dynamics for planar case

Ginibre process

Ginibre energy H(x1,...,xn) = 1

n ∑n i=1 |xi|2 + 1 n2 ∑i=j log 1

|xi−xj|

Ginibre process on D ⊂ Cn = (R2)n reversible for P ∝ e−βnH

22/28

Dynamics of gases of particles Dynamics for planar case

Ginibre process

Ginibre energy H(x1,...,xn) = 1

n ∑n i=1 |xi|2 + 1 n2 ∑i=j log 1

|xi−xj|

Ginibre process on D ⊂ Cn = (R2)n reversible for P ∝ e−βnH dX n

t =

• 2

βn dBn

t −∇H(X n t )dt.

22/28

Dynamics of gases of particles Dynamics for planar case

Ginibre process

Ginibre energy H(x1,...,xn) = 1

n ∑n i=1 |xi|2 + 1 n2 ∑i=j log 1

|xi−xj|

Ginibre process on D ⊂ Cn = (R2)n reversible for P ∝ e−βnH dX i,n

t

=

• 2

βn dBi,n

t

− 2

nX i,n

t dt − 2

n2 ∑

j=i

X j,n

t

− X i,n

t

|X i,n

t

− X j,n

t |2 dt.

22/28

Dynamics of gases of particles Dynamics for planar case

Ginibre process

Ginibre energy H(x1,...,xn) = 1

n ∑n i=1 |xi|2 + 1 n2 ∑i=j log 1

|xi−xj|

Ginibre process on D ⊂ Cn = (R2)n reversible for P ∝ e−βnH dX i,n

t

=

• 2αn

βn dBi,n

t

− 2αn

n X i,n

t dt − 2αn

n2 ∑

j=i

X j,n

t

− X i,n

t

|X i,n

t

− X j,n

t |2 dt.

22/28

Dynamics of gases of particles Dynamics for planar case

Ginibre process

Ginibre energy H(x1,...,xn) = 1

n ∑n i=1 |xi|2 + 1 n2 ∑i=j log 1

|xi−xj|

Ginibre process on D ⊂ Cn = (R2)n reversible for P ∝ e−βnH dX i,n

t

=

• 2αn

βn dBi,n

t

− 2αn

n X i,n

t dt − 2αn

n2 ∑

j=i

X j,n

t

− X i,n

t

|X i,n

t

− X j,n

t |2 dt.

Mean-field interacting particle system X n

t = (X i,n t )1≤i≤n

22/28

Dynamics of gases of particles Dynamics for planar case

Ginibre process

Ginibre energy H(x1,...,xn) = 1

n ∑n i=1 |xi|2 + 1 n2 ∑i=j log 1

|xi−xj|

Ginibre process on D ⊂ Cn = (R2)n reversible for P ∝ e−βnH dX i,n

t

=

• 2αn

βn dBi,n

t

− 2αn

n X i,n

t dt − 2αn

n2 ∑

j=i

X j,n

t

− X i,n

t

|X i,n

t

− X j,n

t |2 dt.

Mean-field interacting particle system X n

t = (X i,n t )1≤i≤n

2D: no convexity / Brascamp–Lieb / Bakry–Émery / Caffarelli

22/28

Dynamics of gases of particles Dynamics for planar case

Ginibre process

Ginibre energy H(x1,...,xn) = 1

n ∑n i=1 |xi|2 + 1 n2 ∑i=j log 1

|xi−xj|

Ginibre process on D ⊂ Cn = (R2)n reversible for P ∝ e−βnH dX i,n

t

=

• 2αn

βn dBi,n

t

− 2αn

n X i,n

t dt − 2αn

n2 ∑

j=i

X j,n

t

− X i,n

t

|X i,n

t

− X j,n

t |2 dt.

Mean-field interacting particle system X n

t = (X i,n t )1≤i≤n

2D: no convexity / Brascamp–Lieb / Bakry–Émery / Caffarelli 1D log-gases are log-concave:

22/28

Dynamics of gases of particles Dynamics for planar case

Ginibre process

Ginibre energy H(x1,...,xn) = 1

n ∑n i=1 |xi|2 + 1 n2 ∑i=j log 1

|xi−xj|

Ginibre process on D ⊂ Cn = (R2)n reversible for P ∝ e−βnH dX i,n

t

=

• 2αn

βn dBi,n

t

− 2αn

n X i,n

t dt − 2αn

n2 ∑

j=i

X j,n

t

− X i,n

t

|X i,n

t

− X j,n

t |2 dt.

Mean-field interacting particle system X n

t = (X i,n t )1≤i≤n

2D: no convexity / Brascamp–Lieb / Bakry–Émery / Caffarelli 1D log-gases are log-concave:

◮ Freeman Dyson – Journal of Mathematical Physics (1962)

A Brownian-motion model for the eigenvalues of a random matrix.

22/28

Dynamics of gases of particles Dynamics for planar case

Ginibre process

Ginibre energy H(x1,...,xn) = 1

n ∑n i=1 |xi|2 + 1 n2 ∑i=j log 1

|xi−xj|

Ginibre process on D ⊂ Cn = (R2)n reversible for P ∝ e−βnH dX i,n

t

=

• 2αn

βn dBi,n

t

− 2αn

n X i,n

t dt − 2αn

n2 ∑

j=i

X j,n

t

− X i,n

t

|X i,n

t

− X j,n

t |2 dt.

Mean-field interacting particle system X n

t = (X i,n t )1≤i≤n

2D: no convexity / Brascamp–Lieb / Bakry–Émery / Caffarelli 1D log-gases are log-concave:

◮ Freeman Dyson – Journal of Mathematical Physics (1962)

A Brownian-motion model for the eigenvalues of a random matrix.

◮ László Erd˝

• s & Horng-Tzer Yau – Courant Lecture Notes (2017)

Dynamical Approach To Random Matrix Theory

22/28

Dynamics of gases of particles Dynamics for planar case

Ginibre process

Ginibre energy H(x1,...,xn) = 1

n ∑n i=1 |xi|2 + 1 n2 ∑i=j log 1

|xi−xj|

Ginibre process on D ⊂ Cn = (R2)n reversible for P ∝ e−βnH dX i,n

t

=

• 2αn

βn dBi,n

t

− 2αn

n X i,n

t dt − 2αn

n2 ∑

j=i

X j,n

t

− X i,n

t

|X i,n

t

− X j,n

t |2 dt.

Mean-field interacting particle system X n

t = (X i,n t )1≤i≤n

2D: no convexity / Brascamp–Lieb / Bakry–Émery / Caffarelli 1D log-gases are log-concave:

◮ Freeman Dyson – Journal of Mathematical Physics (1962)

A Brownian-motion model for the eigenvalues of a random matrix.

◮ László Erd˝

• s & Horng-Tzer Yau – Courant Lecture Notes (2017)

Dynamical Approach To Random Matrix Theory

◮ Michel Lassalle – CRAS (1991)

Polynômes de Hermite généralisés

22/28

Dynamics of gases of particles Dynamics for planar case

Ginibre process

Ginibre energy H(x1,...,xn) = 1

n ∑n i=1 |xi|2 + 1 n2 ∑i=j log 1

|xi−xj|

Ginibre process on D ⊂ Cn = (R2)n reversible for P ∝ e−βnH dX i,n

t

=

• 2αn

βn dBi,n

t

− 2αn

n X i,n

t dt − 2αn

n2 ∑

j=i

X j,n

t

− X i,n

t

|X i,n

t

− X j,n

t |2 dt.

Mean-field interacting particle system X n

t = (X i,n t )1≤i≤n

2D: no convexity / Brascamp–Lieb / Bakry–Émery / Caffarelli 1D log-gases are log-concave:

◮ Freeman Dyson – Journal of Mathematical Physics (1962)

A Brownian-motion model for the eigenvalues of a random matrix.

◮ László Erd˝

• s & Horng-Tzer Yau – Courant Lecture Notes (2017)

Dynamical Approach To Random Matrix Theory

◮ Michel Lassalle – CRAS (1991)

Polynômes de Hermite généralisés

◮ Optimal Poincaré and log-Sobolev constants (C.-Lehec)

22/28

Dynamics of gases of particles Dynamics for planar case

Second moment dynamics

Theorem (Second moment dynamics – Bolley-C.-Fontbona) (Rt)t≥0 = ( 1

nXt2)t≥0 is an ergodic Cox–Ingersoll–Ross process:

dRt =

• 8αn

nβn Rt dBt + 4αn n

• n

βn + n − 1

2n − Rt

• dt.

23/28

Dynamics of gases of particles Dynamics for planar case

Second moment dynamics

Theorem (Second moment dynamics – Bolley-C.-Fontbona) (Rt)t≥0 = ( 1

nXt2)t≥0 is an ergodic Cox–Ingersoll–Ross process:

dRt =

• 8αn

nβn Rt dBt + 4αn n

• n

βn + n − 1

2n − Rt

• dt.

In particular, with γn = Gamma(n + n−1

2n βn,βn), for any t ≥ 0

W1(Law(Rt),γn) ≤ e−4 αn

n t W1(Law(R0),γn). 23/28

Dynamics of gases of particles Dynamics for planar case

Second moment dynamics

Theorem (Second moment dynamics – Bolley-C.-Fontbona) (Rt)t≥0 = ( 1

nXt2)t≥0 is an ergodic Cox–Ingersoll–Ross process:

dRt =

• 8αn

nβn Rt dBt + 4αn n

• n

βn + n − 1

2n − Rt

• dt.

Furthermore for any x ∈ D and t ≥ 0, we have

E(Rt | R0 = r) = re− 4αn

n t+

1

2 + n

βn − 1

2n

• 1−e− 4αn

n t

.

Eigenvectors: ∑N

i=1 ℜ(zi), ∑N i=1 ℑ(zi), ∑N i=1 |zi|2 + cN.

23/28

Dynamics of gases of particles Dynamics for planar case

McKean–Vlasov mean-field limit

Theorem? (MKV Mean-field limit – Bolley-C.-Fontbona)

If σ = limn→∞ αn

βn ∈ [0,∞) then limn→∞ µn

t = µt with

∂tµt = σ∆µt +∇·((∇V +∇g ∗ µt)µt). dX i,n

t

=

• 2αn

βn dBi,n

t

− 2αn

n X i,n t dt − 2αn n ∑j=i X i,n

t −X j,n t

|X j,n

t −X i,n t |2 dt. 24/28

Dynamics of gases of particles Dynamics for planar case

McKean–Vlasov mean-field limit

Theorem? (MKV Mean-field limit – Bolley-C.-Fontbona)

If σ = limn→∞ αn

βn ∈ [0,∞) then limn→∞ µn

t = µt with

∂tµt = σ∆µt +∇·((∇V +∇g ∗ µt)µt). dX i,n

t

=

• 2αn

βn dBi,n

t

− 2αn

n X i,n t dt − 2αn n ∑j=i X i,n

t −X j,n t

|X j,n

t −X i,n t |2 dt.

Empirical measure µn

t = 1

n

n

∑

i=1

δX i,n

t 24/28

Dynamics of gases of particles Dynamics for planar case

McKean–Vlasov mean-field limit

Theorem? (MKV Mean-field limit – Bolley-C.-Fontbona)

If σ = limn→∞ αn

βn ∈ [0,∞) then limn→∞ µn

t = µt with

∂tµt = σ∆µt +∇·((∇V +∇g ∗ µt)µt). dX i,n

t

=

• 2αn

βn dBi,n

t

− 2αn

n X i,n t dt − 2αn n ∑j=i X i,n

t −X j,n t

|X j,n

t −X i,n t |2 dt.

Empirical measure µn

t = 1

n

n

∑

i=1

δX i,n

t

Some sort of law of large numbers

24/28

Dynamics of gases of particles Dynamics for planar case

McKean–Vlasov mean-field limit

Theorem? (MKV Mean-field limit – Bolley-C.-Fontbona)

If σ = limn→∞ αn

βn ∈ [0,∞) then limn→∞ µn

t = µt with

∂tµt = σ∆µt +∇·((∇V +∇g ∗ µt)µt). dX i,n

t

=

• 2αn

βn dBi,n

t

− 2αn

n X i,n t dt − 2αn n ∑j=i X i,n

t −X j,n t

|X j,n

t −X i,n t |2 dt.

Empirical measure µn

t = 1

n

n

∑

i=1

δX i,n

t

Some sort of law of large numbers Regimes: (αn,βn) = (n,n2) and (αn,βn) = (n,n)

24/28

Dynamics of gases of particles Dynamics for planar case

McKean–Vlasov mean-field limit

Theorem? (MKV Mean-field limit – Bolley-C.-Fontbona)

If σ = limn→∞ αn

βn ∈ [0,∞) then limn→∞ µn

t = µt with

∂tµt = σ∆µt +∇·((∇V +∇g ∗ µt)µt). dX i,n

t

=

• 2αn

βn dBi,n

t

− 2αn

n X i,n t dt − 2αn n ∑j=i X i,n

t −X j,n t

|X j,n

t −X i,n t |2 dt.

Empirical measure µn

t = 1

n

n

∑

i=1

δX i,n

t

Some sort of law of large numbers Regimes: (αn,βn) = (n,n2) and (αn,βn) = (n,n) Carrillo–McCann–Villani, Fournier–Hauray–Mischler, Serfaty–Duerinckx, . . .

24/28

Dynamics of gases of particles Dynamics for planar case

Numerical analysis and stochastic simulation

Overdamped Langevin dynamics dXt = −α∇H(Xt)dt + α β dBt

25/28

Dynamics of gases of particles Dynamics for planar case

Numerical analysis and stochastic simulation

Overdamped Langevin dynamics dXt = −α∇H(Xt)dt + α β dBt Ergodic theorem lim

t→∞

1 t t

0 δXsds = e−βH

25/28

Dynamics of gases of particles Dynamics for planar case

Numerical analysis and stochastic simulation

Overdamped Langevin dynamics dXt = −α∇H(Xt)dt + α β dBt Ergodic theorem lim

t→∞

1 t t

0 δXsds = e−βH

→ Metropolis Adjusted Langevin Algorithm (MALA)

25/28

Dynamics of gases of particles Dynamics for planar case

Numerical analysis and stochastic simulation

Overdamped Langevin dynamics dXt = −α∇H(Xt)dt + α β dBt Ergodic theorem lim

t→∞

1 t t

0 δXsds = e−βH

→ Metropolis Adjusted Langevin Algorithm (MALA) Underdamped Langevin dynamics (adding momentum/inertia)

• dXt

= ∇U(Yt)dt dYt = −∇H(Xt)−γ∇U(Yt)dt +

• γ

β dBt.

25/28

Dynamics of gases of particles Dynamics for planar case

Numerical analysis and stochastic simulation

Overdamped Langevin dynamics dXt = −α∇H(Xt)dt + α β dBt Ergodic theorem lim

t→∞

1 t t

0 δXsds = e−βH

→ Metropolis Adjusted Langevin Algorithm (MALA) Underdamped Langevin dynamics (adding momentum/inertia)

• dXt

= ∇U(Yt)dt dYt = −∇H(Xt)−γ∇U(Yt)dt +

• γ

β dBt.

Ergodic theorem lim

t→∞

1 t t

0 δ(Xs,Ys)ds = e−βH ⊗e−γU

25/28

Dynamics of gases of particles Dynamics for planar case

Numerical analysis and stochastic simulation

Overdamped Langevin dynamics dXt = −α∇H(Xt)dt + α β dBt Ergodic theorem lim

t→∞

1 t t

0 δXsds = e−βH

→ Metropolis Adjusted Langevin Algorithm (MALA) Underdamped Langevin dynamics (adding momentum/inertia)

• dXt

= ∇U(Yt)dt dYt = −∇H(Xt)−γ∇U(Yt)dt +

• γ

β dBt.

Ergodic theorem lim

t→∞

1 t t

0 δ(Xs,Ys)ds = e−βH ⊗e−γU

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

25/28

Dynamics of gases of particles Dynamics for planar case

Numerical analysis and stochastic simulation

Overdamped Langevin dynamics dXt = −α∇H(Xt)dt + α β dBt Ergodic theorem lim

t→∞

1 t t

0 δXsds = e−βH

→ Metropolis Adjusted Langevin Algorithm (MALA) Underdamped Langevin dynamics (adding momentum/inertia)

• dXt

= ∇U(Yt)dt dYt = −∇H(Xt)−γ∇U(Yt)dt +

• γ

β dBt.

Ergodic theorem lim

t→∞

1 t t

0 δ(Xs,Ys)ds = e−βH ⊗e−γU

→ Hamiltonian or Hybrid Monte Carlo (HMC, C.–Ferré–Stoltz) → Geometric ergodicity via Lyapunov (Lu–Mattingly)

25/28

Dynamics of gases of particles Dynamics for planar case

Eight trajectories for a Dyson Ornstein-Uhlenbeck HMC

26/28

Dynamics of gases of particles Dynamics for planar case

Equilibrium measures

27/28

Dynamics of gases of particles Dynamics for planar case

Equilibrium measures

27/28

Dynamics of gases of particles Dynamics for planar case

Equilibrium measures

27/28

Dynamics of gases of particles Dynamics for planar case

Equilibrium measures

27/28

Dynamics of gases of particles Dynamics for planar case

Universal Gumbel fluctuation for edge of beta Ginibre?

• 4nγn
• |λ|max − 1−
• γn

4n

• law

− →

n→∞ Gumbel.

γn = log(n)− 2log(log(n))−log(2π).

28/28