[PPT] - Concentration for Coulomb gases and Coulomb transport inequalities PowerPoint Presentation

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Concentration for Coulomb gases

and Coulomb transport inequalities Djalil Chafaï1, Adrien Hardy2, Mylène Maïda2

1Université Paris-Dauphine, 2Université de Lille

Probability and Analysis B˛ edlewo – May 18, 2017

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Concentration for Coulomb gases

Motivation: concentration for Ginibre ensemble

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

plot(eig(randcg(500,500)/sqrt(500)))

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Concentration for Coulomb gases Electrostatics

Outline

Electrostatics Coulomb gas model Probability metrics and Coulomb transport inequality Concentration of measure for Coulomb gases

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SLIDE 4

Concentration for Coulomb gases Electrostatics

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.

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SLIDE 5

Concentration for Coulomb gases Electrostatics

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.

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Concentration for Coulomb gases Electrostatics

Coulomb energy and equilibrium measure

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

g(x − y)µ(dx)µ(dy) ∈ R∪{+∞}.

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SLIDE 7

Concentration for Coulomb gases Electrostatics

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).

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SLIDE 8

Concentration for Coulomb gases Electrostatics

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) µV = arginf EV

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SLIDE 9

Concentration for Coulomb gases Electrostatics

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) µV = arginf EV If V is strong then µV is compactly supported with density

1 2cd

∆V

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Concentration for Coulomb gases Electrostatics

Examples of equilibrium measures

d g V

µV

1 2

∞1intervalc(x)

arcsine 1 2 x2 semicircle 2 2

|x|2

uniform on a disc

3 d

x2

uniform on a ball

2 d radial radial in a ring

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Concentration for Coulomb gases Coulomb gas model

Outline

Electrostatics Coulomb gas model Probability metrics and Coulomb transport inequality Concentration of measure for Coulomb gases

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Concentration for Coulomb gases Coulomb gas model

Coulomb gas or one component plasma

Energy of N Coulomb charges in Rd:

HN(x1,...,xN) = N

N

∑

i=1

V(xi)+∑

i=j

g(xi − xj)

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Concentration for Coulomb gases Coulomb gas model

Coulomb gas or one component plasma

Energy of N Coulomb charges in Rd:

HN(x1,...,xN) = N2 V(x)µN(dx)+

x=y

g(x −y)µN(dx)µN(dy)

Empirical measure: µN = 1

N ∑N i=1 δxi

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Concentration for Coulomb gases Coulomb gas model

Coulomb gas or one component plasma

Energy of N Coulomb charges in Rd:

HN(x1,...,xN) = N2 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

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Concentration for Coulomb gases Coulomb gas model

Coulomb gas or one component plasma

Energy of N Coulomb charges in Rd:

HN(x1,...,xN) = N2 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

Boltzmann–Gibbs measure PN

V,β on (Rd)N:

exp

− β

2 HN(x1,...,xN)

ZV,β

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Concentration for Coulomb gases Coulomb gas model

Coulomb gas or one component plasma

Energy of N Coulomb charges in Rd:

HN(x1,...,xN) = N2 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

Boltzmann–Gibbs measure PN

V,β on (Rd)N:

exp

−β

2 N2E = V (µN)

Z N

V,β

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Concentration for Coulomb gases Coulomb gas model

Coulomb gas or one component plasma

Energy of N Coulomb charges in Rd:

HN(x1,...,xN) = N2 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

Boltzmann–Gibbs measure PN

V,β on (Rd)N:

exp

−β

2 N2E = V (µN)

Z N

V,β

V must be strong enough at infinity to ensure integrability

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Concentration for Coulomb gases Coulomb gas model

Coulomb gas or one component plasma

Energy of N Coulomb charges in Rd:

HN(x1,...,xN) = N2 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

Boltzmann–Gibbs measure PN

V,β on (Rd)N:

exp

−β

2 N2E = V (µN)

Z N

V,β

V must be strong enough at infinity to ensure integrability PN,β is neither product nor log-concave

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Concentration for Coulomb gases Coulomb gas model

Empirical measure and equilibrium measure

Random empirical measure under PN

V,β :

µN = 1

N

∑

i=1

δxi.

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Concentration for Coulomb gases Coulomb gas model

Empirical measure and equilibrium measure

Random empirical measure under PN

V,β :

µN = 1

N

∑

i=1

δxi. Under mild assumptions on V, with probability one, µN − →

N→∞ µV.

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Concentration for Coulomb gases Coulomb gas model

Empirical measure and equilibrium measure

Random empirical measure under PN

V,β :

µN = 1

N

∑

i=1

δxi. Under mild assumptions on V, with probability one, µN − →

N→∞ µV.

Large Deviation Principle (Gozlan-C.-Zitt 2014) logPN

V,β

dist(µN,µV) ≥ r
N2

− →

N→∞ −β

2 inf

dist(µ,µV )≥r

EV(µ)−EV(µV)
.

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Concentration for Coulomb gases Coulomb gas model

Empirical measure and equilibrium measure

Random empirical measure under PN

V,β :

µN = 1

N

∑

i=1

δxi. Under mild assumptions on V, with probability one, µN − →

N→∞ µV.

Large Deviation Principle (Gozlan-C.-Zitt 2014) logPN

V,β

dist(µN,µV) ≥ r
N2

− →

N→∞ −β

2 inf

dist(µ,µV )≥r

EV(µ)−EV(µV)
.

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

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

Outline

Electrostatics Coulomb gas model Probability metrics and Coulomb transport inequality Concentration of measure for Coulomb gases

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

Probability metrics and topologies

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

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

Probability metrics and topologies

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

E (µ −ν)

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

Probability metrics and topologies

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

E (µ −ν)

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

fLip1 f∞1

f(x)(µ −ν)(dx),

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

Probability metrics and topologies

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

E (µ −ν)

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

fLip1 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.

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

Probability metrics and topologies

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

E (µ −ν)

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

fLip1 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 .

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

Probability metrics and topologies

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

E (µ −ν)

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

fLip1 f∞1

f(x)(µ −ν)(dx),

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

fLip1

f(x)(µ −ν)(dx).

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

Probability metrics and topologies

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

E (µ −ν)

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

fLip1 f∞1

f(x)(µ −ν)(dx),

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

fLip1

f(x)(µ −ν)(dx).

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

Local Coulomb transport inequality

Theorem (Transport type inequality) W1(µ,ν)2 ≤ CD E (µ −ν).

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

Local Coulomb transport inequality

Theorem (Transport type inequality)

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

W1(µ,ν)2 ≤ CD E (µ −ν).

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

Local Coulomb transport inequality

Theorem (Transport type inequality)

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))

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

Local Coulomb transport inequality

Theorem (Transport type inequality)

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 Popescu local free transport inequality to any dim. d

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

Idea of proof of Coulomb transport inequality

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

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

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

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

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 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

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

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 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).

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SLIDE 39

Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

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 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).

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

Coulomb transport inequality for equilibrium measures

Theorem (Transport type inequality) dBL(µ,µV)2 ≤ CBL

EV(µ)−EV(µV)
.

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

Coulomb transport inequality for equilibrium measures

Theorem (Transport type inequality)

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

dBL(µ,µV)2 ≤ CBL

EV(µ)−EV(µV)
.

Moreover if V has at least quadratic growth then

W1(µ,µV)2 ≤ CW1

EV(µ)−EV(µV)
.

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

Coulomb transport inequality for equilibrium measures

Theorem (Transport type inequality)

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

dBL(µ,µV)2 ≤ CBL

EV(µ)−EV(µV)
.

Moreover if V has at least quadratic growth then

W1(µ,µV)2 ≤ CW1

EV(µ)−EV(µV)
.

Free transport inequalities (d = 2 and V = +∞ on Rc)

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

Coulomb transport inequality for equilibrium measures

Theorem (Transport type inequality)

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

dBL(µ,µV)2 ≤ CBL

EV(µ)−EV(µV)
.

Moreover if V has at least quadratic growth then

W1(µ,µV)2 ≤ CW1

EV(µ)−EV(µV)
.

Free transport inequalities (d = 2 and V = +∞ on Rc) Extends Maïda-Maurel-Segala & Popescu to any dimension d

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Concentration for Coulomb gases Probability metrics and Coulomb transport inequality

Coulomb transport inequality for equilibrium measures

Theorem (Transport type inequality)

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

dBL(µ,µV)2 ≤ CBL

EV(µ)−EV(µV)
.

Moreover if V has at least quadratic growth then

W1(µ,µV)2 ≤ CW1

EV(µ)−EV(µV)
.

Free transport inequalities (d = 2 and V = +∞ on Rc) Extends Maïda-Maurel-Segala & Popescu to any dimension d Growth condition is optimal for W1

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Concentration for Coulomb gases Concentration of measure for Coulomb gases

Outline

Electrostatics Coulomb gas model Probability metrics and Coulomb transport inequality Concentration of measure for Coulomb gases

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SLIDE 46

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Concentration of measure for Coulomb gases

Theorem (Concentration inequality)

If V does has reasonable growth then for every β,N,r

PN

V,β

dBL(µN,µV) ≥ r
e−aβN2r 2

.

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

LDP shows that order in N is optimal

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SLIDE 47

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Concentration of measure for Coulomb gases

Theorem (Concentration inequality)

If V does has reasonable growth then for every β,N,r

PN

V,β

dBL(µN,µV) ≥ r
e−aβN2r 2+1d=2( β

4 N logN)+bβN2−2/d+c(β)N.

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

LDP shows that order in N is optimal Explicit constants a,b,c if V is quadratic

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SLIDE 48

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Concentration of measure for Coulomb gases

Theorem (Concentration inequality)

If V does has reasonable growth then for every β,N,r

PN

V,β

dBL(µN,µV) ≥ r
e−aβN2r 2+1d=2( β

4 N logN)+bβN2−2/d+c(β)N.

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

LDP shows that order in N is optimal Explicit constants a,b,c if V is quadratic Implies Wasserstein convergence: PN

V,β

W1(µN,µV) ≥ r
e−cN2r2,

r

logN

N

if d = 2, N−1/d if d 3.

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Concentration for Coulomb gases Concentration of measure for Coulomb gases

Concentration of measure for Coulomb gases

Theorem (Concentration inequality)

If V does has reasonable growth then for every β,N,r

PN

V,β

dBL(µN,µV) ≥ r
e−aβN2r 2+1d=2( β

4 N logN)+bβN2−2/d+c(β)N.

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

LDP shows that order in N is optimal Explicit constants a,b,c if V is quadratic Implies Wasserstein convergence: PN

V,β

W1(µN,µV) ≥ r
e−cN2r2,

r

logN

N

if d = 2, N−1/d if d 3.

Idea of proof of concentration

Starting point PN

V,β(W1(µN,µV) r) =

1 Z N

V,β

W1(µN,µV )r e− β

2 N2E = V (µN)dx. 17/24

SLIDE 51

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Idea of proof of concentration

Starting point PN

V,β(W1(µN,µV) r) =

1 Z N

V,β

W1(µN,µV )r e− β

2 N2E = V (µN)dx.

Normalizing constant

1 Z N

V,β

exp

N2 β

2 EV(µV)− N

β

2 E (µV)−S(µV)

.

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SLIDE 52

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Idea of proof of concentration

Starting point PN

V,β(W1(µN,µV) r) =

1 Z N

V,β

W1(µN,µV )r e− β

2 N2E = V (µN)dx.

Normalizing constant

1 Z N

V,β

exp

N2 β

2 EV(µV)− N

β

2 E (µV)−S(µV)

.

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).

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SLIDE 53

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Idea of proof of concentration

Starting point PN

V,β(W1(µN,µV) r) =

1 Z N

V,β

W1(µN,µV )r e− β

2 N2E = V (µN)dx.

Normalizing constant

1 Z N

V,β

exp

N2 β

2 EV(µV)− N

β

2 E (µV)−S(µV)

.

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(µV) − 1 CW2 1(µ(ε) N ,µV).

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SLIDE 54

Concentration for Coulomb gases Concentration of measure for Coulomb gases

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

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Concentration for Coulomb gases Concentration of measure for Coulomb gases

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

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Concentration for Coulomb gases Concentration of measure for Coulomb gases

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 N logN+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 limN→∞ W1(µN,µ•) = 0 a.s.

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SLIDE 57

Concentration for Coulomb gases Concentration of measure for Coulomb gases

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 N logN+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 limN→∞ W1(µN,µ•) = 0 a.s. Open problem: Bernoulli ±1 random matrices (universality)

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SLIDE 58

Concentration for Coulomb gases Concentration of measure for Coulomb gases

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(randcg(500,500)/sqrt(500)))

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SLIDE 59

Concentration for Coulomb gases Concentration of measure for Coulomb gases

That’s all folks! Thank you for your attention Dzi˛ ekuj˛ e za uwag˛ e

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SLIDE 60

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Exponential tightness

Theorem (Tightness)

For any r ≥ r0

PN

V,β(supp(µN) ⊂ Br) = PN V,β

max

1≤i≤N |xi| ≥ r

≤ e−cNV∗(r),

where V∗(r) = min|x|r V(x).

Follows by using an argument by Borot and Guionnet

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SLIDE 61

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Exponential tightness

Theorem (Tightness)

For any r ≥ r0

PN

V,β(supp(µN) ⊂ Br) = PN V,β

max

1≤i≤N |xi| ≥ r

≤ e−cNV∗(r),

where V∗(r) = min|x|r V(x).

Follows by using an argument by Borot and Guionnet Gives that almost surely limN→∞ max1≤i≤N |xi| < ∞.

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SLIDE 62

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Exponential tightness

Theorem (Tightness)

For any r ≥ r0

PN

V,β(supp(µN) ⊂ Br) = PN V,β

max

1≤i≤N |xi| ≥ r

≤ e−cNV∗(r),

where V∗(r) = min|x|r V(x).

Follows by using an argument by Borot and Guionnet Gives that almost surely limN→∞ max1≤i≤N |xi| < ∞. Gives Wp versions of convergence and concentration Wp

p(µ,ν) (2M)p−1W1(µ,ν) M(2M)p−1dBL(µ,ν).

For p = 2 we get PN

V,β(W2(µN,µV) ≥ r) 2e−cN3/2r 2.

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SLIDE 63

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Convergence in Wasserstein distance

Corollary (Wasserstein convergence)

If V superquadratic and βN βV

logN

N

then under PN

V,βN a.s.

lim

N→∞W1(µN,µV) = 0.

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SLIDE 64

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Convergence at mesoscopic scale

Corollary (Mesoscopic convergence) If d = 2 then PN

V,β

dBL
τNs

x0 µN,τNs x0 µV

CNs
logN

N

e−cN logN,

Test functions are global, not local as in Rougerie-Serfaty

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SLIDE 65

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Convergence at mesoscopic scale

Corollary (Mesoscopic convergence) If d = 2 then PN

V,β

dBL
τNs

x0 µN,τNs x0 µV

CNs
logN

N

e−cN logN,

If d 3 then PN

V,β

dBL
τNs

x0 µN,τNs x0 µV

CNs−1/d

e−cN2−2/d. Test functions are global, not local as in Rougerie-Serfaty

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SLIDE 66

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Convergence at mesoscopic scale

Corollary (Mesoscopic convergence) If d = 2 then PN

V,β

dBL
τNs

x0 µN,τNs x0 µV

CNs
logN

N

e−cN logN,

If d 3 then PN

V,β

dBL
τNs

x0 µN,τNs x0 µV

CNs−1/d

e−cN2−2/d. If V superquadratic then dBL can be replaced by W1. Test functions are global, not local as in Rougerie-Serfaty

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SLIDE 67

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Notes and comments

Wp2 versions? Popescu free transport inequalities

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SLIDE 68

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Notes and comments

Wp2 versions? Popescu free transport inequalities Hardy-Littlewood-Sobolev inequalities (Keller-Segel PDE)

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SLIDE 69

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Notes and comments

Wp2 versions? Popescu free transport inequalities Hardy-Littlewood-Sobolev inequalities (Keller-Segel PDE) Classical transport inequalities with Coulomb distance

24/24

SLIDE 70

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Notes and comments

Wp2 versions? Popescu free transport inequalities Hardy-Littlewood-Sobolev inequalities (Keller-Segel PDE) Classical transport inequalities with Coulomb distance Varying V and conditional gases. PN

V,β(· | xN) = PN−1

VN,β with
VN =

N N − 1V + 2 N − 1g(xN −·) [covered by our work since g is superharmonic]

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SLIDE 71

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Notes and comments

Wp2 versions? Popescu free transport inequalities Hardy-Littlewood-Sobolev inequalities (Keller-Segel PDE) Classical transport inequalities with Coulomb distance Varying V and conditional gases. PN

V,β(· | xN) = PN−1

VN,β with
VN =

N N − 1V + 2 N − 1g(xN −·) [covered by our work since g is superharmonic]

Usage for CLT with GFF in all dimensions (RV, M+, LS, B+)

24/24

SLIDE 72

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Notes and comments

Wp2 versions? Popescu free transport inequalities Hardy-Littlewood-Sobolev inequalities (Keller-Segel PDE) Classical transport inequalities with Coulomb distance Varying V and conditional gases. PN

V,β(· | xN) = PN−1

VN,β with
VN =

N N − 1V + 2 N − 1g(xN −·) [covered by our work since g is superharmonic]

Usage for CLT with GFF in all dimensions (RV, M+, LS, B+) Weakly confining potentials and heavy-tailed µV

24/24

SLIDE 73

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Notes and comments

Wp2 versions? Popescu free transport inequalities Hardy-Littlewood-Sobolev inequalities (Keller-Segel PDE) Classical transport inequalities with Coulomb distance Varying V and conditional gases. PN

V,β(· | xN) = PN−1

VN,β with
VN =

N N − 1V + 2 N − 1g(xN −·) [covered by our work since g is superharmonic]

Usage for CLT with GFF in all dimensions (RV, M+, LS, B+) Weakly confining potentials and heavy-tailed µV Universality of concentration for random matrices

24/24

SLIDE 74

Concentration for Coulomb gases Concentration of measure for Coulomb gases

Notes and comments

Wp2 versions? Popescu free transport inequalities Hardy-Littlewood-Sobolev inequalities (Keller-Segel PDE) Classical transport inequalities with Coulomb distance Varying V and conditional gases. PN

V,β(· | xN) = PN−1

VN,β with
VN =

N N − 1V + 2 N − 1g(xN −·) [covered by our work since g is superharmonic]

Usage for CLT with GFF in all dimensions (RV, M+, LS, B+) Weakly confining potentials and heavy-tailed µV Universality of concentration for random matrices Crossover and Sanov regime (Allez-Bouchaud-Guionnet)

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