Concentration for Coulomb gases

Transcription

1 1/32 and Coulomb transport inequalities Djalil Chafaï 1, Adrien Hardy 2, Mylène Maïda 2 1 Université Paris-Dauphine, 2 Université de Lille November 4, 2016 IHP Paris Groupe de travail MEGA

2 2/32 Motivation Outline Motivation Electrostatics Coulomb gas model Probability metrics and Coulomb transport inequality Concentration of measure for Coulomb gases

3 3/32 Motivation Concentration of measure P( F(Z ) EF(Z ) r) 2e 1 2 r 2 Sub-Gaussian concentration (Z Gaussian, F Lipschitz)

4 3/32 Motivation Concentration of measure f (N,r) P( Z N EZ N r) 2e Sub-Gaussian concentration (Z Gaussian, F Lipschitz) Dependence with respect to N

5 3/32 Motivation Concentration of measure f (N,r) P( Z N EZ N r) 2e Sub-Gaussian concentration (Z Gaussian, F Lipschitz) Dependence with respect to N Examples

6 3/32 Motivation Concentration of measure f (N,r) P( Z N EZ N r) 2e Sub-Gaussian concentration (Z Gaussian, F Lipschitz) Dependence with respect to N Examples Standard: Z N = X X N

7 3/32 Motivation Concentration of measure f (N,r) P( Z N EZ N r) 2e Sub-Gaussian concentration (Z Gaussian, F Lipschitz) Dependence with respect to N Examples Standard: Z N = X X N RMT: Z N = f (λ 1 (X)) + + f (λ N (X))

8 3/32 Motivation Concentration of measure f (N,r) P( Z N EZ N r) 2e Sub-Gaussian concentration (Z Gaussian, F Lipschitz) Dependence with respect to N Examples Standard: Z N = X X N RMT: Z N = f (λ 1 (X)) + + f (λ N (X)) TSP: Z N = inf σ ΣN N i=1 X σ(i+1) X σ(i)

9 3/32 Motivation Concentration of measure f (N,r) P( Z N EZ N r) 2e Sub-Gaussian concentration (Z Gaussian, F Lipschitz) Dependence with respect to N Examples Standard: Z N = X X N RMT: Z N = f (λ 1 (X)) + + f (λ N (X)) TSP: Z N = inf σ ΣN N i=1 X σ(i+1) X σ(i) High dimensional phenomena, combinatorial optimization

10 3/32 Motivation Concentration of measure f (N,r) P( Z N EZ N r) 2e Sub-Gaussian concentration (Z Gaussian, F Lipschitz) Dependence with respect to N Examples Standard: Z N = X X N RMT: Z N = f (λ 1 (X)) + + f (λ N (X)) TSP: Z N = inf σ ΣN N i=1 X σ(i+1) X σ(i) High dimensional phenomena, combinatorial optimization Talagrand principle

11 3/32 Motivation Concentration of measure f (N,r) P( Z N EZ N r) 2e Sub-Gaussian concentration (Z Gaussian, F Lipschitz) Dependence with respect to N Examples Standard: Z N = X X N RMT: Z N = f (λ 1 (X)) + + f (λ N (X)) TSP: Z N = inf σ ΣN N i=1 X σ(i+1) X σ(i) High dimensional phenomena, combinatorial optimization Talagrand principle Erdős complete convergence to deterministic object

12 3/32 Motivation Concentration of measure f (N,r) P( Z N EZ N r) 2e Sub-Gaussian concentration (Z Gaussian, F Lipschitz) Dependence with respect to N Examples Standard: Z N = X X N RMT: Z N = f (λ 1 (X)) + + f (λ N (X)) TSP: Z N = inf σ ΣN N i=1 X σ(i+1) X σ(i) High dimensional phenomena, combinatorial optimization Talagrand principle Erdős complete convergence to deterministic object Books: Steele, Ledoux, Boucheron-Lugosi-Massart

13 4/32 Motivation Gaussian exactly solvable model Ginibre G = (G jk ) 1 j,k N iid C Gaussian of variance 1 2N

14 4/32 Motivation Gaussian exactly solvable model Ginibre G = (G jk ) 1 j,k N iid C Gaussian of variance 1 2N The matrix G has density on C N2 e N N j,k=1 G jk 2 = e NTr(GG )

15 4/32 Motivation Gaussian exactly solvable model Ginibre G = (G jk ) 1 j,k N iid C Gaussian of variance 1 2N The matrix G has density on C N2 e N N j,k=1 G jk 2 = e NTr(GG ) Change of variable: G = UTU (U, T = D + N)

16 4/32 Motivation Gaussian exactly solvable model Ginibre G = (G jk ) 1 j,k N iid C Gaussian of variance 1 2N The matrix G has density on C N2 e N N j,k=1 G jk 2 = e NTr(GG ) Change of variable: G = UTU (U, T = D + N) Tr(GG ) = Tr(TT ) = Tr(DD ) + Tr(NN )

17 4/32 Motivation Gaussian exactly solvable model Ginibre G = (G jk ) 1 j,k N iid C Gaussian of variance 1 2N The matrix G has density on C N2 e N N j,k=1 G jk 2 = e NTr(GG ) Change of variable: G = UTU (U, T = D + N) Tr(GG ) = Tr(TT ) = Tr(DD ) + Tr(NN ) (λ 1 (G),..., λ N (G)) has density ( ) N ϕ N (z 1,..., z N ) exp N z r 2 z j z k 2. r=1 1 j<k N

18 4/32 Motivation Gaussian exactly solvable model Ginibre G = (G jk ) 1 j,k N iid C Gaussian of variance 1 2N The matrix G has density on C N2 e N N j,k=1 G jk 2 = e NTr(GG ) Change of variable: G = UTU (U, T = D + N) Tr(GG ) = Tr(TT ) = Tr(DD ) + Tr(NN ) (λ 1 (G),..., λ N (G)) has density ( ) N ϕ N (z 1,..., z N ) exp N z r 2 z j z k 2. r=1 Neither product nor log-concave 1 j<k N

19 4/32 Motivation Gaussian exactly solvable model Ginibre G = (G jk ) 1 j,k N iid C Gaussian of variance 1 2N The matrix G has density on C N2 e N N j,k=1 G jk 2 = e NTr(GG ) Change of variable: G = UTU (U, T = D + N) Tr(GG ) = Tr(TT ) = Tr(DD ) + Tr(NN ) (λ 1 (G),..., λ N (G)) has density ( ) N ϕ N (z 1,..., z N ) exp N z r 2 z j z k 2. r=1 1 j<k N Neither product nor log-concave Determinantal (Pemantle-Peres, Breuer-Duits)

20 5/32 Motivation First order global asymptotics Empirical spectral distribution µ G := 1 N N k=1 δ λ k (G)

21 5/32 Motivation First order global asymptotics Empirical spectral distribution µ G := 1 N N k=1 δ λ k (G) Mehta: density of mean empirical spectral distribution Eµ G : ϕ (1) N 1 e N z 2 N l z 2l N (z) = π l! l=0

22 5/32 Motivation First order global asymptotics Empirical spectral distribution µ G := 1 N N k=1 δ λ k (G) Mehta: density of mean empirical spectral distribution Eµ G : Mehta: mean circular law: ϕ (1) N 1 e N z 2 N l z 2l N (z) = π l! l=0 ϕ (1) N (z) N 1 [0,1] ( z ). π

23 5/32 Motivation First order global asymptotics Empirical spectral distribution µ G := 1 N N k=1 δ λ k (G) Mehta: density of mean empirical spectral distribution Eµ G : Mehta: mean circular law: ϕ (1) N 1 e N z 2 N l z 2l N (z) = π l! l=0 ϕ (1) N (z) N 1 [0,1] ( z ). π Silverstein: N E((µ Gf Eµ G f ) 4 ) < strong law: a.s. µ G w n µ.

24 5/32 Motivation First order global asymptotics Empirical spectral distribution µ G := 1 N N k=1 δ λ k (G) Mehta: density of mean empirical spectral distribution Eµ G : Mehta: mean circular law: ϕ (1) N 1 e N z 2 N l z 2l N (z) = π l! l=0 ϕ (1) N (z) N 1 [0,1] ( z ). π Silverstein: N E((µ Gf Eµ G f ) 4 ) < strong law: a.s. µ G w n µ. Open problem: concentration P(d(µ G, µ ) r)? e cn2 r 2

25 /32 Motivation Sub-Gaussian concentration of measure Gaussian Unitary Ensemble (GUE) H = (H jk ) 1 j,k N e NTr(H2 ) e N N k=1 λ2 k (λ j λ k ) 2 1 j<k N

26 /32 Motivation Sub-Gaussian concentration of measure Gaussian Unitary Ensemble (GUE) H = (H jk ) 1 j,k N e NTr(H2 ) e N N k=1 λ2 k (λ j λ k ) 2 1 j<k N Hoffman-Wielandt inequality for H, H Herm N N min σ Σ N N (λ k (H) λ σ(k) (H )) 2 k=1 N (H jk H jk )2. j,k=1

27 /32 Motivation Sub-Gaussian concentration of measure Gaussian Unitary Ensemble (GUE) H = (H jk ) 1 j,k N e NTr(H2 ) e N N k=1 λ2 k (λ j λ k ) 2 1 j<k N Hoffman-Wielandt inequality for H, H Herm N N NW 2 (µ H, µ H ) 2 H H 2 HS. Sub-Gaussian concentration inequality for GUE P( W 2 (µ H, Eµ H ) EW 2 (µ H, Eµ H ) r) 2e cn2 r 2.

28 /32 Motivation Sub-Gaussian concentration of measure Gaussian Unitary Ensemble (GUE) H = (H jk ) 1 j,k N e NTr(H2 ) e N N k=1 λ2 k (λ j λ k ) 2 1 j<k N Hoffman-Wielandt inequality for H, H Herm N N NW 2 (µ H, µ H ) 2 H H 2 HS. Sub-Gaussian concentration inequality for GUE P( W 2 (µ H, Eµ H ) EW 2 (µ H, Eµ H ) r) 2e cn2 r 2. Maïda-Maurel-Segala: P(W 1 (µ H, µ ) r) e cn2 r 2.

29 /32 Motivation Sub-Gaussian concentration of measure Open problem for Ginibre: P(W 1 (µ G, µ ) r) e cn2 r 2.

30 /32 Motivation Sub-Gaussian concentration of measure Open problem for Ginibre: P(W 1 (µ G, µ ) r) e cn2 r 2. Meckes and Meckes: E(W p (µ G, µ ))) by coupling

31 /32 Motivation Sub-Gaussian concentration of measure Open problem for Ginibre: P(W 1 (µ G, µ ) r) e cn2 r 2. Meckes and Meckes: E(W p (µ G, µ ))) by coupling Ortega-Cerda et al: E(W 1 (µ S, µ ))) by complex transport

32 /32 Motivation Sub-Gaussian concentration of measure Open problem for Ginibre: P(W 1 (µ G, µ ) r) e cn2 r 2. Meckes and Meckes: E(W p (µ G, µ ))) by coupling Ortega-Cerda et al: E(W 1 (µ S, µ ))) by complex transport Coulomb gas = Coulomb Boltzmann Gibbs measure e N N k=1 z k 2 1 j<k N z j z k 2

33 /32 Motivation Sub-Gaussian concentration of measure Open problem for Ginibre: P(W 1 (µ G, µ ) r) e cn2 r 2. Meckes and Meckes: E(W p (µ G, µ ))) by coupling Ortega-Cerda et al: E(W 1 (µ S, µ ))) by complex transport Coulomb gas = Coulomb Boltzmann Gibbs measure N exp N z k 2 1 log z j z k k=1 1 j k N Exchangeable but neither product nor log-concave

34 /32 Motivation Sub-Gaussian concentration of measure Open problem for Ginibre: P(W 1 (µ G, µ ) r) e cn2 r 2. Meckes and Meckes: E(W p (µ G, µ ))) by coupling Ortega-Cerda et al: E(W 1 (µ S, µ ))) by complex transport Coulomb gas = Coulomb Boltzmann Gibbs measure ( )) exp ( N 2 z 2 1 µ N (dz) + log z w µ N(dz)µ N (dw) Exchangeable but neither product nor log-concave Empirical measure µ N := 1 N N k=1 δ z k

35 /32 Motivation Sub-Gaussian concentration of measure Open problem for Ginibre: P(W 1 (µ G, µ ) r) e cn2 r 2. Meckes and Meckes: E(W p (µ G, µ ))) by coupling Ortega-Cerda et al: E(W 1 (µ S, µ ))) by complex transport Coulomb gas = Coulomb Boltzmann Gibbs measure ( )) exp ( N 2 V dµ N + g(z w)µ N (dz)µ N (dw) Exchangeable but neither product nor log-concave Empirical measure µ N := 1 N N k=1 δ z k

36 8/32 Electrostatics Outline Motivation Electrostatics Coulomb gas model Probability metrics and Coulomb transport inequality Concentration of measure for Coulomb gases

37 /32 Electrostatics Coulomb kernel in mathematical physics Coulomb kernel in R d, d 2, log 1 if d = 2, x R d x g(x) := 1 x d 2 if d 3.

38 /32 Electrostatics Coulomb kernel in mathematical physics Coulomb kernel in R d, d 2, log 1 if d = 2, x R d x g(x) := 1 x d 2 if d 3. Fundamental solution of Poisson s equation { 2π if d = 2, g = c d δ 0 where c d := (d 2) S d 1 if d 3.

39 0/32 Electrostatics Coulomb energy and metric Probability measures on R d with compact support

40 0/32 Electrostatics Coulomb energy and metric Probability measures on R d with compact support Coulomb energy: E(µ) := g(x y)µ(dx)µ(dy) R {+ }.

41 0/32 Electrostatics Coulomb energy and metric Probability measures on R d with compact support Coulomb energy: E(µ) := g(x y)µ(dx)µ(dy) R {+ }. Coulomb metric: (µ, ν) E(µ ν).

42 1/32 Electrostatics Confinement and equilibrium measure External potential V : R d R {+ } growing at infinity

43 1/32 Electrostatics Confinement and equilibrium measure External potential V : R d R {+ } growing at infinity Coulomb energy with confining potential E V (µ) = V (x)µ(dx) + E(µ) = (V (x) + (g µ)(x))µ(dx).

44 1/32 Electrostatics Confinement and equilibrium measure External potential V : R d R {+ } growing at infinity Coulomb energy with confining potential E V (µ) = V (x)µ(dx) + E(µ) = (V (x) + (g µ)(x))µ(dx). External and internal electric fields: V + g µ

45 1/32 Electrostatics Confinement and equilibrium measure External potential V : R d R {+ } growing at infinity Coulomb energy with confining potential E V (µ) = V (x)µ(dx) + E(µ) = (V (x) + (g µ)(x))µ(dx). External and internal electric fields: V + g µ Equilibrium probability measure µ V := arg inf E V

46 1/32 Electrostatics Confinement and equilibrium measure External potential V : R d R {+ } growing at infinity Coulomb energy with confining potential E V (µ) = V (x)µ(dx) + E(µ) = (V (x) + (g µ)(x))µ(dx). External and internal electric fields: V + g µ Equilibrium probability measure µ V := arg inf E V µ V is compactly supported and has density 1 2c d V

47 12/32 Electrostatics Examples of equilibrium measures d g V µ V interval c(x) arcsine 1 2 x 2 semicircle 2 2 x 2 uniform on a disc 3 d x 2 uniform on a ball 2 d radial radial in a ring

48 13/32 Coulomb gas model Outline Motivation Electrostatics Coulomb gas model Probability metrics and Coulomb transport inequality Concentration of measure for Coulomb gases

49 14/32 Coulomb gas model Coulomb gas or one component plasma Interaction energy of N Coulomb charges in R d : H N (x 1,..., x N ) := N N i=1 V (x i ) + i j g(x i x j ).

50 14/32 Coulomb gas model Coulomb gas or one component plasma Interaction energy of N Coulomb charges in R d : H N (x 1,..., x N ) := N N i=1 V (x i ) + i j g(x i x j ). Boltzmann Gibbs probability measure on (R d ) N dp N V,β (x 1,..., x N ) dx 1 dx N ( exp β ) 2 H N(x 1,..., x N )

51 14/32 Coulomb gas model Coulomb gas or one component plasma Interaction energy of N Coulomb charges in R d : H N (x 1,..., x N ) := N N i=1 V (x i ) + i j g(x i x j ). Boltzmann Gibbs probability measure on (R d ) N dp N V,β (x 1,..., x N ) dx 1 dx N ( exp β ) 2 H N(x 1,..., x N ) V must be strong enough at infinity to ensure integrability.

52 5/32 Coulomb gas model Empirical measure and equilibrium measure Random empirical measure under P N V,β : ˆµ N := 1 N N δ xi. i=1

53 5/32 Coulomb gas model Empirical measure and equilibrium measure Random empirical measure under P N V,β : ˆµ N := 1 N N δ xi. i=1 Under mild assumptions on V, with probability one, µ N N µ V.

54 5/32 Coulomb gas model Empirical measure and equilibrium measure Random empirical measure under P N V,β : ˆµ N := 1 N N δ xi. i=1 Under mild assumptions on V, with probability one, N 2 µ N N µ V. Large Deviation Principle (BAG, HP, BAZ, CGZ, S, B) ( ) log P N V,β d(µ N, µ V ) r β ( inf EV (µ) E V (µ V ) ). N 2 d(µ,µ V ) r

55 16/32 Coulomb gas model Quantitative or non asymptotic estimates The LDP gives for any r > 0 and any N N 0, ( ) e cr N2 P N V,β d(µ N, µ V ) r e Cr N2.

56 16/32 Coulomb gas model Quantitative or non asymptotic estimates The LDP gives for any r > 0 and any N N 0, ( ) e cr N2 P N V,β d(µ N, µ V ) r e Cr N2. Sub-Gaussian concentration of measure: C quadratic in r?

57 16/32 Coulomb gas model Quantitative or non asymptotic estimates The LDP gives for any r > 0 and any N N 0, ( ) e cr N2 P N V,β d(µ N, µ V ) r e Cr N2. Sub-Gaussian concentration of measure: C quadratic in r? Other distances such as W p?

58 16/32 Coulomb gas model Quantitative or non asymptotic estimates The LDP gives for any r > 0 and any N N 0, ( ) e cr N2 P N V,β d(µ N, µ V ) r e Cr N2. Sub-Gaussian concentration of measure: C quadratic in r? Other distances such as W p? Yes for one-dimensional log-gas: Maïda-Maurel-Segala

59 16/32 Coulomb gas model Quantitative or non asymptotic estimates The LDP gives for any r > 0 and any N N 0, ( ) e cr N2 P N V,β d(µ N, µ V ) r e Cr N2. Sub-Gaussian concentration of measure: C quadratic in r? Other distances such as W p? Yes for one-dimensional log-gas: Maïda-Maurel-Segala Nothing known otherwise (nothing for Ginibre ensemble!)

60 7/32 Coulomb gas model Key observation Write P N V,β with µ N: ( ) dp N V,β (x 1,..., x N ) exp β 2 N2 E V (µ N) = dx 1 dx N ZV N,β where E V (µ N) := V (x)µ N (dx) + g(x y)µ N (dx)µ N (dy). x y

61 7/32 Coulomb gas model Key observation Write P N V,β with µ N: ( ) dp N V,β (x 1,..., x N ) exp β 2 N2 E V (µ N) = dx 1 dx N ZV N,β where E V (µ N) := V (x)µ N (dx) + g(x y)µ N (dx)µ N (dy). x y Serfaty et al: rewrite E V (µ N) E V (µ V ) with L 2 norm of electric field of µ N µ V. Leads to renormalized energy.

62 7/32 Coulomb gas model Key observation Write P N V,β with µ N: ( ) dp N V,β (x 1,..., x N ) exp β 2 N2 E V (µ N) = dx 1 dx N ZV N,β where E V (µ N) := V (x)µ N (dx) + g(x y)µ N (dx)µ N (dy). x y Serfaty et al: rewrite E V (µ N) E V (µ V ) with L 2 norm of electric field of µ N µ V. Leads to renormalized energy. Alternative: compare E V (µ N) E V (µ V ) with W 1 (µ N, µ V ).

63 18/32 Probability metrics and Coulomb transport inequality Outline Motivation Electrostatics Coulomb gas model Probability metrics and Coulomb transport inequality Concentration of measure for Coulomb gases

64 19/32 Probability metrics and Coulomb transport inequality Probability metrics Coulomb divergence E V (µ) E V (µ V )

65 19/32 Probability metrics and Coulomb transport inequality Probability metrics Coulomb divergence E V (µ) E V (µ V ) Coulomb metric E(µ ν)

66 19/32 Probability metrics and Coulomb transport inequality Probability metrics Coulomb divergence E V (µ) E V (µ V ) Coulomb metric E(µ ν) Bounded-Lipschitz or Fortet Mourier distance d BL (µ, ν) := sup f Lip 1 f 1 f (x)(µ ν)(dx),

67 19/32 Probability metrics and Coulomb transport inequality Probability metrics Coulomb divergence E V (µ) E V (µ V ) Coulomb metric E(µ ν) Bounded-Lipschitz or Fortet Mourier distance d BL (µ, ν) := sup f Lip 1 f 1 f (x)(µ ν)(dx), (Monge-Kantorovich-)Wasserstein distance W p (µ, ν) := inf E( X Y p ) 1/p. (X,Y ) X µ,y ν

68 9/32 Probability metrics and Coulomb transport inequality Probability metrics Coulomb divergence E V (µ) E V (µ V ) Coulomb metric E(µ ν) Bounded-Lipschitz or Fortet Mourier distance d BL (µ, ν) := sup f Lip 1 f 1 f (x)(µ ν)(dx), (Monge-Kantorovich-)Wasserstein distance ( 1/p. W p (µ, ν) := x y p π(dx, dy)) inf π Π(µ,ν)

69 9/32 Probability metrics and Coulomb transport inequality Probability metrics Coulomb divergence E V (µ) E V (µ V ) Coulomb metric E(µ ν) Bounded-Lipschitz or Fortet Mourier distance d BL (µ, ν) := sup f Lip 1 f 1 f (x)(µ ν)(dx), Kantorovich-Rubinstein duality W 1 (µ, ν) = sup f Lip 1 f (x)(µ ν)(dx).

70 Probability metrics and Coulomb transport inequality Probability metrics Coulomb divergence E V (µ) E V (µ V ) Coulomb metric E(µ ν) 9/32 Bounded-Lipschitz or Fortet Mourier distance d BL (µ, ν) := sup f Lip 1 f 1 f (x)(µ ν)(dx), Kantorovich-Rubinstein duality d BL (µ, ν) W 1 (µ, ν) = sup f Lip 1 Topologies f (x)(µ ν)(dx).

71 0/32 Probability metrics and Coulomb transport inequality Local Coulomb transport inequality Theorem (Transport type inequality CHM 2016) D R d compact, supp(µ + ν) D, E(µ) < and E(ν) <, W 1 (µ, ν) 2 C D E(µ ν). Optimal C D is Vol(B 4Vol(D) )

72 0/32 Probability metrics and Coulomb transport inequality Local Coulomb transport inequality Theorem (Transport type inequality CHM 2016) D R d compact, supp(µ + ν) D, E(µ) < and E(ν) <, W 1 (µ, ν) 2 C D E(µ ν). Optimal C D is Vol(B 4Vol(D) ) Extends Popescu local free transport inequality to any d

73 21/32 Probability metrics and Coulomb transport inequality Coulomb transport inequality for equilibrium measures Theorem (Transport type inequality CHM 2016) We have for any probability measure µ ) d BL (µ, µ V ) 2 C BL (E V (µ) E V (µ V ). Moreover if V is superquadratic then W 1 (µ, µ V ) 2 ( C W1 EV (µ) E V (µ V ) ). Free transport inequalities for d = 2 and V = + on R c

74 21/32 Probability metrics and Coulomb transport inequality Coulomb transport inequality for equilibrium measures Theorem (Transport type inequality CHM 2016) We have for any probability measure µ ) d BL (µ, µ V ) 2 C BL (E V (µ) E V (µ V ). Moreover if V is superquadratic then W 1 (µ, µ V ) 2 ( C W1 EV (µ) E V (µ V ) ). Free transport inequalities for d = 2 and V = + on R c Extends Maïda-Maurel-Segala, Popescu

75 21/32 Probability metrics and Coulomb transport inequality Coulomb transport inequality for equilibrium measures Theorem (Transport type inequality CHM 2016) We have for any probability measure µ ) d BL (µ, µ V ) 2 C BL (E V (µ) E V (µ V ). Moreover if V is superquadratic then W 1 (µ, µ V ) 2 ( C W1 EV (µ) E V (µ V ) ). Free transport inequalities for d = 2 and V = + on R c Extends Maïda-Maurel-Segala, Popescu Growth condition is optimal for W 1

76 22/32 Concentration of measure for Coulomb gases Outline Motivation Electrostatics Coulomb gas model Probability metrics and Coulomb transport inequality Concentration of measure for Coulomb gases

77 3/32 Concentration of measure for Coulomb gases Concentration of measure for Coulomb gases Theorem (Concentration inequality CHM 2016) If V does not grows too fast then ( ) d BL (µ N, µ V ) r e aβn2 r 2. P N V,β Moreover if V superquadratic then W 1 instead of d BL. LDP shows that order in N is optimal

78 3/32 Concentration of measure for Coulomb gases Concentration of measure for Coulomb gases Theorem (Concentration inequality CHM 2016) If V does not grows too fast then ( ) d BL (µ N, µ V ) r e aβn2 r 2 +1 d=2 ( β 4 N log N)+bβN2 2/d +c(β)n. P N V,β Moreover if V superquadratic then W 1 instead of d BL. LDP shows that order in N is optimal Explicit constants a, b, c if V sub-quadratic

79 3/32 Concentration of measure for Coulomb gases Concentration of measure for Coulomb gases Theorem (Concentration inequality CHM 2016) If V does not grows too fast then ( ) d BL (µ N, µ V ) r e aβn2 r 2 +1 d=2 ( β 4 N log N)+bβN2 2/d +c(β)n. P N V,β Moreover if V superquadratic then W 1 instead of d BL. LDP shows that order in N is optimal Explicit constants a, b, c if V sub-quadratic Extends Maïda-Maurel-Segala bound to any dimension: P N V,β ( ) W 1 (µ N, µ V ) r e cn2 r 2, r { log N N if d = 2, N 1/d if d 3.

80 24/32 Concentration of measure for Coulomb gases Convergence in Wasserstein distance Corollary (Wasserstein convergence CHM 2016) If V superquadratic and β N β V log N N then under PN V,β N a.s. lim N W 1(µ N, µ V ) = 0.

81 25/32 Concentration of measure for Coulomb gases Convergence at mesoscopic scale Corollary (Mesoscopic convergence CHM 2016) If d = 2 then ( P N ( ) V,β d BL τ N s x 0 µ N, τx Ns 0 µ V CN s log N N ) e cn log N, Test functions are global, not local as in Rougerie-Serfaty

82 25/32 Concentration of measure for Coulomb gases Convergence at mesoscopic scale Corollary (Mesoscopic convergence CHM 2016) If d = 2 then ( P N ( ) V,β d BL τ N s x 0 µ N, τx Ns 0 µ V CN s log N N ) e cn log N, If d 3 then ( ( ) ) d BL τ N s x 0 µ N, τx Ns 0 µ V CN s 1/d e cn2 2/d. P N V,β Test functions are global, not local as in Rougerie-Serfaty

83 25/32 Concentration of measure for Coulomb gases Convergence at mesoscopic scale Corollary (Mesoscopic convergence CHM 2016) If d = 2 then ( P N ( ) V,β d BL τ N s x 0 µ N, τx Ns 0 µ V CN s log N N ) e cn log N, If d 3 then ( ( ) ) d BL τ N s x 0 µ N, τx Ns 0 µ V CN s 1/d e cn2 2/d. P N V,β If V superquadratic then d BL can be replaced by W 1. Test functions are global, not local as in Rougerie-Serfaty

84 6/32 Concentration of measure for Coulomb gases Concentration for spectrum of Ginibre matrices Corollary (Concentration for Ginibre CHM 2016) If G is N N with iid Gaussian entries of variance 1 2N then ( ) P W 1 (µ G, µ ) r e 1 4C N2 r N log N+N[ 1 C log π]. Open problem: universality, even for ±1

85 6/32 Concentration of measure for Coulomb gases Concentration for spectrum of Ginibre matrices Corollary (Concentration for Ginibre CHM 2016) If G is N N with iid Gaussian entries of variance 1 2N then ( ) P W 1 (µ G, µ ) r e 1 4C N2 r N log N+N[ 1 C log π]. Open problem: universality, even for ±1 Provides W 1 convergence

86 27/32 Concentration of measure for Coulomb gases Exponential tightness Theorem (Tightness CHM 2016) For any r r 0 P N V,β (supp(µ N) B r ) = P N V,β( max 1 i N x i r ) e cnv (r), where V (r) := min x r V (x). Follows by using an argument by Borot and Guionnet

87 27/32 Concentration of measure for Coulomb gases Exponential tightness Theorem (Tightness CHM 2016) For any r r 0 P N V,β (supp(µ N) B r ) = P N V,β( max 1 i N x i 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 lim N max 1 i N x i <.

88 27/32 Concentration of measure for Coulomb gases Exponential tightness Theorem (Tightness CHM 2016) For any r r 0 P N V,β (supp(µ N) B r ) = P N V,β( max 1 i N x i 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 lim N max 1 i N x i <. Gives W p versions of convergence and concentration W p p(µ, ν) (2M) p 1 W 1 (µ, ν) M(2M) p 1 d BL (µ, ν). For p = 2 we get P N V,β (W 2(µ N, µ V ) r) 2e cn3/2 r 2.

89 28/32 Concentration of measure for Coulomb gases Notes and comments W p 2 versions? Popescu free transport inequalities

90 28/32 Concentration of measure for Coulomb gases Notes and comments W p 2 versions? Popescu free transport inequalities Hardy-Littlewood-Sobolev inequalities (Keller-Segel PDE)

91 28/32 Concentration of measure for Coulomb gases Notes and comments W p 2 versions? Popescu free transport inequalities Hardy-Littlewood-Sobolev inequalities (Keller-Segel PDE) Classical transport inequalities with Coulomb distance

92 28/32 Concentration of measure for Coulomb gases Notes and comments W p 2 versions? Popescu free transport inequalities Hardy-Littlewood-Sobolev inequalities (Keller-Segel PDE) Classical transport inequalities with Coulomb distance Varying V and conditional gases. P N V,β ( x N) = P N 1 Ṽ N,β with Ṽ N := N N 1 V + 2 N 1 g(x N ) [covered by our work since g is superharmonic]

93 28/32 Concentration of measure for Coulomb gases Notes and comments W p 2 versions? Popescu free transport inequalities Hardy-Littlewood-Sobolev inequalities (Keller-Segel PDE) Classical transport inequalities with Coulomb distance Varying V and conditional gases. P N V,β ( x N) = P N 1 Ṽ N,β with Ṽ N := N N 1 V + 2 N 1 g(x N ) [covered by our work since g is superharmonic] Usage for CLT with GFF in all dimensions (VR, M+, LS, B+)

94 28/32 Concentration of measure for Coulomb gases Notes and comments W p 2 versions? Popescu free transport inequalities Hardy-Littlewood-Sobolev inequalities (Keller-Segel PDE) Classical transport inequalities with Coulomb distance Varying V and conditional gases. P N V,β ( x N) = P N 1 Ṽ N,β with Ṽ N := N N 1 V + 2 N 1 g(x N ) [covered by our work since g is superharmonic] Usage for CLT with GFF in all dimensions (VR, M+, LS, B+) Weakly confining potentials and heavy-tailed µ V

95 28/32 Concentration of measure for Coulomb gases Notes and comments W p 2 versions? Popescu free transport inequalities Hardy-Littlewood-Sobolev inequalities (Keller-Segel PDE) Classical transport inequalities with Coulomb distance Varying V and conditional gases. P N V,β ( x N) = P N 1 Ṽ N,β with Ṽ N := N N 1 V + 2 N 1 g(x N ) [covered by our work since g is superharmonic] Usage for CLT with GFF in all dimensions (VR, M+, LS, B+) Weakly confining potentials and heavy-tailed µ V Universality of concentration for random matrices

96 28/32 Concentration of measure for Coulomb gases Notes and comments W p 2 versions? Popescu free transport inequalities Hardy-Littlewood-Sobolev inequalities (Keller-Segel PDE) Classical transport inequalities with Coulomb distance Varying V and conditional gases. P N V,β ( x N) = P N 1 Ṽ N,β with Ṽ N := N N 1 V + 2 N 1 g(x N ) [covered by our work since g is superharmonic] Usage for CLT with GFF in all dimensions (VR, M+, LS, B+) Weakly confining potentials and heavy-tailed µ V Universality of concentration for random matrices Crossover and Sanov regime (Allez-Bouchaud-Guionnet)

97 29/32 Concentration of measure for Coulomb gases That s all folks! Thank you for your attention.

98 0/32 Concentration of measure for Coulomb gases Idea of proof of Coulomb transport inequality Potential: if U µ (x) := g µ(x) then U µ (x) = c d µ

99 0/32 Concentration of measure for Coulomb gases Idea of proof of Coulomb transport inequality Potential: if U µ (x) := g µ(x) then U µ (x) = c d µ Electric field: U µ (x). Carré du champ : U µ 2

100 0/32 Concentration of measure for Coulomb gases Idea of proof of Coulomb transport inequality Potential: if U µ (x) := g µ(x) then U µ (x) = c d µ Electric field: U µ (x). Carré du champ : U µ 2 Integration by parts + Schwarz s inequality in R d and L 2 c d f (x)(µ ν)(dx) = f (x) U µ ν (x)dx f (x) U µ ν (x) dx f Lip D + U µ ν (x) dx f Lip ( D + U µ ν (x) 2 dx) 1/2.

101 1/32 Concentration of measure for Coulomb gases Idea... Continued Again by integration by parts U µ ν (x) 2 dx = = c d = c d E(µ ν). U µ ν (x) U µ ν (x)dx U µ ν (x)(µ ν)(dx) Finally W 1 (µ, ν) 2 D + c d E(µ ν).

102 32/32 Concentration of measure for Coulomb gases Idea of proof of concentration dp N V,β (W 1(µ N, µ V ) r) = 1 ZV N e β 2 E (µ N ) dx.,β W 1 (µ N,µ V ) r Normalizing constant 1 Z N V,β exp { N 2 β 2 E V (µ V ) N ( )} β 2 E(µ V ) S(µ V ).

103 32/32 Concentration of measure for Coulomb gases Idea of proof of concentration dp N V,β (W 1(µ N, µ V ) r) = 1 ZV N e β 2 E (µ N ) dx.,β W 1 (µ N,µ V ) r Normalizing constant 1 Z N V,β exp { N 2 β 2 E V (µ V ) N ( )} β 2 E(µ V ) S(µ V ). Regularization: g superharmonic, µ (ε) N := µ N λ ε, E (µ N ) N 2 E V (µ (ε) N ) + NE(λ ε) + N N (V λ ε V )(x i ). i=1

104 Concentration of measure for Coulomb gases Idea of proof of concentration dp N V,β (W 1(µ N, µ V ) r) = 1 ZV N e β 2 E (µ N ) dx.,β W 1 (µ N,µ V ) r Normalizing constant 1 Z N V,β exp { N 2 β 2 E V (µ V ) N ( )} β 2 E(µ V ) S(µ V ). 32/32 Regularization: g superharmonic, µ (ε) N := µ N λ ε, E (µ N ) N 2 E V (µ (ε) N ) + NE(λ ε) + N N (V λ ε V )(x i ). Coulomb transport E V (µ (ε) N ) + E V (µ V ) 1 C W2 1 (µ(ε) N, µ V ). i=1