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

Concentration for Coulomb gases Concentration for Coulomb gases and Coulomb transport inequalities Djalil Chafaï 1 , Adrien Hardy 2 , Mylène Maïda 2 1 Université Paris-Dauphine, 2 Université de Lille Probability and Analysis B˛ edlewo – May 18, 2017 1/24

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))) 2/24

Concentration for Coulomb gases Electrostatics Outline Electrostatics Coulomb gas model Probability metrics and Coulomb transport inequality Concentration of measure for Coulomb gases 3/24

Concentration for Coulomb gases Electrostatics Coulomb kernel in mathematical physics � Coulomb kernel in R d , d � 2,  log 1  if d = 2,   | x | x ∈ R d �→ g ( x ) = 1    if d ≥ 3 . | x | d − 2 4/24

Concentration for Coulomb gases Electrostatics Coulomb kernel in mathematical physics � Coulomb kernel in R d , d � 2,  log 1  if d = 2,   | x | x ∈ R d �→ g ( x ) = 1    if d ≥ 3 . | x | d − 2 � Fundamental solution of Poisson’s equation � 2 π if d = 2 , ∆ g = − c d δ 0 c d = where ( d − 2 ) | S d − 1 | if d ≥ 3 . 4/24

Concentration for Coulomb gases Electrostatics Coulomb energy and equilibrium measure � Coulomb energy of probability measure µ on R d : �� E ( µ ) = g ( x − y ) µ ( d x ) µ ( d y ) ∈ R ∪{ + ∞ } . 5/24

Concentration for Coulomb gases Electrostatics Coulomb energy and equilibrium measure � Coulomb energy of probability measure µ on R d : �� E ( µ ) = g ( x − y ) µ ( d x ) µ ( d y ) ∈ R ∪{ + ∞ } . � Coulomb energy with confining potential (external field) � E V ( µ ) = E ( µ )+ V ( x ) µ ( d x ) . 5/24

Concentration for Coulomb gases Electrostatics Coulomb energy and equilibrium measure � Coulomb energy of probability measure µ on R d : �� E ( µ ) = g ( x − y ) µ ( d x ) µ ( d y ) ∈ R ∪{ + ∞ } . � Coulomb energy with confining potential (external field) � E V ( µ ) = E ( µ )+ V ( x ) µ ( d x ) . � Equilibrium probability measure (electrostatics) µ V = arginf E V 5/24

Concentration for Coulomb gases Electrostatics Coulomb energy and equilibrium measure � Coulomb energy of probability measure µ on R d : �� E ( µ ) = g ( x − y ) µ ( d x ) µ ( d y ) ∈ R ∪{ + ∞ } . � Coulomb energy with confining potential (external field) � E V ( µ ) = E ( µ )+ V ( x ) µ ( d x ) . � Equilibrium probability measure (electrostatics) µ V = arginf E V � If V is strong then µ V is compactly supported with density 1 ∆ V 2 c d 5/24

Concentration for Coulomb gases Electrostatics Examples of equilibrium measures d g V µ V ∞ 1 interval c ( x ) 1 2 arcsine x 2 1 2 semicircle | x | 2 2 2 uniform on a disc � x � 2 � 3 d uniform on a ball � 2 d radial radial in a ring 6/24

Concentration for Coulomb gases Coulomb gas model Outline Electrostatics Coulomb gas model Probability metrics and Coulomb transport inequality Concentration of measure for Coulomb gases 7/24

Concentration for Coulomb gases Coulomb gas model Coulomb gas or one component plasma � Energy of N Coulomb charges in R d : N ∑ V ( x i )+ ∑ H N ( x 1 ,..., x N ) = N g ( x i − x j ) i = 1 i � = j 8/24

Concentration for Coulomb gases Coulomb gas model Coulomb gas or one component plasma � Energy of N Coulomb charges in R d : H N ( x 1 ,..., x N ) = N 2 � � �� � V ( x ) µ N ( d x )+ g ( x − y ) µ N ( d x ) µ N ( d y ) x � = y � Empirical measure: µ N = 1 N ∑ N i = 1 δ x i 8/24

Concentration for Coulomb gases Coulomb gas model Coulomb gas or one component plasma � Energy of N Coulomb charges in R d : H N ( x 1 ,..., x N ) = N 2 � � �� � V ( x ) µ N ( d x )+ g ( x − y ) µ N ( d x ) µ N ( d y ) x � = y � �� � E � = V ( µ N ) � Empirical measure: µ N = 1 N ∑ N i = 1 δ x i 8/24

Concentration for Coulomb gases Coulomb gas model Coulomb gas or one component plasma � Energy of N Coulomb charges in R d : H N ( x 1 ,..., x N ) = N 2 � � �� � V ( x ) µ N ( d x )+ g ( x − y ) µ N ( d x ) µ N ( d y ) x � = y � �� � E � = V ( µ N ) � Empirical measure: µ N = 1 N ∑ N i = 1 δ x i � Boltzmann–Gibbs measure P N V , β on ( R d ) N : � � − β exp 2 H N ( x 1 ,..., x N ) Z V , β 8/24

Concentration for Coulomb gases Coulomb gas model Coulomb gas or one component plasma � Energy of N Coulomb charges in R d : H N ( x 1 ,..., x N ) = N 2 � � �� � V ( x ) µ N ( d x )+ g ( x − y ) µ N ( d x ) µ N ( d y ) x � = y � �� � E � = V ( µ N ) � Empirical measure: µ N = 1 N ∑ N i = 1 δ x i � Boltzmann–Gibbs measure P N V , β on ( R d ) N : � � 2 N 2 E � = − β exp V ( µ N ) Z N V , β 8/24

Concentration for Coulomb gases Coulomb gas model Coulomb gas or one component plasma � Energy of N Coulomb charges in R d : H N ( x 1 ,..., x N ) = N 2 � � �� � V ( x ) µ N ( d x )+ g ( x − y ) µ N ( d x ) µ N ( d y ) x � = y � �� � E � = V ( µ N ) � Empirical measure: µ N = 1 N ∑ N i = 1 δ x i � Boltzmann–Gibbs measure P N V , β on ( R d ) N : � � 2 N 2 E � = − β exp V ( µ N ) Z N V , β � V must be strong enough at infinity to ensure integrability 8/24

Concentration for Coulomb gases Coulomb gas model Coulomb gas or one component plasma � Energy of N Coulomb charges in R d : H N ( x 1 ,..., x N ) = N 2 � � �� � V ( x ) µ N ( d x )+ g ( x − y ) µ N ( d x ) µ N ( d y ) x � = y � �� � E � = V ( µ N ) � Empirical measure: µ N = 1 N ∑ N i = 1 δ x i � Boltzmann–Gibbs measure P N V , β on ( R d ) N : � � 2 N 2 E � = − β exp V ( µ N ) Z N V , β � V must be strong enough at infinity to ensure integrability � P N , β is neither product nor log-concave 8/24

Concentration for Coulomb gases Coulomb gas model Empirical measure and equilibrium measure � Random empirical measure under P N V , β : N µ N = 1 ∑ δ x i . N i = 1 9/24

Concentration for Coulomb gases Coulomb gas model Empirical measure and equilibrium measure � Random empirical measure under P N V , β : N µ N = 1 ∑ δ x i . N i = 1 � Under mild assumptions on V , with probability one, µ N − N → ∞ µ V . → 9/24

Concentration for Coulomb gases Coulomb gas model Empirical measure and equilibrium measure � Random empirical measure under P N V , β : N µ N = 1 ∑ δ x i . N i = 1 � Under mild assumptions on V , with probability one, µ N − N → ∞ µ V . → � Large Deviation Principle (Gozlan-C.-Zitt 2014) � � log P N dist ( µ N , µ V ) ≥ r � � N → ∞ − β V , β − → inf E V ( µ ) − E V ( µ V ) . N 2 2 dist ( µ , µ V ) ≥ r 9/24

Concentration for Coulomb gases Coulomb gas model Empirical measure and equilibrium measure � Random empirical measure under P N V , β : N µ N = 1 ∑ δ x i . N i = 1 � Under mild assumptions on V , with probability one, µ N − N → ∞ µ V . → � Large Deviation Principle (Gozlan-C.-Zitt 2014) � � log P N dist ( µ N , µ V ) ≥ r � � N → ∞ − β V , β − → inf E V ( µ ) − E V ( µ V ) . N 2 2 dist ( µ , µ V ) ≥ r � Quantitative estimates? How to relate dist and E V ( · ) − E V ( µ V ) ? 9/24

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 10/24

Concentration for Coulomb gases Probability metrics and Coulomb transport inequality Probability metrics and topologies � Coulomb divergence (Large Deviations rate function) E V ( µ ) − E V ( µ V ) 11/24

Concentration for Coulomb gases Probability metrics and Coulomb transport inequality Probability metrics and topologies � Coulomb divergence (Large Deviations rate function) E V ( µ ) − E V ( µ V ) � Coulomb metric � E ( µ − ν ) 11/24

Concentration for Coulomb gases Probability metrics and Coulomb transport inequality Probability metrics and topologies � Coulomb divergence (Large Deviations rate function) E V ( µ ) − E V ( µ V ) � Coulomb metric � E ( µ − ν ) � Bounded-Lipschitz or Fortet–Mourier distance � d BL ( µ , ν ) = sup f ( x )( µ − ν )( d x ) , � f � Lip � 1 � f � ∞ � 1 11/24

Concentration for Coulomb gases Probability metrics and Coulomb transport inequality Probability metrics and topologies � Coulomb divergence (Large Deviations rate function) E V ( µ ) − E V ( µ V ) � Coulomb metric � E ( µ − ν ) � Bounded-Lipschitz or Fortet–Mourier distance � d BL ( µ , ν ) = sup f ( x )( µ − ν )( d x ) , � f � Lip � 1 � f � ∞ � 1 � (Monge-Kantorovich-)Wasserstein distance E ( | X − Y | p ) 1 / p . W p ( µ , ν ) = inf ( X , Y ) X ∼ µ , Y ∼ ν 11/24