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

Dynamics of a planar Coulomb gas Poincaré inequality Comparison to Gaussianity via convexity Theorem (Brascamp–Lieb 1976) If µ p d x q “ e ´ H p x q d x, ∇ 2 H ą 0 on R d , then for any smooth f : R d Ñ R , Z Var µ p f q ď E µ xp ∇ 2 H q ´ 1 ∇ f , ∇ f y � Proof by induction on dimension d � Ornstein–Uhlenbeck: ∇ 2 H “ I d � Convexity: ∇ 2 H ě ρ I d ą 0 gives Poincaré with 1 { ρ � H convex means that µ p d x q “ e ´ H d x is log-concave 8/26

Dynamics of a planar Coulomb gas Poincaré inequality Comparison to Gaussianity via convexity Theorem (Brascamp–Lieb 1976) If µ p d x q “ e ´ H p x q d x, ∇ 2 H ą 0 on R d , then for any smooth f : R d Ñ R , Z Var µ p f q ď E µ xp ∇ 2 H q ´ 1 ∇ f , ∇ f y � Proof by induction on dimension d � Ornstein–Uhlenbeck: ∇ 2 H “ I d � Convexity: ∇ 2 H ě ρ I d ą 0 gives Poincaré with 1 { ρ � H convex means that µ p d x q “ e ´ H d x is log-concave � Jensen divergence: Var µ p f q “ E µ Φ p f q´ Φ p E µ f q , Φ p u q “ u 2 8/26

Dynamics of a planar Coulomb gas Poincaré inequality Comparison to Gaussianity via convexity Theorem (Bakry–Émery 1984) If µ p d x q “ e ´ H p x q d x, ∇ 2 H ě ρ I d ą 0 , then for any convex Φ : I Ñ R Z with p u , v q ÞÑ Φ 2 p u q v 2 convex and any smooth f : R d Ñ I E µ Φ p f q´ Φ p E µ f q ď E µ p Φ 2 p f q| ∇ f | 2 q ρ 9/26

Dynamics of a planar Coulomb gas Poincaré inequality Comparison to Gaussianity via convexity Theorem (Bakry–Émery 1984) If µ p d x q “ e ´ H p x q d x, ∇ 2 H ě ρ I d ą 0 , then for any convex Φ : I Ñ R Z with p u , v q ÞÑ Φ 2 p u q v 2 convex and any smooth f : R d Ñ I E µ Φ p f q´ Φ p E µ f q ď E µ p Φ 2 p f q| ∇ f | 2 q ρ � Proof by semigroup interpolation e p t ´ s q G p Φ p e sG f qq , e tG f “ E f p X t q 9/26

Dynamics of a planar Coulomb gas Poincaré inequality Comparison to Gaussianity via convexity Theorem (Bakry–Émery 1984) If µ p d x q “ e ´ H p x q d x, ∇ 2 H ě ρ I d ą 0 , then for any convex Φ : I Ñ R Z with p u , v q ÞÑ Φ 2 p u q v 2 convex and any smooth f : R d Ñ I E µ Φ p f q´ Φ p E µ f q ď E µ p Φ 2 p f q| ∇ f | 2 q ρ � Proof by semigroup interpolation e p t ´ s q G p Φ p e sG f qq , e tG f “ E f p X t q � Ornstein–Uhlenbeck: ∇ 2 H “ I d 9/26

Dynamics of a planar Coulomb gas Poincaré inequality Comparison to Gaussianity via convexity Theorem (Bakry–Émery 1984) If µ p d x q “ e ´ H p x q d x, ∇ 2 H ě ρ I d ą 0 , then for any convex Φ : I Ñ R Z with p u , v q ÞÑ Φ 2 p u q v 2 convex and any smooth f : R d Ñ I E µ Φ p f q´ Φ p E µ f q ď E µ p Φ 2 p f q| ∇ f | 2 q ρ � Proof by semigroup interpolation e p t ´ s q G p Φ p e sG f qq , e tG f “ E f p X t q � Ornstein–Uhlenbeck: ∇ 2 H “ I d � Poincaré: I “ R , Φ p u q “ u 2 9/26

Dynamics of a planar Coulomb gas Poincaré inequality Comparison to Gaussianity via convexity Theorem (Bakry–Émery 1984) If µ p d x q “ e ´ H p x q d x, ∇ 2 H ě ρ I d ą 0 , then for any convex Φ : I Ñ R Z with p u , v q ÞÑ Φ 2 p u q v 2 convex and any smooth f : R d Ñ I E µ Φ p f q´ Φ p E µ f q ď E µ p Φ 2 p f q| ∇ f | 2 q ρ � Proof by semigroup interpolation e p t ´ s q G p Φ p e sG f qq , e tG f “ E f p X t q � Ornstein–Uhlenbeck: ∇ 2 H “ I d � Poincaré: I “ R , Φ p u q “ u 2 � Beckner: I “ R ` , Φ p u q “ u p , 1 ă p ď 2 9/26

Dynamics of a planar Coulomb gas Poincaré inequality Comparison to Gaussianity via convexity Theorem (Bakry–Émery 1984) If µ p d x q “ e ´ H p x q d x, ∇ 2 H ě ρ I d ą 0 , then for any convex Φ : I Ñ R Z with p u , v q ÞÑ Φ 2 p u q v 2 convex and any smooth f : R d Ñ I E µ Φ p f q´ Φ p E µ f q ď E µ p Φ 2 p f q| ∇ f | 2 q ρ � Proof by semigroup interpolation e p t ´ s q G p Φ p e sG f qq , e tG f “ E f p X t q � Ornstein–Uhlenbeck: ∇ 2 H “ I d � Poincaré: I “ R , Φ p u q “ u 2 � Beckner: I “ R ` , Φ p u q “ u p , 1 ă p ď 2 � Logarithmic Sobolev: I “ R ` , Φ p u q “ u log p u q 9/26

Dynamics of a planar Coulomb gas Poincaré inequality Comparison to Gaussianity via convexity Theorem (Caffarelli 2000) If µ p d x q “ e ´ H p x q d x, ∇ 2 H ě ρ I d ą 0 , then µ is the image of N p 0 , I d q Z by a Lipschitz function F : R d Ñ R d with } F } Lip ď 1 ? ρ . 10/26

Dynamics of a planar Coulomb gas Poincaré inequality Comparison to Gaussianity via convexity Theorem (Caffarelli 2000) If µ p d x q “ e ´ H p x q d x, ∇ 2 H ě ρ I d ą 0 , then µ is the image of N p 0 , I d q Z by a Lipschitz function F : R d Ñ R d with } F } Lip ď 1 ? ρ . � Proof by Monge–Ampère equation: f “ det p DF q g 10/26

Dynamics of a planar Coulomb gas Poincaré inequality Comparison to Gaussianity via convexity Theorem (Caffarelli 2000) If µ p d x q “ e ´ H p x q d x, ∇ 2 H ě ρ I d ą 0 , then µ is the image of N p 0 , I d q Z by a Lipschitz function F : R d Ñ R d with } F } Lip ď 1 ? ρ . � Proof by Monge–Ampère equation: f “ det p DF q g � Gives Poincaré from the Gaussian by transportation: Var µ p f q “ Var N p 0 , I d q p f p F qq ď E N p 0 , I d q p| ∇ f p F qq| 2 q ď E µ p| ∇ f | 2 q ρ 10/26

Dynamics of a planar Coulomb gas Poincaré inequality Comparison to Gaussianity via convexity Theorem (Caffarelli 2000) If µ p d x q “ e ´ H p x q d x, ∇ 2 H ě ρ I d ą 0 , then µ is the image of N p 0 , I d q Z by a Lipschitz function F : R d Ñ R d with } F } Lip ď 1 ? ρ . � Proof by Monge–Ampère equation: f “ det p DF q g � Gives Poincaré from the Gaussian by transportation: Var µ p f q “ Var N p 0 , I d q p f p F qq ď E N p 0 , I d q p| ∇ f p F qq| 2 q ď E µ p| ∇ f | 2 q ρ � Gives also any Φ -Sobolev inequality from the Gaussian! 10/26

Dynamics of a planar Coulomb gas Poincaré inequality KLS conjecture Conjecture (Kannan–Lovász–Simonovits 1995) There exists a universal constant C ą 0 such that for any dimension d ě 1 and any smooth H : R d Ñ R with ∇ 2 H ě 0 and Cov “ I d , µ p d x q “ e ´ H p x q d x satisfies to a Poincaré inequality with constant C. Z � . . . 11/26

Dynamics of a planar Coulomb gas Poincaré inequality KLS conjecture Conjecture (Kannan–Lovász–Simonovits 1995) There exists a universal constant C ą 0 such that for any dimension d ě 1 and any smooth H : R d Ñ R with ∇ 2 H ě 0 and Cov “ I d , µ p d x q “ e ´ H p x q d x satisfies to a Poincaré inequality with constant C. Z � . . . � KLS/Bobkov true with d 1 { 2 11/26

Dynamics of a planar Coulomb gas Poincaré inequality KLS conjecture Conjecture (Kannan–Lovász–Simonovits 1995) There exists a universal constant C ą 0 such that for any dimension d ě 1 and any smooth H : R d Ñ R with ∇ 2 H ě 0 and Cov “ I d , µ p d x q “ e ´ H p x q d x satisfies to a Poincaré inequality with constant C. Z � . . . � KLS/Bobkov true with d 1 { 2 � . . . , Bourgain, . . . , Klartag, . . . , Eldan, . . . 11/26

Dynamics of a planar Coulomb gas Poincaré inequality KLS conjecture Conjecture (Kannan–Lovász–Simonovits 1995) There exists a universal constant C ą 0 such that for any dimension d ě 1 and any smooth H : R d Ñ R with ∇ 2 H ě 0 and Cov “ I d , µ p d x q “ e ´ H p x q d x satisfies to a Poincaré inequality with constant C. Z � . . . � KLS/Bobkov true with d 1 { 2 � . . . , Bourgain, . . . , Klartag, . . . , Eldan, . . . � Lee–Vempala 2016: true with d 1 { 4 11/26

Dynamics of a planar Coulomb gas Poincaré inequality KLS conjecture Conjecture (Kannan–Lovász–Simonovits 1995) There exists a universal constant C ą 0 such that for any dimension d ě 1 and any smooth H : R d Ñ R with ∇ 2 H ě 0 and Cov “ I d , µ p d x q “ e ´ H p x q d x satisfies to a Poincaré inequality with constant C. Z � . . . � KLS/Bobkov true with d 1 { 2 � . . . , Bourgain, . . . , Klartag, . . . , Eldan, . . . � Lee–Vempala 2016: true with d 1 { 4 � . . . 11/26

Dynamics of a planar Coulomb gas Dyson Process Outline Poincaré inequality Dyson Process Ginibre process 12/26

Dynamics of a planar Coulomb gas Dyson Process Gaussian Hermitian Random Matrices “ R n 2 “ R d � Herm n ˆ n ” R n ` 2 n 2 ´ n 2 13/26

Dynamics of a planar Coulomb gas Dyson Process Gaussian Hermitian Random Matrices “ R n 2 “ R d � Herm n ˆ n ” R n ` 2 n 2 ´ n 2 � Boltzmann–Gibbs measure 2 Tr p M 2 q µ p d M q “ e ´ n d M Z 13/26

Dynamics of a planar Coulomb gas Dyson Process Gaussian Hermitian Random Matrices “ R n 2 “ R d � Herm n ˆ n ” R n ` 2 n 2 ´ n 2 � Boltzmann–Gibbs measure 2 Tr p M 2 q µ p d M q “ e ´ n d M Z � Stochastic Differential Equation à la Ornstein–Uhlenbeck d M t “ ´ nM t d t ` d B t . 13/26

Dynamics of a planar Coulomb gas Dyson Process Gaussian Hermitian Random Matrices “ R n 2 “ R d � Herm n ˆ n ” R n ` 2 n 2 ´ n 2 � Boltzmann–Gibbs measure 2 Tr p M 2 q µ p d M q “ e ´ n d M Z � Stochastic Differential Equation à la Ornstein–Uhlenbeck d M t “ ´ α n nM t d t `? α n d B t . 13/26

Dynamics of a planar Coulomb gas Dyson Process Gaussian Hermitian Random Matrices “ R n 2 “ R d � Herm n ˆ n ” R n ` 2 n 2 ´ n 2 � Boltzmann–Gibbs measure 2 Tr p M 2 q µ p d M q “ e ´ n d M Z � Stochastic Differential Equation à la Ornstein–Uhlenbeck d M t “ ´ M t d t ` d B t ? n . 13/26

Dynamics of a planar Coulomb gas Dyson Process Gaussian Hermitian Random Matrices “ R n 2 “ R d � Herm n ˆ n ” R n ` 2 n 2 ´ n 2 � Boltzmann–Gibbs measure 2 Tr p M 2 q µ p d M q “ e ´ n d M Z � Stochastic Differential Equation à la Ornstein–Uhlenbeck d M t “ ´ M t d t ` d B t ? n . � Change of variable: if spec p M q “ t x 1 ,..., x n u , M “ UDU ˚ with D “ diag p x 1 ,..., x n q 13/26

Dynamics of a planar Coulomb gas Dyson Process Gaussian Hermitian Random Matrices “ R n 2 “ R d � Herm n ˆ n ” R n ` 2 n 2 ´ n 2 � Boltzmann–Gibbs measure 2 Tr p M 2 q µ p d M q “ e ´ n d M Z � Stochastic Differential Equation à la Ornstein–Uhlenbeck d M t “ ´ M t d t ` d B t ? n . � Change of variable: if spec p M q “ t x 1 ,..., x n u , M “ UDU ˚ with D “ diag p x 1 ,..., x n q � Stochastic process of spectrum? 13/26

Dynamics of a planar Coulomb gas Dyson Process Gaussian Unitary Ensemble and Dyson Process � State space D “ tp x 1 ,..., x n q P R n : x 1 ă ¨¨¨ ă x n u 14/26

Dynamics of a planar Coulomb gas Dyson Process Gaussian Unitary Ensemble and Dyson Process � State space D “ tp x 1 ,..., x n q P R n : x 1 ă ¨¨¨ ă x n u � Boltzmann–Gibbs measure via change of variable ř n i ś i “ 1 x 2 e ´ n i ă j p x j ´ x i q 2 2 µ p d x q “ d x Z 14/26

Dynamics of a planar Coulomb gas Dyson Process Gaussian Unitary Ensemble and Dyson Process � State space D “ tp x 1 ,..., x n q P R n : x 1 ă ¨¨¨ ă x n u � Boltzmann–Gibbs measure via change of variable ř n i ´ 2 ř i “ 1 x 2 1 ´ n i ă j log 2 xj ´ xi µ p d x q “ e d x Z 14/26

Dynamics of a planar Coulomb gas Dyson Process Gaussian Unitary Ensemble and Dyson Process � State space D “ tp x 1 ,..., x n q P R n : x 1 ă ¨¨¨ ă x n u � Boltzmann–Gibbs measure via change of variable ř n i ´ 2 ř i “ 1 x 2 1 ´ n i ă j log 2 xj ´ xi µ p d x q “ e d x Z � Dyson Ornstein–Uhlenbeck process via Itô formula ˆ ˙ ÿ t ` 2 1 dt ` d B i t d X i X i t “ ´ ? X j n t ´ X i n i ă j t 14/26

Dynamics of a planar Coulomb gas Dyson Process Gaussian Unitary Ensemble and Dyson Process � State space D “ tp x 1 ,..., x n q P R n : x 1 ă ¨¨¨ ă x n u � Boltzmann–Gibbs measure via change of variable ř n i ´ 2 ř i “ 1 x 2 1 ´ n i ă j log 2 xj ´ xi µ p d x q “ e d x Z � Dyson Ornstein–Uhlenbeck process via Itô formula ˆ ˙ ÿ t ` 2 1 dt ` d B i t d X i X i t “ ´ ? X j n t ´ X i n i ă j t � Well-posedness: . . . , Rogers–Shi, . . . 14/26

Dynamics of a planar Coulomb gas Dyson Process Gaussian Unitary Ensemble and Dyson Process � State space D “ tp x 1 ,..., x n q P R n : x 1 ă ¨¨¨ ă x n u � Boltzmann–Gibbs measure via change of variable ř n i ´ 2 ř i “ 1 x 2 1 ´ n i ă j log 2 xj ´ xi µ p d x q “ e d x Z � Dyson Ornstein–Uhlenbeck process via Itô formula ˆ ˙ ÿ t ` 2 1 dt ` d B i t d X i X i t “ ´ ? X j n t ´ X i n i ă j t � Well-posedness: . . . , Rogers–Shi, . . . � Poincaré and log-Sobolev: . . . , Erd˝ os–Yau et al, . . . 14/26

Dynamics of a planar Coulomb gas Dyson Process James Dyson (1947 –) 15/26

Dynamics of a planar Coulomb gas Dyson Process Freeman Dyson (1923 –) 15/26

Dynamics of a planar Coulomb gas Dyson Process Freeman Dyson (1923 –) � Freeman Dyson A Brownian-motion model for the eigenvalues of a random matrix Journal of Mathematical Physics 3 (1962) 1191–1198. 15/26

Dynamics of a planar Coulomb gas Dyson Process Freeman Dyson (1923 –) � Freeman Dyson A Brownian-motion model for the eigenvalues of a random matrix Journal of Mathematical Physics 3 (1962) 1191–1198. � Greg Anderson & Alice Guionnet & Ofer Zeitouni An introduction to random matrices (CUP 2009) 15/26

Dynamics of a planar Coulomb gas Dyson Process Freeman Dyson (1923 –) � Freeman Dyson A Brownian-motion model for the eigenvalues of a random matrix Journal of Mathematical Physics 3 (1962) 1191–1198. � Greg Anderson & Alice Guionnet & Ofer Zeitouni An introduction to random matrices (CUP 2009) � László Erd˝ os & Horng-Tzer Yau Dynamical Approach To Random Matrix Theory (AMS 2017) 15/26

Dynamics of a planar Coulomb gas Dyson Process Optimal Poincaré constant (mind the gap!) � Boltzmann–Gibbs measure ÿ n ÿ µ p d x q “ e ´ H p x q 1 H p x q “ n x 2 with i ` 2 d x log 2 x j ´ x i Z i “ 1 i ă j 16/26

Dynamics of a planar Coulomb gas Dyson Process Optimal Poincaré constant (mind the gap!) � Boltzmann–Gibbs measure ÿ n ÿ µ p d x q “ e ´ H p x q 1 H p x q “ n x 2 with i ` 2 d x log 2 x j ´ x i Z i “ 1 i ă j � Log-concavity ∇ 2 H p x q ě n . 16/26

Dynamics of a planar Coulomb gas Dyson Process Optimal Poincaré constant (mind the gap!) � Boltzmann–Gibbs measure ÿ n ÿ µ p d x q “ e ´ H p x q 1 H p x q “ n x 2 with i ` 2 d x log 2 x j ´ x i Z i “ 1 i ă j � Log-concavity ∇ 2 H p x q ě n . � Brascamp–Lieb or Bakry–Émery or Caffarelli Var µ p f q ď E µ p| ∇ f | 2 q . n 16/26

Dynamics of a planar Coulomb gas Dyson Process Optimal Poincaré constant (mind the gap!) � Boltzmann–Gibbs measure ÿ n ÿ µ p d x q “ e ´ H p x q 1 H p x q “ n x 2 with i ` 2 d x log 2 x j ´ x i Z i “ 1 i ă j � Log-concavity ∇ 2 H p x q ě n . � Brascamp–Lieb or Bakry–Émery or Caffarelli Var µ p f q ď E µ p| ∇ f | 2 q . n � Equality achieved for f p x q “ x 1 `¨¨¨` x n (compute traces) 16/26

Dynamics of a planar Coulomb gas Dyson Process Optimal Poincaré constant (mind the gap!) � Boltzmann–Gibbs measure ÿ n ÿ µ p d x q “ e ´ H p x q 1 H p x q “ n x 2 with i ` 2 d x log 2 x j ´ x i Z i “ 1 i ă j � Log-concavity ∇ 2 H p x q ě n . � Brascamp–Lieb or Bakry–Émery or Caffarelli Var µ p f q ď E µ p| ∇ f | 2 q . n � Equality achieved for f p x q “ x 1 `¨¨¨` x n (compute traces) � Lipschitz deformation of Gaussian (Hoffman–Wielandt) 16/26

Dynamics of a planar Coulomb gas Ginibre process Outline Poincaré inequality Dyson Process Ginibre process 17/26

Dynamics of a planar Coulomb gas Ginibre process Ginibre process � Boltzmann–Gibbs measure on Mat n ˆ n p C q µ p M q “ e ´ n Tr p MM ˚ q d M Z 18/26

Dynamics of a planar Coulomb gas Ginibre process Ginibre process � Boltzmann–Gibbs measure on Mat n ˆ n p C q µ p M q “ e ´ n Tr p MM ˚ q d M Z � Schur unitary decomposition: if t x 1 ,..., x n u “ spec p M q , M “ UTU ˚ with and T “ D ` N D “ diag p x 1 ,..., x n q . 18/26

Dynamics of a planar Coulomb gas Ginibre process Ginibre process � Boltzmann–Gibbs measure on Mat n ˆ n p C q µ p M q “ e ´ n Tr p MM ˚ q d M Z � Schur unitary decomposition: if t x 1 ,..., x n u “ spec p M q , M “ UTU ˚ with and T “ D ` N D “ diag p x 1 ,..., x n q . � Lack of normality is generic: µ pt N “ 0 uq “ 0 18/26

Dynamics of a planar Coulomb gas Ginibre process Ginibre process � Boltzmann–Gibbs measure on Mat n ˆ n p C q µ p M q “ e ´ n Tr p MM ˚ q d M Z � Schur unitary decomposition: if t x 1 ,..., x n u “ spec p M q , M “ UTU ˚ with and T “ D ` N D “ diag p x 1 ,..., x n q . � Lack of normality is generic: µ pt N “ 0 uq “ 0 � Process on spectrum melts N and D ( Ñ Bourgade–Dubach) 18/26

Dynamics of a planar Coulomb gas Ginibre process Ginibre process � Boltzmann–Gibbs measure on Mat n ˆ n p C q µ p M q “ e ´ n Tr p MM ˚ q d M Z � Schur unitary decomposition: if t x 1 ,..., x n u “ spec p M q , M “ UTU ˚ with and T “ D ` N D “ diag p x 1 ,..., x n q . � Lack of normality is generic: µ pt N “ 0 uq “ 0 � Process on spectrum melts N and D ( Ñ Bourgade–Dubach) � How about an O.-U. like diffusion leaving invariant µ ? 18/26

Dynamics of a planar Coulomb gas Ginibre process � State space D “ C n zY i ‰ j tp x 1 ,..., x n q P C n : x i “ x j u 19/26

Dynamics of a planar Coulomb gas Ginibre process � State space D “ C n zY i ‰ j tp x 1 ,..., x n q P C n : x i “ x j u � Boltzmann–Gibbs measure via change of variable µ p d x q “ e ´ n ř n i “ 1 | x i | 2 ź | x i ´ x j | 2 d x Z i ă j 19/26

Dynamics of a planar Coulomb gas Ginibre process � State space D “ C n zY i ‰ j tp x 1 ,..., x n q P C n : x i “ x j u � Boltzmann–Gibbs measure via change of variable ´ n ř n i “ 1 | x i | 2 ´ 2 ř 1 i ă j log | xi ´ xj | µ p d x q “ e d x Z 19/26

Dynamics of a planar Coulomb gas Ginibre process � State space D “ C n zY i ‰ j tp x 1 ,..., x n q P C n : x i “ x j u � Boltzmann–Gibbs measure via change of variable ´ n ř n i “ 1 | x i | 2 ´ 2 ř 1 i ă j log | xi ´ xj | µ p d x q “ e d x Z � Ginibre process on C n “ p R 2 q n ÿ X j t d t ´ 2 t ´ X i t | 2 d t ` d B i t t d X i t “ ´ 2 X i ? n . t ´ X j n | X i i ‰ j 19/26

Dynamics of a planar Coulomb gas Ginibre process � State space D “ C n zY i ‰ j tp x 1 ,..., x n q P C n : x i “ x j u � Boltzmann–Gibbs measure via change of variable ´ ¯ ř n ř 1 i “ 1 | x i | 2 ` 1 1 ´ β n i ă j log n 2 n | xi ´ xj | µ p d x q “ e d x Z � Ginibre process on C n “ p R 2 q n c α n ÿ X j t ´ X i t “ ´ 2 α n t d t ´ 2 α n d X i n X i t d B i t | 2 d t ` t . t ´ X j n | X i β n j ‰ i 19/26

Dynamics of a planar Coulomb gas Ginibre process � State space D “ C n zY i ‰ j tp x 1 ,..., x n q P C n : x i “ x j u � Boltzmann–Gibbs measure via change of variable ´ ¯ ř n ř 1 i “ 1 | x i | 2 ` 1 1 ´ β n i ă j log n 2 n | xi ´ xj | µ p d x q “ e d x Z � Ginibre process on C n “ p R 2 q n c α n ÿ X j t ´ X i t “ ´ 2 α n t d t ´ 2 α n d X i n X i t d B i t | 2 d t ` t . t ´ X j n | X i β n j ‰ i � RMT: β n “ n 2 19/26

Dynamics of a planar Coulomb gas Ginibre process � State space D “ C n zY i ‰ j tp x 1 ,..., x n q P C n : x i “ x j u � Boltzmann–Gibbs measure via change of variable ´ ¯ ř n ř 1 i “ 1 | x i | 2 ` 1 1 ´ β n i ă j log n 2 n | xi ´ xj | µ p d x q “ e d x Z � Ginibre process on C n “ p R 2 q n c α n ÿ X j t ´ X i t “ ´ 2 α n t d t ´ 2 α n d X i n X i t d B i t | 2 d t ` t . t ´ X j n | X i β n j ‰ i � RMT: β n “ n 2 � No convexity / Brascamp–Lieb / Bakry–Émery / Caffarelli 19/26

Dynamics of a planar Coulomb gas Ginibre process � State space D “ C n zY i ‰ j tp x 1 ,..., x n q P C n : x i “ x j u � Boltzmann–Gibbs measure via change of variable ´ ¯ ř n ř 1 i “ 1 | x i | 2 ` 1 1 ´ β n i ă j log n 2 n | xi ´ xj | µ p d x q “ e d x Z � Ginibre process on C n “ p R 2 q n c α n ÿ X j t ´ X i t “ ´ 2 α n t d t ´ 2 α n d X i n X i t d B i t | 2 d t ` t . t ´ X j n | X i β n j ‰ i � RMT: β n “ n 2 � No convexity / Brascamp–Lieb / Bakry–Émery / Caffarelli � No Hoffman–Wielandt for non-normal matrices 19/26

Dynamics of a planar Coulomb gas Ginibre process Well posedness � Explosion time T B D “ lim R Ñ8 T R where T R “ inf t t ě 0 : H p X t q ą R u 20/26

Dynamics of a planar Coulomb gas Ginibre process Well posedness � Explosion time T B D “ lim R Ñ8 T R where T R “ inf t t ě 0 : dist p X t , B D q ď 1 { R u 20/26

Dynamics of a planar Coulomb gas Ginibre process Well posedness � Explosion time T B D “ lim R Ñ8 T R where " * t ´ X j | X i i ‰ j | X i t ě 0 : max t | ě R or min t | ď 1 { R T R “ inf i 20/26

Dynamics of a planar Coulomb gas Ginibre process Well posedness Theorem (Well-posedness) For all X 0 “ x P D, n ě 2 , β n ą 0 , we have P p T B D “ `8q “ 1 . 20/26

Dynamics of a planar Coulomb gas Ginibre process Well posedness Theorem (Well-posedness) For all X 0 “ x P D, n ě 2 , β n ą 0 , we have P p T B D “ `8q “ 1 . � No constraint on β in contrast with Rogers–Shi for Dyson O.–U. ! 20/26

Dynamics of a planar Coulomb gas Ginibre process Well posedness Theorem (Well-posedness) For all X 0 “ x P D, n ě 2 , β n ą 0 , we have P p T B D “ `8q “ 1 . � No constraint on β in contrast with Rogers–Shi for Dyson O.–U. ! � Positivity and coercivity x P D H p x q ą 0 and inf x ÑB D H p x q “ `8 lim 20/26

Dynamics of a planar Coulomb gas Ginibre process Well posedness Theorem (Well-posedness) For all X 0 “ x P D, n ě 2 , β n ą 0 , we have P p T B D “ `8q “ 1 . � No constraint on β in contrast with Rogers–Shi for Dyson O.–U. ! � Positivity and coercivity x P D H p x q ą 0 and inf x ÑB D H p x q “ `8 lim � Cutoff W p x q “ r W p x q on | x | ă R with r W smooth 20/26

Dynamics of a planar Coulomb gas Ginibre process Well posedness Theorem (Well-posedness) For all X 0 “ x P D, n ě 2 , β n ą 0 , we have P p T B D “ `8q “ 1 . � No constraint on β in contrast with Rogers–Shi for Dyson O.–U. ! � Positivity and coercivity x P D H p x q ą 0 and inf x ÑB D H p x q “ `8 lim � Cutoff W p x q “ r W p x q on | x | ă R with r W smooth � Itô formula ˆż t ^ T ˙ E x p H p X t ^ T qq´ H p x q “ E x GH p X s q d s . 0 20/26

Dynamics of a planar Coulomb gas Ginibre process Well posedness Theorem (Well-posedness) For all X 0 “ x P D, n ě 2 , β n ą 0 , we have P p T B D “ `8q “ 1 . � No constraint on β in contrast with Rogers–Shi for Dyson O.–U. ! � Positivity and coercivity x P D H p x q ą 0 and inf x ÑB D H p x q “ `8 lim � Cutoff W p x q “ r W p x q on | x | ă R with r W smooth � Itô formula ˆż t ^ T ˙ E x p H p X t ^ T qq´ H p x q “ E x GH p X s q d s . 0 � R 1 T R ď t ď H p X t ^ T R q and GH ď c n on D 20/26

Dynamics of a planar Coulomb gas Ginibre process Poincaré inequality Theorem (Poincaré inequality) For any n, the law µ n satisfies a Poincaré inequality. 21/26

Dynamics of a planar Coulomb gas Ginibre process Poincaré inequality Theorem (Poincaré inequality) For any n, the law µ n satisfies a Poincaré inequality. � Proof using Lyapunov criterion 21/26

Dynamics of a planar Coulomb gas Ginibre process Poincaré inequality Theorem (Poincaré inequality) For any n, the law µ n satisfies a Poincaré inequality. � Proof using Lyapunov criterion � Bakry–Barthe–Cattiaux–Guillin Lyapunov approach ÿ n ÿ H p x q “ 1 | x i | 2 ` 1 1 log n 2 | x i ´ x j | n i “ 1 j ‰ i Gf “ α n ∆ f ´ α n ∇ H ¨ ∇ f β n 21/26

Dynamics of a planar Coulomb gas Ginibre process Poincaré inequality Theorem (Poincaré inequality) For any n, the law µ n satisfies a Poincaré inequality. � Proof using Lyapunov criterion � Bakry–Barthe–Cattiaux–Guillin Lyapunov approach ÿ n ÿ H p x q “ 1 | x i | 2 ` 1 1 log n 2 | x i ´ x j | n i “ 1 j ‰ i Gf “ α n ∆ f ´ α n ∇ H ¨ ∇ f β n G Ψ ď ´ c Ψ ` c 1 1 K 21/26

Dynamics of a planar Coulomb gas Ginibre process Poincaré inequality Theorem (Poincaré inequality) For any n, the law µ n satisfies a Poincaré inequality. � Proof using Lyapunov criterion � Bakry–Barthe–Cattiaux–Guillin Lyapunov approach ÿ n ÿ H p x q “ 1 | x i | 2 ` 1 1 log n 2 | x i ´ x j | n i “ 1 j ‰ i Gf “ α n ∆ f ´ α n ∇ H ¨ ∇ f β n G Ψ ď ´ c Ψ ` c 1 1 K Ψ “ e γ H 21/26

Dynamics of a planar Coulomb gas Ginibre process Poincaré inequality Theorem (Poincaré inequality) For any n, the law µ n satisfies a Poincaré inequality. � Proof using Lyapunov criterion � Bakry–Barthe–Cattiaux–Guillin Lyapunov approach ÿ n ÿ H p x q “ 1 | x i | 2 ` 1 1 log n 2 | x i ´ x j | n i “ 1 j ‰ i Gf “ α n ∆ f ´ α n ∇ H ¨ ∇ f β n G Ψ ď ´ c Ψ ` c 1 1 K Ψ “ e γ H 21/26

Dynamics of a planar Coulomb gas Ginibre process Uniform Poincaré for the one particle marginal Theorem (Uniform Poincaré for one-particle) If β n “ n 2 then the one-particle marginal of µ is log-concave and satisfies a Poincaré inequality with a constant uniform in n. 22/26

Dynamics of a planar Coulomb gas Ginibre process Uniform Poincaré for the one particle marginal Theorem (Uniform Poincaré for one-particle) If β n “ n 2 then the one-particle marginal of µ is log-concave and satisfies a Poincaré inequality with a constant uniform in n. � If β n “ n 2 then one particle marginal of µ has density z P C ÞÑ ϕ p z q “ e ´ n | z | 2 n ´ 1 n ℓ | z | 2 ℓ ÿ . π ℓ ! ℓ “ 0 22/26

Dynamics of a planar Coulomb gas Ginibre process Uniform Poincaré for the one particle marginal Theorem (Uniform Poincaré for one-particle) If β n “ n 2 then the one-particle marginal of µ is log-concave and satisfies a Poincaré inequality with a constant uniform in n. � If β n “ n 2 then one particle marginal of µ has density z P C ÞÑ ϕ p z q “ e ´ n | z | 2 n ´ 1 n ℓ | z | 2 ℓ ÿ . π ℓ ! ℓ “ 0 � Circular law ˇ ˇ ˇ ˇ 1 t| z |ď 1 u ˇ ˇ ˇ “ 0 . n Ñ8 sup lim ˇ ϕ p z q´ π z P K 22/26

Dynamics of a planar Coulomb gas Ginibre process Uniform Poincaré for the one particle marginal Theorem (Uniform Poincaré for one-particle) If β n “ n 2 then the one-particle marginal of µ is log-concave and satisfies a Poincaré inequality with a constant uniform in n. � If β n “ n 2 then one particle marginal of µ has density z P C ÞÑ ϕ p z q “ e ´ n | z | 2 n ´ 1 n ℓ | z | 2 ℓ ÿ . π ℓ ! ℓ “ 0 � Circular law ˇ ˇ ˇ ˇ 1 t| z |ď 1 u ˇ ˇ ˇ “ 0 . n Ñ8 sup lim ˇ ϕ p z q´ π z P K � The function z ÞÑ log ř n ´ 1 | z | 2 ℓ is concave! ℓ “ 0 ℓ ! 22/26