Concentration for Coulomb gases and Coulomb transport inequalities

Transcription

1 Concentration for Coulomb gases and Coulomb transport inequalities Mylène Maïda U. Lille, Laboratoire Paul Painlevé Joint work with Djalil Chafaï and Adrien Hardy U. Paris-Dauphine and U. Lille ICERM, Providence - February 2018

2 Outline of the talk 2

3 Outline of the talk Coulomb gases : definition and known results 2

4 Outline of the talk Coulomb gases : definition and known results Concentration inequalities 2

5 Outline of the talk Coulomb gases : definition and known results Concentration inequalities Outline of the proof and Coulomb transport inequalities 2

6 We consider the Poisson equation Coulomb gases (d 2) g = c d δ 0. The fundamental solution is given by { log x for d = 2, g(x) := 1 for d 3. x d 2 A gas of N particles interacting according to the Coulomb law would have an energy given by H N (x 1,..., x N ) := i j N g(x i x j ) + N V (x i ). i=1 3

7 We denote by P N V,β the Gibbs measure on (Rd ) N associated to this energy : dp N V,β(x 1,..., x N ) = 1 ZV N e β 2 H N (x 1,...,x N ) dx 1,..., dx N,β 4

8 We denote by P N V,β the Gibbs measure on (Rd ) N associated to this energy : dp N V,β(x 1,..., x N ) = 1 ZV N e β 2 H N (x 1,...,x N ) dx 1,..., dx N,β Example (Ginibre) : let M N be an N by N matrix with iid entries with law N C (0, 1 N ), then the eigenvalues have joint law PN x 2,2 with dp N x 2,2 (x 1,..., x N ) i<j x i x j 2 e N N i=1 x i 2 4

9 Global asymptotics of the empirical measure 5

10 Global asymptotics of the empirical measure Our main subject of study is the empirical measure ˆµ N := 1 N N δ xi. i=1 5

11 Global asymptotics of the empirical measure Our main subject of study is the empirical measure ˆµ N := 1 N N δ xi. i=1 One can rewrite H N (x 1,..., x N ) = N 2 E V (ˆµ N) ( := N 2 g(x y)ˆµ N (dx)ˆµ N (dy) + x y ) V (x)ˆµ N (dx). 5

12 Global asymptotics of the empirical measure Our main subject of study is the empirical measure ˆµ N := 1 N N δ xi. i=1 One can rewrite H N (x 1,..., x N ) = N 2 E V (ˆµ N) ( := N 2 g(x y)ˆµ N (dx)ˆµ N (dy) + x y ) V (x)ˆµ N (dx). More generally, one can define, for any µ P(R d ), E V (µ) := (g(x y) + 12 V (x) + 12 ) V (y) µ(dx)µ(dy). 5

13 If V is admissible, there exists a unique minimizer µ V of the functional E V and it is compactly suppported. 6

14 If V is admissible, there exists a unique minimizer µ V of the functional E V and it is compactly suppported. If V is continuous, one can check that almost surely ˆµ N converges weakly to µ V. 6

15 If V is admissible, there exists a unique minimizer µ V of the functional E V and it is compactly suppported. If V is continuous, one can check that almost surely ˆµ N converges weakly to µ V. A large deviation principle, due to Chafaï, Gozlan and Zitt is also available : 6

16 If V is admissible, there exists a unique minimizer µ V of the functional E V and it is compactly suppported. If V is continuous, one can check that almost surely ˆµ N converges weakly to µ V. A large deviation principle, due to Chafaï, Gozlan and Zitt is also available : for d a distance that metrizes the weak topology (for example Fortet-Mourier) one has in particular 1 N 2 log PN V,β(d(ˆµ N, µ V ) r) β inf N 2 (E V (µ) E V (µ V )). d(µ,µ V ) r 6

17 If V is admissible, there exists a unique minimizer µ V of the functional E V and it is compactly suppported. If V is continuous, one can check that almost surely ˆµ N converges weakly to µ V. A large deviation principle, due to Chafaï, Gozlan and Zitt is also available : for d a distance that metrizes the weak topology (for example Fortet-Mourier) one has in particular 1 N 2 log PN V,β(d(ˆµ N, µ V ) r) β inf N 2 (E V (µ) E V (µ V )). d(µ,µ V ) r What about concentration? 6

18 If V is admissible, there exists a unique minimizer µ V of the functional E V and it is compactly suppported. If V is continuous, one can check that almost surely ˆµ N converges weakly to µ V. A large deviation principle, due to Chafaï, Gozlan and Zitt is also available : for d a distance that metrizes the weak topology (for example Fortet-Mourier) one has in particular 1 N 2 log PN V,β(d(ˆµ N, µ V ) r) β inf N 2 (E V (µ) E V (µ V )). d(µ,µ V ) r What about concentration? Local behavior extensively using several variations of the concept of renormalized energy (see in particular Simona s talk this morning). 6

19 Concentration estimates 7

20 Concentration estimates We will consider both the bounded Lipschitz distance d BL and the Wassertein W 1 distance, where we recall that d BL (µ, ν) = f d(µ ν); W 1 (µ, ν) = sup f d(µ ν) f Lip 1 sup f 1 f Lip 1 7

21 Concentration estimates We will consider both the bounded Lipschitz distance d BL and the Wassertein W 1 distance, where we recall that d BL (µ, ν) = f d(µ ν); W 1 (µ, ν) = sup f d(µ ν) f Lip 1 sup f 1 f Lip 1 Theorem If V is C 2 and V and V satisfy some growth conditions, 7

22 Concentration estimates We will consider both the bounded Lipschitz distance d BL and the Wassertein W 1 distance, where we recall that d BL (µ, ν) = f d(µ ν); W 1 (µ, ν) = sup f d(µ ν) f Lip 1 sup f 1 f Lip 1 Theorem If V is C 2 and V and V satisfy some growth conditions,then there exist a > 0, b R, c(β) such that for all N 2 and for all r > 0, P N V,β(d(ˆµ N, µ V ) r) e aβn2 r 2 +1 d=2 β 4 N log N+bβN2 2 d +c(β)n 7

23 More insight about the growth conditions : 8

24 More insight about the growth conditions : V has to grow not faster then V 8

25 More insight about the growth conditions : V has to grow not faster then V as soon as the model is well defined, the concentration estimate holds for d BL 8

26 More insight about the growth conditions : V has to grow not faster then V as soon as the model is well defined, the concentration estimate holds for d BL if moreover V (x) c x κ, for some κ > 0, we can say more about c(β) near 0 and near 8

27 More insight about the growth conditions : V has to grow not faster then V as soon as the model is well defined, the concentration estimate holds for d BL if moreover V (x) c x κ, for some κ > 0, we can say more about c(β) near 0 and near if moreover V (x) c x 2, the concentration estimate holds for W 1 8

28 More insight about the growth conditions : V has to grow not faster then V as soon as the model is well defined, the concentration estimate holds for d BL if moreover V (x) c x κ, for some κ > 0, we can say more about c(β) near 0 and near if moreover V (x) c x 2, the concentration estimate holds for W 1 the latter allows to get the almost sure convergence of W 1 (ˆµ N, µ V ) to zero down to β log N N 8

29 More insight about the growth conditions : V has to grow not faster then V as soon as the model is well defined, the concentration estimate holds for d BL if moreover V (x) c x κ, for some κ > 0, we can say more about c(β) near 0 and near if moreover V (x) c x 2, the concentration estimate holds for W 1 the latter allows to get the almost sure convergence of W 1 (ˆµ N, µ V ) to zero down to β log N N if the potential is subquadratic, a, b and c(β) can be made more explicit. 8

30 A few more comments : 9

31 A few more comments : Possible { rewriting : there exist r 0, C > 0, such that for all N 2, if log N r r 0 N if d = 2 r 0 N 1/d if d 3, P N V,β(d(ˆµ N, µ V ) r) e CN2 r 2. 9

32 A few more comments : Possible { rewriting : there exist r 0, C > 0, such that for all N 2, if log N r r 0 N if d = 2 r 0 N 1/d if d 3, P N V,β(d(ˆµ N, µ V ) r) e CN2 r 2. thanks to the large deviation results of CGZ, we know that we are in the right scale 9

33 A few more comments : Possible { rewriting : there exist r 0, C > 0, such that for all N 2, if log N r r 0 N if d = 2 r 0 N 1/d if d 3, P N V,β(d(ˆµ N, µ V ) r) e CN2 r 2. thanks to the large deviation results of CGZ, we know that we are in the right scale for Ginibre, the constants can be computed explicitely ; improves on previous results based on determinantal structure (can we use the Gaussian nature of the entries?) 9

34 A few more comments : Possible { rewriting : there exist r 0, C > 0, such that for all N 2, if log N r r 0 N if d = 2 r 0 N 1/d if d 3, P N V,β(d(ˆµ N, µ V ) r) e CN2 r 2. thanks to the large deviation results of CGZ, we know that we are in the right scale for Ginibre, the constants can be computed explicitely ; improves on previous results based on determinantal structure (can we use the Gaussian nature of the entries?) non optimal local laws can be deduced 9

35 Outline of the proof 10

36 Outline of the proof Special case when V = δ K, for K a compact set of R d. 10

37 Outline of the proof Special case when V = δ K, for K a compact set of R d. First ingredient : lower bound on the partition function. There exists C such that Z N V,β e β 2 N2 E V (µ V ) NC. 10

38 Outline of the proof Special case when V = δ K, for K a compact set of R d. First ingredient : lower bound on the partition function. There exists C such that Z N V,β e β 2 N2 E V (µ V ) NC. For A (R d ) N, P N V,β(A) = 1 e β 2 H N (x 1,...,x N ) dx 1... dx N Z N V,β A 10

39 Outline of the proof Special case when V = δ K, for K a compact set of R d. First ingredient : lower bound on the partition function. There exists C such that Z N V,β e β 2 N2 E V (µ V ) NC. For A (R d ) N, P N V,β(A) = 1 Z N V,β e NC A A e β 2 H N (x 1,...,x N ) dx 1... dx N e β 2 N2 (E V (ˆµ N ) E V (µ V )) dx 1... dx N 10

40 Outline of the proof Special case when V = δ K, for K a compact set of R d. First ingredient : lower bound on the partition function. There exists C such that Z N V,β e β 2 N2 E V (µ V ) NC. For A (R d ) N, P N V,β(A) = 1 Z N V,β e NC A A e β 2 H N (x 1,...,x N ) dx 1... dx N e β 2 N2 (E V (ˆµ N ) E V (µ V )) dx 1... dx N e NC e β 2 N2 inf A (E V (ˆµ N ) E V (µ V )) (volk) N 10

41 Outline of the proof Special case when V = δ K, for K a compact set of R d. First ingredient : lower bound on the partition function. There exists C such that Z N V,β e β 2 N2 E V (µ V ) NC. For A (R d ) N, P N V,β(A) = 1 Z N V,β e NC A A e β 2 H N (x 1,...,x N ) dx 1... dx N e β 2 N2 (E V (ˆµ N ) E V (µ V )) dx 1... dx N e NC e β 2 N2 inf A (E V (ˆµ N ) E V (µ V )) (volk) N We want to take A := {d(ˆµ N, µ V ) r}. 10

42 Coulomb transport inequalities 11

43 Coulomb transport inequalities We aim at an inequality of the type : for any µ P(R d ), d(µ, µ V ) 2 C V (E V (µ) E V (µ V )). 11

44 Coulomb transport inequalities We aim at an inequality of the type : for any µ P(R d ), d(µ, µ V ) 2 C V (E V (µ) E V (µ V )). This inequality is the Coulomb counterpart of Talagrand T 1 inequality : ν satisfies T 1 iif there exists C > 0 such that for any µ P(R d ), W 1 (µ, ν) 2 CH(µ ν). 11

45 Coulomb transport inequalities We aim at an inequality of the type : for any µ P(R d ), d(µ, µ V ) 2 C V (E V (µ) E V (µ V )). This inequality is the Coulomb counterpart of Talagrand T 1 inequality : ν satisfies T 1 iif there exists C > 0 such that for any µ P(R d ), W 1 (µ, ν) 2 CH(µ ν). In 1D, previous results by Biane-Voiculescu, Hiai-Petz-Ueda, Ledoux-Popescu, M.-Maurel-Segala 11

46 Coulomb transport inequalities We aim at an inequality of the type : for any µ P(R d ), d(µ, µ V ) 2 C V (E V (µ) E V (µ V )). This inequality is the Coulomb counterpart of Talagrand T 1 inequality : ν satisfies T 1 iif there exists C > 0 such that for any µ P(R d ), W 1 (µ, ν) 2 CH(µ ν). In 1D, previous results by Biane-Voiculescu, Hiai-Petz-Ueda, Ledoux-Popescu, M.-Maurel-Segala To point out what is specific to the Coulombian nature of the interaction, we will show the following local version of our inequality : 11

47 Coulomb transport inequalities We aim at an inequality of the type : for any µ P(R d ), d(µ, µ V ) 2 C V (E V (µ) E V (µ V )). This inequality is the Coulomb counterpart of Talagrand T 1 inequality : ν satisfies T 1 iif there exists C > 0 such that for any µ P(R d ), W 1 (µ, ν) 2 CH(µ ν). In 1D, previous results by Biane-Voiculescu, Hiai-Petz-Ueda, Ledoux-Popescu, M.-Maurel-Segala To point out what is specific to the Coulombian nature of the interaction, we will show the following local version of our inequality : Proposition For any compact set D of R d, there exists C D such that for any µ, ν P(D) such that E(µ) < and E(ν) <, W 1 (µ, ν) 2 C D E(µ ν). 11

48 Proof of the Proposition 12

49 Proof of the Proposition If µ and ν have their support in D, there exists D + such that W 1 (µ, ν) = f d(µ ν) sup f Lip 1 f C(D +) 12

50 Proof of the Proposition If µ and ν have their support in D, there exists D + such that W 1 (µ, ν) = f d(µ ν) sup f Lip 1 f C(D +) By a density argument, one can asumme that η := µ ν has a smooth density h, 12

51 Proof of the Proposition If µ and ν have their support in D, there exists D + such that W 1 (µ, ν) = f d(µ ν) sup f Lip 1 f C(D +) By a density argument, one can asumme that η := µ ν has a smooth density h, let U η := g h. 12

52 Proof of the Proposition If µ and ν have their support in D, there exists D + such that W 1 (µ, ν) = f d(µ ν) sup f Lip 1 f C(D +) By a density argument, one can asumme that η := µ ν has a smooth density h, let U η := g h. From the Poisson equation, we know that for any smooth function ϕ, ϕ(y)g(y)dy = c d ϕ(0). 12

53 Proof of the Proposition If µ and ν have their support in D, there exists D + such that W 1 (µ, ν) = f d(µ ν) sup f Lip 1 f C(D +) By a density argument, one can asumme that η := µ ν has a smooth density h, let U η := g h. From the Poisson equation, we know that for any smooth function ϕ, ϕ(y)g(y)dy = c d ϕ(0). Choosing ϕ(y) = h(x y), we get that h(x y)g(y)dy = c d h(x) 12

54 Proof of the Proposition If µ and ν have their support in D, there exists D + such that W 1 (µ, ν) = f d(µ ν) sup f Lip 1 f C(D +) By a density argument, one can asumme that η := µ ν has a smooth density h, let U η := g h. From the Poisson equation, we know that for any smooth function ϕ, ϕ(y)g(y)dy = c d ϕ(0). Choosing ϕ(y) = h(x y), we get that h(x y)g(y)dy = c d h(x) But we also have h(x y)g(y)dy = g(x y)h(y)dy = U η (x). 12

55 Therefore, for any Lipschitz function with support in D + f dη = 1 f (x) U η (x)dx = 1 f (x) U η (x)dx c d c d 13

56 Therefore, for any Lipschitz function with support in D + f dη = 1 f (x) U η (x)dx = 1 f (x) U η (x)dx c d c d We can now conclude as f (x) U η (x)dx D f U η U η + D + ( ) 1/2 vol(d + ) U η 2. 13

57 Therefore, for any Lipschitz function with support in D + f dη = 1 f (x) U η (x)dx = 1 f (x) U η (x)dx c d c d We can now conclude as f (x) U η (x)dx D f U η U η + D + ( ) 1/2 vol(d + ) U η 2. But U η 2 = c d E(η). 13

58 Thank you for your attention! 14