Superconcentration inequalities for centered Gaussian stationnary processes

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1 Superconcentration inequalities for centered Gaussian stationnary processes Kevin Tanguy Toulouse University June 21, / 22

2 Outline What is superconcentration? Convergence of extremes (Gaussian case). Superconcentration inequality for stationary Gaussian sequences. Tools and sketch of the proof. 2 / 22

3 What is superconcentration? 2 / 22

4 X = (X 1,..., X n ) N (0, Γ) 3 / 22

5 X = (X 1,..., X n ) N (0, Γ) with E [ Xi 2 ] = 1. 3 / 22

6 X = (X 1,..., X n ) N (0, Γ) with E [ Xi 2 ] = 1. Set M n = max i X i. 3 / 22

7 X = (X 1,..., X n ) N (0, Γ) with E [ Xi 2 ] = 1. Set M n = max i X i. Variance upper bound Var(M n )? 3 / 22

8 X = (X 1,..., X n ) N (0, Γ) with E [ Xi 2 ] = 1. Set M n = max i X i. Variance upper bound Var(M n )? Classical concentration theory Var(M n ) max Var(X i ). i 3 / 22

9 X = (X 1,..., X n ) N (0, Γ) with E [ Xi 2 ] = 1. Set M n = max i X i. Variance upper bound Var(M n )? Classical concentration theory Var(M n ) max Var(X i ). i sharp inequality, 3 / 22

10 X = (X 1,..., X n ) N (0, Γ) with E [ Xi 2 ] = 1. Set M n = max i X i. Variance upper bound Var(M n )? Classical concentration theory Var(M n ) max Var(X i ). i sharp inequality, does not depend on Γ. 3 / 22

11 Take Γ = Id ( X i s independent). 4 / 22

12 Take Γ = Id ( X i s independent). Var(M n ) 1 4 / 22

13 Take Γ = Id ( X i s independent). In fact, Var(M n ) Var(M n ) 1 C log n, C > 0 4 / 22

14 Classical theory suboptimal 5 / 22

15 Classical theory suboptimal Chatterjee s terminology : superconcentration phenomenon. 5 / 22

16 Classical theory suboptimal Chatterjee s terminology : superconcentration phenomenon. Lot of different models Largest eigenvalue in random matrix theory. 5 / 22

17 Classical theory suboptimal Chatterjee s terminology : superconcentration phenomenon. Lot of different models Largest eigenvalue in random matrix theory. 6 / 22

18 Classical theory suboptimal Chatterjee s terminology : superconcentration phenomenon. Lot of different models Largest eigenvalue in random matrix theory. Branching random walk. 6 / 22

19 Branching random walk. Take a binary tree of depth N. Put X e i.i.d. N (0, 1) on each edge e. Take a path π from the top to the bottom of the tree, X π = e π X e. 7 / 22

20 Branching random walk. Take a binary tree of depth N. Put X e i.i.d. N (0, 1) on each edge e. Take a path π from the top to the bottom of the tree, X π = e π X e. Classical theory : Var(max π X π ) N (X π N (0, N)). 7 / 22

21 Branching random walk. Take a binary tree of depth N. Put X e i.i.d. N (0, 1) on each edge e. Take a path π from the top to the bottom of the tree, X π = e π X e. Classical theory : Var(max π X π ) N (X π N (0, N)). In fact, Var(max π X π ) C [Bramson-Ding-Zeitouni]. 7 / 22

22 Classical theory suboptimal Chatterjee s terminology : superconcentration phenomenon. Lot of different models Largest eigenvalue in random matrix theory. Branching random walk. 8 / 22

23 Classical theory suboptimal Chatterjee s terminology : superconcentration phenomenon. Lot of different models Largest eigenvalue in random matrix theory. Branching random walk. Discrete Gaussian free field on Z d. 8 / 22

24 Classical theory suboptimal Chatterjee s terminology : superconcentration phenomenon. Lot of different models Largest eigenvalue in random matrix theory. Branching random walk. Discrete Gaussian free field on Z d. Free energy in spin glasses theory (SK model). 8 / 22

25 Classical theory suboptimal Chatterjee s terminology : superconcentration phenomenon. Lot of different models Largest eigenvalue in random matrix theory. Branching random walk. Discrete Gaussian free field on Z d. Free energy in spin glasses theory (SK model). First passage in percolation theory. 8 / 22

26 Classical theory suboptimal Chatterjee s terminology : superconcentration phenomenon. Lot of different models Largest eigenvalue in random matrix theory. Branching random walk. Discrete Gaussian free field on Z d. Free energy in spin glasses theory (SK model). First passage in percolation theory. Stationary Gaussian sequences. 8 / 22

27 What is superconcentration? Convergence of extremes (Gaussian case). 9 / 22

28 Stationary Gaussian sequences Theorem [Berman 64] (X i ) i 0 centered normalized stationary Gaussian sequence with covariance function φ. 10 / 22

29 Stationary Gaussian sequences Theorem [Berman 64] (X i ) i 0 centered normalized stationary Gaussian sequence with covariance function φ. Assume φ(n) = o(1/ log n) (n ), 10 / 22

30 Stationary Gaussian sequences Theorem [Berman 64] (X i ) i 0 centered normalized stationary Gaussian sequence with covariance function φ. Assume φ(n) = o(1/ log n) (n ), then a n (M n b n ) d G, (n ) 10 / 22

31 Stationary Gaussian sequences Theorem [Berman 64] (X i ) i 0 centered normalized stationary Gaussian sequence with covariance function φ. Assume φ(n) = o(1/ log n) (n ), then a n (M n b n ) d G, (n ) where a n = 2 log n 10 / 22

32 Stationary Gaussian sequences Theorem [Berman 64] (X i ) i 0 centered normalized stationary Gaussian sequence with covariance function φ. Assume φ(n) = o(1/ log n) (n ), then a n (M n b n ) d G, (n ) where a n = 2 log n and P(G t) = 1 e e t, t R (Gumbel). 10 / 22

33 Stationary Gaussian sequences Theorem [Berman 64] (X i ) i 0 centered normalized stationary Gaussian sequence with covariance function φ. Assume φ(n) = o(1/ log n) (n ), then a n (M n b n ) d G, (n ) where a n = 2 log n and P(G t) = 1 e e t, t R (Gumbel). Note : P(G t) e t 10 / 22

34 Results from classical theory Var(M n ) / 22

35 Results from classical theory Var(M n ) 1. Question : Correct bound? 11 / 22

36 Results from classical theory Var(M n ) 1. Question : Correct bound? Proposition [Chatterjee 14] Var(M n ) C log n. 11 / 22

37 What is superconcentration? Convergence of extremes (Gaussian case). Superconcentration inequality for stationary Gaussian sequences. 11 / 22

38 Question : (super)concentration inequality reflecting convergence of extremes? 12 / 22

39 Question : (super)concentration inequality reflecting convergence of extremes? Similar as Y 1,..., Y n i.i.d Rademacher ( P(Y1 = 1) = P(Y 1 = 1) = 1/2 ) 12 / 22

40 Question : (super)concentration inequality reflecting convergence of extremes? Similar as Y 1,..., Y n i.i.d Rademacher n 1/2 n i=1 X i d N (0, 1) ( P(Y1 = 1) = P(Y 1 = 1) = 1/2 ) (CLT) 12 / 22

41 Question : (super)concentration inequality reflecting convergence of extremes? Similar as Y 1,..., Y n i.i.d Rademacher n 1/2 n i=1 X i d N (0, 1) ( P(Y1 = 1) = P(Y 1 = 1) = 1/2 ) (CLT) P(n 1/2 n i=1 X i t) e t2 /2, t / 22

42 (Super)concentration inequality? Gaussian concentration inequality [Borel-Sudakov-Tsirelson 76] Take X N (0, Id) and F : R n R Lipschitz. 13 / 22

43 (Super)concentration inequality? Gaussian concentration inequality [Borel-Sudakov-Tsirelson 76] Take X N (0, Id) and F : R n R Lipschitz. P( F (X ) E[F (X )] ) t 2e t2 /2 F 2 Lip, t 0 13 / 22

44 (Super)concentration inequality? Gaussian concentration inequality [Borel-Sudakov-Tsirelson 76] Take X N (0, Id) and F : R n R Lipschitz. P( F (X ) E[F (X )] ) t 2e t2 /2 F 2 Lip, t 0 If F (x) = max i x i, F 2 Lip 1 ( P a n M n E[M n ] ) t 2e t2 2an 2, t 0, a n = 2 log n 13 / 22

45 (Super)concentration inequality? Gaussian concentration inequality [Borel-Sudakov-Tsirelson 76] Take X N (0, Id) and F : R n R Lipschitz. P( F (X ) E[F (X )] ) t 2e t2 /2 F 2 Lip, t 0 If F (x) = max i x i, F 2 Lip 1 ( P a n M n E[M n ] ) t 2e t2 2an 2, t 0, a n = 2 log n does not reflect Gumbel asymptotics. 13 / 22

46 Question : (super)concentration inequality reflecting convergence of extremes? 14 / 22

47 Question : (super)concentration inequality reflecting convergence of extremes? Theorem [T. 15] (X i ) i 0 centered stationary Gaussian sequence, covariance function φ. Assume φ(n) = o(1/ log n) (n ) and technicals hypothesis, then P ( M n E[M n ] t) 6e ct log n, t / 22

48 Question : (super)concentration inequality reflecting convergence of extremes? Theorem [T. 15] (X i ) i 0 centered stationary Gaussian sequence, covariance function φ. Assume φ(n) = o(1/ log n) (n ) and technicals hypothesis, then P ( M n E[M n ] t) 6e ct log n, t 0. Remark : same inequality holds with b n instead of E [M n ]. 14 / 22

49 Question : (super)concentration inequality reflecting convergence of extremes? Theorem [T. 15] (X i ) i 0 centered stationary Gaussian sequence, covariance function φ. Assume φ(n) = o(1/ log n) (n ) and technicals hypothesis, then P ( M n E[M n ] t) 6e ct log n, t 0. Remark : same inequality holds with b n instead of E [M n ]. ( ) P log n Mn b n t Ce ct, t / 22

50 Question : (super)concentration inequality reflecting convergence of extremes? Theorem [T. 15] (X i ) i 0 centered stationary Gaussian sequence, covariance function φ. Assume φ(n) = o(1/ log n) (n ) and technicals hypothesis, then P ( M n E[M n ] t) 6e ct log n, t 0. Remark : same inequality holds with b n instead of E [M n ]. ( ) P log n Mn b n t Ce ct, t 0. Reflects Gumbel asymptotics 14 / 22

51 Recall Theorem [Berman 64 ] (X i ) i 0 centered normalized stationary Gaussian sequence with covariance function φ. Assume φ(n) = o(log n) (n ), then a n (M n b n ) d G, (n ) where a n = 2 log n and P(G t) = 1 e e t, t R (Gumbel). Note : P(G t) e t 15 / 22

52 Theorem [T. 15 ] (X i ) i 0 centered stationary Gaussian, covariance function φ. Assume φ(n) = o(log n) (n ) and technicals hypothesis, then P ( M n E[M n ] t) 6e ct log n, t 0. Remark : same inequality holds with b n instead of E [M n ]. Reflects Gumbel asymptotics. 16 / 22

53 Theorem [T. 15 ] (X i ) i 0 centered stationary Gaussian, covariance function φ. Assume φ(n) = o(log n) (n ) and technicals hypothesis, then P ( M n E[M n ] t) 6e ct log n, t 0. Remark : same inequality holds with b n instead of E [M n ]. Reflects Gumbel asymptotics. Implies Var(max i X i ) C log n (optimal). 16 / 22

54 Tools Proof? 17 / 22

55 Tools Proof? Chatterjee s scheme of proof for the variance at the exponential level. 17 / 22

56 Tools Proof? Chatterjee s scheme of proof for the variance at the exponential level. General theorem implies superconcentration inequality for Gaussian stationary sequences. 17 / 22

57 Key steps of the proof Main steps Semigroup representation of the variance 18 / 22

58 Key steps of the proof Main steps Semigroup representation of the variance Hypercontractivity 18 / 22

59 Key steps of the proof Main steps Semigroup representation of the variance Hypercontractivity Proper use of a covering C of {1,... n}. 18 / 22

60 Interpolation, independant case Ornstein-Uhlenbeck semigroup P t f (x) = f (xe t + 1 e 2t y)dγ(y), t 0, x R n. R n 19 / 22

61 Interpolation, independant case Ornstein-Uhlenbeck semigroup P t f (x) = f (xe t + 1 e 2t y)dγ(y), t 0, x R n. R n Heat equation : t P t f = LP t f where L = x. 19 / 22

62 Interpolation, independant case Ornstein-Uhlenbeck semigroup P t f (x) = f (xe t + 1 e 2t y)dγ(y), t 0, x R n. R n Heat equation : t P t f = LP t f where L = x. Var(f ) = E[f 2 ] E[f ] 2 = 19 / 22

63 Interpolation, independant case Ornstein-Uhlenbeck semigroup P t f (x) = f (xe t + 1 e 2t y)dγ(y), t 0, x R n. R n Heat equation : t P t f = LP t f where L = x. Var(f ) = E[f 2 ] E[f ] 2 = E [ (P 0 f ) 2] ( E[P f ] ) 2 19 / 22

64 Interpolation, independant case Ornstein-Uhlenbeck semigroup P t f (x) = f (xe t + 1 e 2t y)dγ(y), t 0, x R n. R n Heat equation : t P t f = LP t f where L = x. Var(f ) = E[f 2 ] E[f ] 2 = E [ (P 0 f ) 2] ( E[P f ] ) 2 = 0 d dt E[ (P t f ) 2] dt 19 / 22

65 Interpolation, independant case Ornstein-Uhlenbeck semigroup P t f (x) = f (xe t + 1 e 2t y)dγ(y), t 0, x R n. R n Heat equation : t P t f = LP t f where L = x. Var(f ) = E[f 2 ] E[f ] 2 = E [ (P 0 f ) 2] ( E[P f ] ) 2 = = d dt E[ (P t f ) 2] dt E [ (P t f )( t P t f ) ] dt = / 22

66 Interpolation, independant case Ornstein-Uhlenbeck semigroup P t f (x) = f (xe t + 1 e 2t y)dγ(y), t 0, x R n. R n Heat equation : t P t f = LP t f where L = x. Var(f ) = E[f 2 ] E[f ] 2 = E [ (P 0 f ) 2] ( E[P f ] ) 2 = = 2 = 2 0 d dt E[ (P t f ) 2] dt 0 n e 2t 0 i=1 E [ (P t f )( t P t f ) ] dt =... E [ (P t i f ) 2] dt 19 / 22

67 Hypercontractivity P t hypercontractive, with p t = 1 + e 2t < 2. E [ (P t h) 2] 1/2 E[h p t ] 1/pt, 20 / 22

68 Hypercontractivity P t hypercontractive, with p t = 1 + e 2t < 2. E [ (P t h) 2] 1/2 E[h p t ] 1/pt, Apply to h = i (e θmn ), θ R and combined with concentration Lemma. 20 / 22

69 Hypercontractivity P t hypercontractive, with p t = 1 + e 2t < 2. E [ (P t h) 2] 1/2 E[h p t ] 1/pt, Apply to h = i (e θmn ), θ R and combined with concentration Lemma. General case similar, with further step to treat correlation. 20 / 22

70 Conclusion Similar results Gaussian stationary sequences with other covariance behavior. Gaussian stationary processes on R d. discrete Gaussian free field on Z d, d 3 21 / 22

71 Conclusion Similar results Gaussian stationary sequences with other covariance behavior. Gaussian stationary processes on R d. discrete Gaussian free field on Z d, d 3 (infinite group of finite type+volume growth condition ok). 21 / 22

72 Conclusion Similar results Gaussian stationary sequences with other covariance behavior. Gaussian stationary processes on R d. discrete Gaussian free field on Z d, d 3 (infinite group of finite type+volume growth condition ok). Uniform measure on the sphere S n 1 R n. Log-concave measures on R n with convexity assumptions. 21 / 22

73 Conclusion Hypercontractivity relevant Gaussian processes independent case (variance 1 log n ). 22 / 22

74 Conclusion Hypercontractivity relevant Gaussian processes independent case (variance 1 log n ). discrete Gaussian free field on Z 2 completely different behavior. Var(M n ) = O(1) [Bramson-Ding-Zeitouni] convergence in distribution Gumbel randomly shifted [Bramson-Ding-Zeitouni 15]. Hypercontractivity alone doesn t work. 22 / 22

75 Thank you for your attention. 22 / 22

Concentration inequalities: basics and some new challenges

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