Top eigenvalue of a random matrix: Large deviations

Transcription

1 Top eigenvalue of a random matrix: Large deviations Satya N. Majumdar Laboratoire de Physique Théorique et Modèles Statistiques,CNRS, Université Paris-Sud, France

2 First Appearence of Random Matrices

3 Covariance Matrix 1 X = 2 phys. X 11 X 21 math X 12 X 22 in general (MxN) 3 X 31 X 33 X t = X 11 X 21 X 31 X 12 X 22 X33 in general (NxM) W= X t X = X 11 + X + 21 X 31 X 11 X 12 + X 21 X 22 + X 31 X 33 X 12 X 11 + X 22 X 21 + X 33 X 31 X X 22 + X33 (NxN) COVARIANCE MATRIX (unnormalized)

4 Covariance Matrix 1 X = 2 phys. X 11 X 21 math X 12 X 22 in general (MxN) 3 X 31 X 33 X t = X 11 X 21 X 31 X 12 X 22 X33 in general (NxM) W= X t X = X 11 + X + 21 X 31 X 11 X 12 + X 21 X 22 + X 31 X 33 X 12 X 11 + X 22 X 21 + X 33 X 31 X X 22 + X33 (NxN) COVARIANCE MATRIX (unnormalized) Null model random data: X random (M N) matrix W = X t X random N N matrix (Wishart, 1928)

5 RMT in Nuclear Physics: Eugene Wigner

6

7 Random Matrices in Nuclear Physics spectra of heavy nuclei U E 232 Th E WIGNER ( 50) : replace complex H by random matrix DYSON, GAUDIN, MEHTA,...

8 Applications of Random Matrices Physics: nuclear physics, quantum chaos, disorder and localization, mesoscopic transport, optics/lasers, quantum entanglement, neural networks, gauge theory, QCD, matrix models, cosmology, string theory, statistical physics (growth models, interface, directed polymers...),... Mathematics: Riemann zeta function (number theory), free probability theory, combinatorics and knot theory, determinantal points processes, integrable systems,... Statistics: multivariate statistis, principal component analysis (PCA), image processing, data compression, Bayesian model selection,... Information Theory: signal processing, wireless communications,.. Biology: sequence matching, RNA folding, gene expression network... Economics and Finance: time series analysis,... Recent Ref: The Oxford Handbook of Random Matrix Theory ed. by G. Akemann, J. Baik and P. Di Francesco (21)

9 Spectral Statistics in Random Matrix Theory (RMT) Working model: real, symmetric N N Gaussian random matrix J 11 J J 1N J = J 12 J J 2N J 1N J 2N... J NN Prob.[J] exp 1 2 i,j J 2 ij = exp [ 1 2 Tr ( J 2)] invariant under rotation

10 Spectral Statistics in Random Matrix Theory (RMT) Working model: real, symmetric N N Gaussian random matrix J 11 J J 1N J = J 12 J J 2N J 1N J 2N... J NN Prob.[J] exp 1 2 i,j J 2 ij = exp [ 1 2 Tr ( J 2)] invariant under rotation N real eigenvalues: λ 1, λ 2,..., λ N strongly correlated Spectral statistics in RMT statistics of {λ 1, λ 2,..., λ N }

11 Top Eigenvalue of a random matrix λ max Recent excitements in statistical physics & mathematics on λ max the top eigenvalue of a random matrix

12 Top Eigenvalue of a random matrix λ max Recent excitements in statistical physics & mathematics on λ max the top eigenvalue of a random matrix large (left) Pr( λ max ) λ max typical TRACY WIDOM large (right) Typical fluctuations (small) Tracy-Widom distribution ubiquitous [directed polymer, random permutation, growth models, KPZ equation, sequence alignment,...]

13 Top Eigenvalue of a random matrix λ max Recent excitements in statistical physics & mathematics on λ max the top eigenvalue of a random matrix large (left) Pr( λ max ) λ max typical TRACY WIDOM large (right) Typical fluctuations (small) Tracy-Widom distribution ubiquitous [directed polymer, random permutation, growth models, KPZ equation, sequence alignment,...] This talk Atypical rare fluctuations large deviation functions

14 Top Eigenvalue of a random matrix λ max Recent excitements in statistical physics & mathematics on λ max the top eigenvalue of a random matrix large (left) Pr( λ max ) λ max typical TRACY WIDOM large (right) Typical fluctuations (small) Tracy-Widom distribution ubiquitous [directed polymer, random permutation, growth models, KPZ equation, sequence alignment,...] This talk Atypical rare fluctuations large deviation functions 3-rd order phase transition

15 Plan: Top eigenvalue λ max of a Gaussian random matrix stability of a large complex system

16 Plan: Top eigenvalue λ max of a Gaussian random matrix stability of a large complex system Prob. distr. of λ max Coulomb gas with a wall

17 Plan: Top eigenvalue λ max of a Gaussian random matrix stability of a large complex system Prob. distr. of λ max Coulomb gas with a wall Limiting distribution: Tracy-Widom physics of large deviation tails: left tail (pushed Coulomb gas) right tail (unpushed Coulomb gas)

18 Plan: Top eigenvalue λ max of a Gaussian random matrix stability of a large complex system Prob. distr. of λ max Coulomb gas with a wall Limiting distribution: Tracy-Widom physics of large deviation tails: left tail (pushed Coulomb gas) right tail (unpushed Coulomb gas) 3-rd order phase transition: Pushed Unpushed (Unstable) (Stable)

19 Plan: Top eigenvalue λ max of a Gaussian random matrix stability of a large complex system Prob. distr. of λ max Coulomb gas with a wall Limiting distribution: Tracy-Widom physics of large deviation tails: left tail (pushed Coulomb gas) right tail (unpushed Coulomb gas) 3-rd order phase transition: Pushed Unpushed (Unstable) (Stable) Similar 3-rd order phase transition in Yang-Mills gauge theory and other systems ubiquitous

20 Plan: Top eigenvalue λ max of a Gaussian random matrix stability of a large complex system Prob. distr. of λ max Coulomb gas with a wall Limiting distribution: Tracy-Widom physics of large deviation tails: left tail (pushed Coulomb gas) right tail (unpushed Coulomb gas) 3-rd order phase transition: Pushed Unpushed (Unstable) (Stable) Similar 3-rd order phase transition in Yang-Mills gauge theory and other systems ubiquitous Extension to Wishart matrices Recent experiments in coupled fiber lasers system

21 Plan: Top eigenvalue λ max of a Gaussian random matrix stability of a large complex system Prob. distr. of λ max Coulomb gas with a wall Limiting distribution: Tracy-Widom physics of large deviation tails: left tail (pushed Coulomb gas) right tail (unpushed Coulomb gas) 3-rd order phase transition: Pushed Unpushed (Unstable) (Stable) Similar 3-rd order phase transition in Yang-Mills gauge theory and other systems ubiquitous Extension to Wishart matrices Recent experiments in coupled fiber lasers system Summary and Generalizations

22 I. Why λ max?

23 Stability of a Large Complex System

24 Linear Stability of a Large Complex (Randomly Connected) System Consider a stable non-interacting population of N species with equlibrium density ρ i

25 Linear Stability of a Large Complex (Randomly Connected) System Consider a stable non-interacting population of N species with equlibrium density ρ i Stable: x i = ρ i ρ i small disturbed density dx i /dt = x i relaxes back to 0

26 Linear Stability of a Large Complex (Randomly Connected) System Consider a stable non-interacting population of N species with equlibrium density ρ i Stable: x i = ρ i ρ i small disturbed density dx i /dt = x i relaxes back to 0 Now switch on the interaction between species

27 Linear Stability of a Large Complex (Randomly Connected) System Consider a stable non-interacting population of N species with equlibrium density ρ i Stable: x i = ρ i ρ i small disturbed density dx i /dt = x i relaxes back to 0 Now switch on the interaction between species dx i /dt = x i + α N j=1 J ij x j J ij (N N) random interaction matrix α interaction strength

28 Linear Stability of a Large Complex (Randomly Connected) System Consider a stable non-interacting population of N species with equlibrium density ρ i Stable: x i = ρ i ρ i small disturbed density dx i /dt = x i relaxes back to 0 Now switch on the interaction between species dx i /dt = x i + α N j=1 J ij x j J ij (N N) random interaction matrix α interaction strength Question: What is the probabality that the system remains stable once the interaction is switched on? (R.M. May, Nature, 238, 413, 1972)

29 Stability Criterion d linear stability: dt [x] = [αj I ][x] (J random interaction matrix)

30 Stability Criterion d linear stability: dt [x] = [αj I ][x] (J random interaction matrix) Let {λ 1, λ 2,, λ N } eigenvalues of the matrix J

31 Stability Criterion d linear stability: dt [x] = [αj I ][x] (J random interaction matrix) Let {λ 1, λ 2,, λ N } eigenvalues of the matrix J Stable if αλ i < 1 for all i = 1, 2,, N λ max < 1 α = w stability criterion w inverse interaction strength

32 Stability Criterion d linear stability: dt [x] = [αj I ][x] (J random interaction matrix) Let {λ 1, λ 2,, λ N } eigenvalues of the matrix J Stable if αλ i < 1 for all i = 1, 2,, N λ max < 1 α = w stability criterion w inverse interaction strength Prob.(the system is stable)=prob.[λ max < w] =P(w, N) Cumulative distribution of the top eigenvalue

33 Stable-Unstable Phase Transition as N Assuming that the interaction matrix J ij Real Symmetric Gaussian [ Prob.[J ij ] exp N ] 2 i,j J2 ij exp [ N2 Tr(J2 ) ]

34 Stable-Unstable Phase Transition as N Assuming that the interaction matrix J ij Real Symmetric Gaussian [ Prob.[J ij ] exp N ] 2 i,j J2 ij exp [ N2 Tr(J2 ) ] May observed a sharp phase transition as N : w = 1 α > 2 Stable (weakly interacting) w = 1 α < 2 Unstable (strongly interacting) Prob.(the system is stable)=prob.[λ max < w] =P(w, N) P( w, N) 1 STABLE 0 w UNSTABLE 2 = 1/α

35 Finite but Large N: Prob.(the system is stable)=prob.[λ max < w] =P(w, N) What happens for finite but large N?

36 Finite but Large N: Prob.(the system is stable)=prob.[λ max < w] =P(w, N) What happens for finite but large N? P( w, N) = Prob.[ λ max < w ] 1 STABLE finite but large N 0 UNSTABLE 2 w Is there any thermodynamic sense to this phase transition? What is the analogue of free energy? What is the order of this phase transition?

37 II. Summary of Results

38 For Large but Finite N: Summary of Results P( w, N) = Prob.[ λ max < w ] 1 STABLE finite but large N width of O (N 2/3 ) UNSTABLE 2 w

39 For Large but Finite N: Summary of Results P( w, N) = Prob.[ λ max < w ] 1 STABLE finite but large N width of O (N 2/3 ) UNSTABLE 2 w P(w, N) exp [ N 2 Φ (w) +... ] for 2 w O(1) [ ( F 1 2 N 2/3 w )] 2 for w 2 O(N 2/3 ) 1 exp [ NΦ + (w) +...] for 2 w O(1)

40 For Large but Finite N: Summary of Results P( w, N) = Prob.[ λ max < w ] 1 STABLE finite but large N width of O (N 2/3 ) UNSTABLE 2 w P(w, N) exp [ N 2 Φ (w) +... ] for 2 w O(1) [ ( F 1 2 N 2/3 w )] 2 for w 2 O(N 2/3 ) 1 exp [ NΦ + (w) +...] for 2 w O(1) Crossover function: F 1 (z) Tracy-Widom (1994) Exact tail functions: Φ (w) (Dean & S.M. 2006, S.M. & Vergassola 2009) cf: Th. 6.2 in Ben Arous, Dembo and Guionnet 20 Higher order corrections: (Borot, Eynard, S.M., & Nadal 21, Nadal & S.M. 21, Borot & Nadal 22)

41 Exact Left and Right Large Deviation Function Using Coulomb gas + Saddle point method for large N:

42 Exact Left and Right Large Deviation Function Using Coulomb gas + Saddle point method for large N: Left large deviation function: 1 [ Φ (w) = 36w 2 w 4 (15w + w 3 ) w ( + 27 ln(18) 2 ln(w + )] 6 + w 2 ) where w < 2 [D. S. Dean & S.M., PRL, 97, 1602 (2006)]

43 Exact Left and Right Large Deviation Function Using Coulomb gas + Saddle point method for large N: Left large deviation function: 1 [ Φ (w) = 36w 2 w 4 (15w + w 3 ) w ( + 27 ln(18) 2 ln(w + )] 6 + w 2 ) where w < 2 [D. S. Dean & S.M., PRL, 97, 1602 (2006)] In particular, as w 2 (from left), Φ (w) ( 2 w) 3

44 Exact Left and Right Large Deviation Function Using Coulomb gas + Saddle point method for large N: Left large deviation function: 1 [ Φ (w) = 36w 2 w 4 (15w + w 3 ) w ( + 27 ln(18) 2 ln(w + )] 6 + w 2 ) where w < 2 [D. S. Dean & S.M., PRL, 97, 1602 (2006)] In particular, as w 2 (from left), Φ (w) ( 2 w) 3 Right large deviation function: Φ + (w) = 1 2 w w ln [ w ] w where w > 2 [S.M. & M. Vergassola, PRL, 102, 0606 (2009)]

45 Exact Left and Right Large Deviation Function Using Coulomb gas + Saddle point method for large N: Left large deviation function: 1 [ Φ (w) = 36w 2 w 4 (15w + w 3 ) w ( + 27 ln(18) 2 ln(w + )] 6 + w 2 ) where w < 2 [D. S. Dean & S.M., PRL, 97, 1602 (2006)] In particular, as w 2 (from left), Φ (w) ( 2 w) 3 Right large deviation function: Φ + (w) = 1 2 w w ln [ w ] w where w > 2 [S.M. & M. Vergassola, PRL, 102, 0606 (2009)] As w 2 (from right), Φ + (w) 27/4 3 (w 2) 3/2

46 Large Deviation Functions These large deviation functions Φ ± (w) have been found useful in a large variety of problems: [Fyodorov 2004, Fyodorov & Williams 2007, Bray & Dean 2007, Auffinger, Ben Arous & Cerny 20, Fydorov & Nadal stationary points on random Gaussian surfaces and spin glass landscapes] [Cavagna, Garrahan, Giardina 2000,... Glassy systems] [Susskind 2003, Douglas et. al. 2004, Aazami & Easther 2006, Marsh et. al. 21,... String theory & Cosmology] [Beltrani 2007, Dedieu & Malajovich, 2007, Houdre Random Polynomials, Random Words (Young diagrams)

47 3-rd Order Phase Transition exp { N 2 Φ (w) +... } for w < 2 (unstable) P(w, N) 1 exp { NΦ + (w) +...} for w > 2 (stable)

48 3-rd Order Phase Transition exp { N 2 Φ (w) +... } for w < 2 (unstable) P(w, N) 1 exp { NΦ + (w) +...} for w > 2 (stable) lim 1 Φ (w) ( 2 w) 3 as w 2 ln [P(w, N)] = N N2 0 as w 2 + analogue of the free energy difference

49 3-rd Order Phase Transition exp { N 2 Φ (w) +... } for w < 2 (unstable) P(w, N) 1 exp { NΦ + (w) +...} for w > 2 (stable) lim 1 Φ (w) ( 2 w) 3 as w 2 ln [P(w, N)] = N N2 0 as w 2 + analogue of the free energy difference [ ln P]/N 2 Tracy Widom finite N ( large) 2 N limit ~ ( 2 _ w) 3 w

50 3-rd Order Phase Transition exp { N 2 Φ (w) +... } for w < 2 (unstable) P(w, N) 1 exp { NΦ + (w) +...} for w > 2 (stable) lim 1 Φ (w) ( 2 w) 3 as w 2 ln [P(w, N)] = N N2 0 as w 2 + analogue of the free energy difference Tracy Widom 3-rd derivative discontinuous [ ln P]/N 2 Crossover: N, w 2 keeping finite N ( large) (w 2) N 2/3 fixed 2 N limit ~ ( 2 _ w) 3 [ w P(w, N) F ( 1 2 N 2/3 w 2 )] Tracy-Widom

51 Large N Phase Transition: Phase Diagram 1 N crossover STABLE ( weakly interacting ) UNSTABLE ( strongly interacting ) 0 α = 1 w 1 2

52

53

54 Large N Phase Transition: Phase Diagram U(N) lattice gauge theory in 2 d GROSS WITTEN WADIA transition (1980) 1 N crossover WEAK STRONG 0 g c coupling strength g

55 Large N Phase Transition: Phase Diagram U(N) lattice gauge theory in 2 d GROSS WITTEN WADIA transition (1980) 1 N 1 N crossover crossover WEAK STRONG STABLE ( weakly interacting ) UNSTABLE ( strongly interacting ) 0 g c coupling strength g 0 1 α = 1 2 w

56 Large N Phase Transition: Phase Diagram U(N) lattice gauge theory in 2 d GROSS WITTEN WADIA transition (1980) 1 N 1 N crossover crossover WEAK STRONG STABLE ( weakly interacting ) UNSTABLE ( strongly interacting ) 0 g c coupling strength g 0 1 α = 1 2 w Similar 3-rd order phase transition in U(N) lattice-gauge theory in 2-d Unstable phase Strong coupling phase of Yang-Mills gauge theory Stable phase Weak coupling phase of Yang-Mills gauge theory Tracy-Widom crossover function in the double scaling regime (for finite but large N)

57 III. Coulomb Gas

58 Gaussian Random Matrices (N N) Gaussian random matrix: J [J ij ] Ensembles: Orthogonal (GOE), Unitary (GUE) or Symplectic (GSE)

59 Gaussian Random Matrices (N N) Gaussian random matrix: J [J ij ] Ensembles: Orthogonal (GOE), Unitary (GUE) or Symplectic (GSE) [ Prob[J ij ] exp β 2 N Tr ( J J ) ] )

60 Gaussian Random Matrices (N N) Gaussian random matrix: J [J ij ] Ensembles: Orthogonal (GOE), Unitary (GUE) or Symplectic (GSE) [ Prob[J ij ] exp β 2 N Tr ( J J ) ] ) N real eigenvalues {λ 1, λ 2,..., λ N } correlated random variables

61 Gaussian Random Matrices (N N) Gaussian random matrix: J [J ij ] Ensembles: Orthogonal (GOE), Unitary (GUE) or Symplectic (GSE) [ Prob[J ij ] exp β 2 N Tr ( J J ) ] ) N real eigenvalues {λ 1, λ 2,..., λ N } correlated random variables Joint distribution of eigenvalues (Wigner, 1951) [ P(λ 1, λ 2,..., λ N ) = 1 exp β N Z N 2 N i=1 λ 2 i ] λ j λ k β where the Dyson index β = 1 (GOE), β = 2 (GUE) or β = 4 (GSE) j<k

62 Gaussian Random Matrices (N N) Gaussian random matrix: J [J ij ] Ensembles: Orthogonal (GOE), Unitary (GUE) or Symplectic (GSE) [ Prob[J ij ] exp β 2 N Tr ( J J ) ] ) N real eigenvalues {λ 1, λ 2,..., λ N } correlated random variables Joint distribution of eigenvalues (Wigner, 1951) [ P(λ 1, λ 2,..., λ N ) = 1 exp β N Z N 2 N i=1 λ 2 i ] λ j λ k β where the Dyson index β = 1 (GOE), β = 2 (GUE) or β = 4 (GSE) j<k Z N = Partition Function =... { i [ dλ i } exp β 2 N ] λ j λ k β N λ 2 i i=1 j<k

63 Coulomb Gas Interpretation Z N =... { i dλ i } exp β N N λ 2 i log λ 2 j λ k i=1 j k

64 Coulomb Gas Interpretation Z N =... { i dλ i } exp β N N λ 2 i log λ 2 j λ k i=1 j k 2-d Coulomb gas confined to a line (Dyson) with β inverse temp. λ 1 λ 2 λ 3 λ Ν confining parabolic potential λ

65 Coulomb Gas Interpretation Z N =... { i dλ i } exp β N N λ 2 i log λ 2 j λ k i=1 j k 2-d Coulomb gas confined to a line (Dyson) with β inverse temp. λ 1 λ 2 λ 3 λ Ν confining parabolic potential λ Balance of energy N 2 λ 2 N 2 Typical eigenvalue: λ typ O(1) for large N

66 Spectral Density: Wigner s Semicircle Law Av. density of states: ρ(λ, N) = 1 N N δ(λ λ i ) i=1

67 Spectral Density: Wigner s Semicircle Law Av. density of states: ρ(λ, N) = 1 N Wigner s Semi-circle: ρ(λ, N) N N δ(λ λ i ) i=1 ρ(λ) = 1 π 2 λ 2 WIGNER SEMI CIRCLE ρ( λ) 0 SEA 2 2 λ

68 Spectral Density: Wigner s Semicircle Law Av. density of states: ρ(λ, N) = 1 N Wigner s Semi-circle: ρ(λ, N) N N δ(λ λ i ) i=1 ρ(λ) = 1 π 2 λ 2 WIGNER SEMI CIRCLE ρ( λ) 0 SEA 2 2 λ λ max = 2 for large N. λ max fluctuates from one sample to another. Prob[λ max, N] =?

69 Tracy-Widom distribution for λ max WIGNER SEMI CIRCLE ρ(λ,ν) TRACY WIDOM N 2/3 1 cumulative distribution of P( w, N) = Prob.[ λ max < w ] λ max SEA λ 0 finite but large N width of O (N 2/3 ) Tracy Widom 2 w

70 Tracy-Widom distribution for λ max WIGNER SEMI CIRCLE ρ(λ,ν) TRACY WIDOM N 2/3 1 cumulative distribution of P( w, N) = Prob.[ λ max < w ] λ max SEA λ 0 finite but large N width of O (N 2/3 ) Tracy Widom 2 w λ max = 2 ; typical fluctuation: λ max ρ(λ) dλ 1/N Using ρ(λ) ( 2 λ) 1/2 λ max 2 N 2/3 small [Bowick & Brezin 91, Forrester 93]

71 Tracy-Widom distribution for λ max WIGNER SEMI CIRCLE ρ(λ,ν) TRACY WIDOM N 2/3 1 cumulative distribution of P( w, N) = Prob.[ λ max < w ] λ max SEA λ 0 finite but large N width of O (N 2/3 ) Tracy Widom 2 w λ max = 2 ; typical fluctuation: λ max ρ(λ) dλ 1/N Using ρ(λ) ( 2 λ) 1/2 λ max 2 N 2/3 small [Bowick & Brezin 91, Forrester 93] typical fluctuations are distributed via Tracy-Widom ( 94): cumulative distribution: Prob[λ max w, N] F β ( 2N 2/3 ( w 2 ))

72 Tracy-Widom distribution for λ max WIGNER SEMI CIRCLE ρ(λ,ν) TRACY WIDOM N 2/3 1 cumulative distribution of P( w, N) = Prob.[ λ max < w ] λ max SEA λ 0 finite but large N width of O (N 2/3 ) Tracy Widom 2 w λ max = 2 ; typical fluctuation: λ max ρ(λ) dλ 1/N Using ρ(λ) ( 2 λ) 1/2 λ max 2 N 2/3 small [Bowick & Brezin 91, Forrester 93] typical fluctuations are distributed via Tracy-Widom ( 94): cumulative distribution: Prob[λ max w, N] F β ( 2N 2/3 ( w 2 )) Prob. density (pdf): f β (x) = df β (x)/dx; F β (x) Painlevé-II

73 Tracy-Widom distributions (1994) The scaling function F β (x) has the expression: β = 1: F 1 (x) = exp [ 1 2 [ x (y x)q 2 (y) + q(y) ] dy ] β = 2: F 2 (x) = exp [ x (y x)q 2 (y) dy ] β = 4: F 4 (x) = exp [ 1 2 (y x)q 2 (y) dy ] cosh [ 1 x 2 q(y) dy ] x d 2 q dy 2 = 2 q 3 (y) + y q(y) with q(y) Ai(y) as y Painlevé-II

74 Tracy-Widom distributions (1994) The scaling function F β (x) has the expression: β = 1: F 1 (x) = exp [ 1 2 [ x (y x)q 2 (y) + q(y) ] dy ] β = 2: F 2 (x) = exp [ x (y x)q 2 (y) dy ] β = 4: F 4 (x) = exp [ 1 2 (y x)q 2 (y) dy ] cosh [ 1 x 2 q(y) dy ] x d 2 q dy 2 = 2 q 3 (y) + y q(y) with q(y) Ai(y) as y Painlevé-II Probability density: f β (x) = df β (x)/dx

75 Tracy-Widom Distribution for λ max Probability densities f(x) 0.5 β = β = β = x

76 Tracy-Widom Distribution for λ max Probability densities f(x) 0.5 β = β = β = x Tracy-Widom density f β (x) depends explicitly on β.

77 Tracy-Widom Distribution for λ max Probability densities f(x) 0.5 β = β = β = x Tracy-Widom density f β (x) depends explicitly on β. ] Asymptotics: f β (x) exp [ β24 x 3 as x ] exp [ 2β3 x 3/2 as x

78 Tracy-Widom Distribution for λ max Probability densities f(x) 0.5 β = β = β = x Tracy-Widom density f β (x) depends explicitly on β. ] Asymptotics: f β (x) exp [ β24 x 3 as x ] exp [ 2β3 x 3/2 as x Applications: Growth models, Directed polymer, Sequence Matching... (Baik, Borodin, Calabrese, Comtet, Corwin, Deift, Dotsenko, Dumitriu, Edelman, Ferrari, Forrester, Johansson, Johnstone, Le doussal, Nadal, Nechaev, O Connell, Péché, Prähofer, Quastel, Rains, Rambeau, Rosso, Sano, Sasamoto, Schehr, Spohn, Takeuchi, Virag,...)

79 Probability of Large Deviations of λ max : ρ (λ, Ν) TRACY WIDOM WIGNER SEMI CIRCLE 2/3 N LEFT LARGE DEVIATION λ RIGHT LARGE DEVIATION

80 Probability of Large Deviations of λ max : ρ (λ, Ν) TRACY WIDOM WIGNER SEMI CIRCLE 2/3 N LEFT LARGE DEVIATION λ RIGHT LARGE DEVIATION Tracy-Widom law Prob[λ max w, N] F β ( 2 N 2/3 (w 2) ) describes the prob. of typical (small) fluctuations of O(N 2/3 ) around the mean 2, i.e., when λ max 2 N 2/3

81 Probability of Large Deviations of λ max : ρ (λ, Ν) TRACY WIDOM WIGNER SEMI CIRCLE 2/3 N LEFT LARGE DEVIATION λ RIGHT LARGE DEVIATION Tracy-Widom law Prob[λ max w, N] F β ( 2 N 2/3 (w 2) ) describes the prob. of typical (small) fluctuations of O(N 2/3 ) around the mean 2, i.e., when λ max 2 N 2/3 Q: How to describe the prob. of large (atypical) fluctuations when λ max 2 O(1) Large deviations from mean

82 Large Deviation Tails of λ max WIGNER SEMI CIRCLE ρ (λ, Ν) TRACY WIDOM 2/3 N LEFT LARGE DEVIATION λ RIGHT LARGE DEVIATION

83 Large Deviation Tails of λ max WIGNER SEMI CIRCLE ρ (λ, Ν) TRACY WIDOM 2/3 N LEFT LARGE DEVIATION λ RIGHT LARGE DEVIATION Prob. density of the top eigenvalue: Prob. [λ max = w, N] behaves as: exp [ βn 2 Φ (w) ] for [ ( N 2/3 f β 2 N 2/3 w )] 2 2 w O(1) for w 2 O(N 2/3 ) exp [ βnφ + (w)] for w 2 O(1)

84 IV. Saddle Point Method

85 Distribution of λ max : Saddle Point Method Prob[λ max w, N] = Prob[λ 1 w, λ 2 w,..., λ N w] = Z N(w) Z N ( ) w w Z N (w) =... { dλ i } exp β N 2 N λ 2 i log λ j λ k i i=1 j k

86 Distribution of λ max : Saddle Point Method Prob[λ max w, N] = Prob[λ 1 w, λ 2 w,..., λ N w] = Z N(w) Z N ( ) w w Z N (w) =... { dλ i } exp β N 2 N λ 2 i log λ j λ k i i=1 j k denominator numerator WALL λ λ w

87 Distribution of λ max : Saddle Point Method Prob[λ max w, N] = Prob[λ 1 w, λ 2 w,..., λ N w] = Z N(w) Z N ( ) w w Z N (w) =... { dλ i } exp β N 2 N λ 2 i log λ j λ k i i=1 j k denominator numerator WALL λ λ w

88 Setting up the Saddle Point Method Z N (w) w E ({λ i }) = 1 2N dλ i exp [ βn 2 E ({λ i }) ] i λ 2 i 1 2N 2 log λ j λ k i j k

89 Setting up the Saddle Point Method Z N (w) w E ({λ i }) = 1 2N dλ i exp [ βn 2 E ({λ i }) ] i λ 2 i 1 2N 2 log λ j λ k i j k As N discrete sum continuous integral: E [ρ(λ)] = w λ 2 ρ(λ) dλ w w ln λ λ ρ(λ) ρ(λ ) dλ dλ

90 Setting up the Saddle Point Method Z N (w) w E ({λ i }) = 1 2N dλ i exp [ βn 2 E ({λ i }) ] i λ 2 i 1 2N 2 log λ j λ k i j k As N discrete sum continuous integral: E [ρ(λ)] = w λ 2 ρ(λ) dλ w where the charge density: ρ(λ) = 1 N w ln λ λ ρ(λ) ρ(λ ) dλ dλ i δ(λ λ i)

91 Setting up the Saddle Point Method Z N (w) w E ({λ i }) = 1 2N dλ i exp [ βn 2 E ({λ i }) ] i λ 2 i 1 2N 2 log λ j λ k i j k As N discrete sum continuous integral: E [ρ(λ)] = w λ 2 ρ(λ) dλ w w where the charge density: ρ(λ) = 1 N i δ(λ λ i) { ( Z N (w) Dρ(λ) exp [ β N 2 E [ρ(λ)] + C ln λ λ ρ(λ) ρ(λ ) dλ dλ )} ] ρ(λ)dλ 1 + O(N)

92 Setting up the Saddle Point Method Z N (w) w E ({λ i }) = 1 2N dλ i exp [ βn 2 E ({λ i }) ] i λ 2 i 1 2N 2 log λ j λ k i j k As N discrete sum continuous integral: E [ρ(λ)] = w λ 2 ρ(λ) dλ w w where the charge density: ρ(λ) = 1 N i δ(λ λ i) { ( Z N (w) Dρ(λ) exp [ β N 2 E [ρ(λ)] + C ln λ λ ρ(λ) ρ(λ ) dλ dλ )} ] ρ(λ)dλ 1 + O(N) for large N, minimize the action S[ρ(λ)] = E[ρ(λ)] + C [ ρ(λ)dλ 1] Saddle Point Method: δs δρ = 0 ρ w (λ)

93 Setting up the Saddle Point Method Z N (w) w E ({λ i }) = 1 2N dλ i exp [ βn 2 E ({λ i }) ] i λ 2 i 1 2N 2 log λ j λ k i j k As N discrete sum continuous integral: E [ρ(λ)] = w λ 2 ρ(λ) dλ w w where the charge density: ρ(λ) = 1 N i δ(λ λ i) { ( Z N (w) Dρ(λ) exp [ β N 2 E [ρ(λ)] + C ln λ λ ρ(λ) ρ(λ ) dλ dλ )} ] ρ(λ)dλ 1 + O(N) for large N, minimize the action S[ρ(λ)] = E[ρ(λ)] + C [ ρ(λ)dλ 1] Saddle Point Method: δs δρ = 0 ρ w (λ) Z N (w) exp [ βn 2 S [ρ w (λ)] ]

94 Saddle Point Solution saddle point δs δf λ 2 2 = 0 w ρ w (λ ) ln λ λ dλ + C = 0

95 Saddle Point Solution saddle point δs δf λ 2 2 = 0 w ρ w (λ ) ln λ λ dλ + C = 0 Taking a derivative w.r.t. λ gives a singular integral Eq. w ρ w (λ ) dλ λ = P λ λ for λ [, w] Semi-Hilbert transform force balance condition

96 Saddle Point Solution saddle point δs δf λ 2 2 = 0 w ρ w (λ ) ln λ λ dλ + C = 0 Taking a derivative w.r.t. λ gives a singular integral Eq. w ρ w (λ ) dλ λ = P λ λ for λ [, w] Semi-Hilbert transform force balance condition When w : solution Wigner semi-circle law ρ (λ) = 1 π 2 λ 2

97 Saddle Point Solution saddle point δs δf λ 2 2 = 0 w ρ w (λ ) ln λ λ dλ + C = 0 Taking a derivative w.r.t. λ gives a singular integral Eq. w ρ w (λ ) dλ λ = P λ λ for λ [, w] Semi-Hilbert transform force balance condition When w : solution Wigner semi-circle law ρ (λ) = 1 π 2 λ 2 Exact solution for all w : [D. S. Dean & S.M., PRL, 97, 1602 (2006); PRE, 77, (2008)]

98 Exact Saddle Point Solution Exact solution (D. Dean and S.M., 2006, 2008): 1 π 2 λ 2 for w 2 ρ w (λ) = λ+l(w) 2π [w + L(w) 2λ] for w < 2 w λ where L(w) = [2 w w]/3

99 Exact Saddle Point Solution Exact solution (D. Dean and S.M., 2006, 2008): 1 π 2 λ 2 for w 2 ρ w (λ) = λ+l(w) 2π [w + L(w) 2λ] for w < 2 w λ where L(w) = [2 w w]/3 charge density ρ w (λ) vs. λ for different W W < 2 W= 2 W > 2 w w w L(w) w 2 W= CRITICAL POINT 2 2 pushed critical unpushed (UNSTABLE) (STABLE)

100 Exact Saddle Point Solution Exact solution (D. Dean and S.M., 2006, 2008): 1 π 2 λ 2 for w 2 ρ w (λ) = λ+l(w) 2π [w + L(w) 2λ] for w < 2 w λ where L(w) = [2 w w]/3 charge density ρ w (λ) vs. λ for different W W < 2 W= 2 W > 2 w w w L(w) w 2 W= CRITICAL POINT 2 2 pushed critical unpushed (UNSTABLE) (STABLE)

101 Left Large Deviation Function Prob[λ max w, N] = Z N(w) Z N ( ) exp [ βn 2 {S[ρ w (λ)] S[ρ (λ)]} ] exp [ β N 2 Φ (w) ]

102 Left Large Deviation Function Prob[λ max w, N] = Z N(w) Z N ( ) exp [ βn 2 {S[ρ w (λ)] S[ρ (λ)]} ] exp [ β N 2 Φ (w) ] lim 1 N N 2 ln [P(w, N)] = Φ (w) left large deviation function physically Φ (w) energy cost in pushing the Coulomb gas

103 Left Large Deviation Function Prob[λ max w, N] = Z N(w) Z N ( ) exp [ βn 2 {S[ρ w (λ)] S[ρ (λ)]} ] exp [ β N 2 Φ (w) ] lim 1 N N 2 ln [P(w, N)] = Φ (w) left large deviation function physically Φ (w) energy cost in pushing the Coulomb gas 1 [ Φ (w) = 36w 2 w 4 (15w + w 3 ) w ( + 27 ln(18) 2 ln(w + )] 6 + w 2 ) for w < 2 (Dean & S.M., 2006,2008)

104 Left Large Deviation Function Prob[λ max w, N] = Z N(w) Z N ( ) exp [ βn 2 {S[ρ w (λ)] S[ρ (λ)]} ] exp [ β N 2 Φ (w) ] lim 1 N N 2 ln [P(w, N)] = Φ (w) left large deviation function physically Φ (w) energy cost in pushing the Coulomb gas 1 [ Φ (w) = 36w 2 w 4 (15w + w 3 ) w ( + 27 ln(18) 2 ln(w + )] 6 + w 2 ) for w < 2 (Dean & S.M., 2006,2008) Note also that Φ (w) ( 2 w) 3 as w 2 from below

105 Matching with the left tail of Tracy-Widom: ρ (λ, Ν) TRACY WIDOM WIGNER SEMI CIRCLE 2/3 N LEFT LARGE DEVIATION λ RIGHT LARGE DEVIATION

106 Matching with the left tail of Tracy-Widom: ρ (λ, Ν) TRACY WIDOM WIGNER SEMI CIRCLE 2/3 N LEFT LARGE DEVIATION λ RIGHT LARGE DEVIATION As w 2 from below, Φ (w) ( 2 w) matches with the left tail of the Tracy-Widom distribution Prob.[λ max = w, N] exp [ β N 2 Φ (w) ] [ exp 2 N 2/3 (w 2) 3] β 24

107 Matching with the left tail of Tracy-Widom: ρ (λ, Ν) TRACY WIDOM WIGNER SEMI CIRCLE 2/3 N LEFT LARGE DEVIATION λ RIGHT LARGE DEVIATION As w 2 from below, Φ (w) ( 2 w) matches with the left tail of the Tracy-Widom distribution Prob.[λ max = w, N] exp [ β N 2 Φ (w) ] [ exp 2 N 2/3 (w 2) 3] β 24 recovers the left tail of TW: f β (x) exp[ β 24 x 3 ] as x

108 Right Large Deviation Function: w > 2 For w 2, saddle point solution of the charge density ρ w (λ) sticks to the semi-circle form: ρ sc (λ) = 1 π 2 λ2 for all w 2 Prob[λ max w, N] = Z N (w) Z N ( ) 1 as N

109 Right Large Deviation Function: w > 2 For w 2, saddle point solution of the charge density ρ w (λ) sticks to the semi-circle form: ρ sc (λ) = 1 π 2 λ2 for all w 2 Prob[λ max w, N] = Z N (w) Z N ( ) 1 as N Need a different strategy

110 Right Large Deviation Function: w > 2 For w 2, saddle point solution of the charge density ρ w (λ) sticks to the semi-circle form: ρ sc (λ) = 1 π 2 λ2 for all w 2 Prob[λ max w, N] = Z N (w) Z N ( ) 1 as N Prob. density: p(w, N) = d dw P(w, N) Need a different strategy

111 Right Large Deviation Function: w > 2 For w 2, saddle point solution of the charge density ρ w (λ) sticks to the semi-circle form: ρ sc (λ) = 1 π 2 λ2 for all w 2 Prob[λ max w, N] = Z N (w) Z N ( ) 1 as N Prob. density: p(w, N) = d dw P(w, N) Need a different strategy p(w, N) e βnw 2 /2 w... w eβ N 1 j=1 ln w λ j P N 1 (λ 1, λ 2,..., λ N 1 )

112 Right Large Deviation Function: w > 2 For w 2, saddle point solution of the charge density ρ w (λ) sticks to the semi-circle form: ρ sc (λ) = 1 π 2 λ2 for all w 2 Prob[λ max w, N] = Z N (w) Z N ( ) 1 as N Prob. density: p(w, N) = d dw P(w, N) Need a different strategy p(w, N) e βnw 2 /2 w... w eβ N 1 j=1 ln w λ j P N 1 (λ 1, λ 2,..., λ N 1 ) WIGNER SEMI CIRCLE (N 1)-fold integral λ λ max = w

113 Pulled Coulomb gas p(w, N) e βnw 2 /2 e β j ln w λ j

114 Pulled Coulomb gas p(w, N) e βnw 2 /2 e β j ln w λ j Large N limit: p(w, N) exp [ βn w βn ] ln(w λ) ρ sc (λ) dλ

115 Pulled Coulomb gas p(w, N) e βnw 2 /2 e β j ln w λ j Large N limit: p(w, N) exp WIGNER SEMI CIRCLE [ βn w βn ] ln(w λ) ρ sc (λ) dλ exp[ β N Φ + (w)] λ λ max = w N Φ + (w) = E(w) = w ln(w λ) ρ sc(λ) dλ energy cost in pulling a charge out of the Wigner sea

116 Pulled Coulomb gas p(w, N) e βnw 2 /2 e β j ln w λ j Large N limit: p(w, N) exp WIGNER SEMI CIRCLE [ βn w βn ] ln(w λ) ρ sc (λ) dλ exp[ β N Φ + (w)] λ λ Φ + (w) = 1 2 w w ln max = w N Φ + (w) = E(w) = w ln(w λ) ρ sc(λ) dλ energy cost in pulling a charge out of the Wigner sea [ w ] w (w > 2) [S.M. & Vergassola, PRL, 102, 1602 (2009)]

117 Matching with the right tail of Tracy-Widom ρ (λ, Ν) TRACY WIDOM WIGNER SEMI CIRCLE 2/3 N LEFT LARGE DEVIATION λ RIGHT LARGE DEVIATION

118 Matching with the right tail of Tracy-Widom ρ (λ, Ν) TRACY WIDOM WIGNER SEMI CIRCLE 2/3 N LEFT LARGE DEVIATION λ RIGHT LARGE DEVIATION As w 2 from above, Φ + (w) 27/4 3 (w 2) 3/2 matches with the right tail of the Tracy-Widom distribution Prob.[λ max = w, N] exp [ [ β N Φ + (w)] exp 2β 3 2 N 2/3 (w 2) 3/2]

119 Matching with the right tail of Tracy-Widom ρ (λ, Ν) TRACY WIDOM WIGNER SEMI CIRCLE 2/3 N LEFT LARGE DEVIATION λ RIGHT LARGE DEVIATION As w 2 from above, Φ + (w) 27/4 3 (w 2) 3/2 matches with the right tail of the Tracy-Widom distribution Prob.[λ max = w, N] exp [ [ β N Φ + (w)] exp 2β 3 2 N 2/3 (w 2) 3/2] recovers the right tail of TW: f β (x) exp[ 2β 3 x 3/2 ] as x

120 Comparison with Simulations: ln(p(t)) y=n 1/2 [t (2N) 1/2 ] N N real Gaussian matrix (β = 1): N = 10 squares simulation points red line Tracy-Widom blue line left large deviation function ( N 2 ) green line right large deviation function ( N).

121 Summary and Generalizations ρ (λ, Ν) TRACY WIDOM WIGNER SEMI CIRCLE 2/3 N LEFT LARGE DEVIATION 2 0 λ 2 RIGHT LARGE DEVIATION

122 Summary and Generalizations ρ (λ, Ν) TRACY WIDOM WIGNER SEMI CIRCLE 2/3 N LEFT LARGE DEVIATION 2 0 λ 2 RIGHT LARGE DEVIATION Prob. density of the top eigenvalue: Prob. [λ max = w, N] behaves as: exp [ βn 2 Φ (w) ] for [ ( N 2/3 f β 2 N 2/3 w )] 2 2 w O(1) for w 2 O(N 2/3 ) exp [ βnφ + (w)] for w 2 O(1)

123 3-rd Order Phase Transition Cumulative prob. of λ max : exp { βn 2 Φ (w) +... } for w < 2 P (λ max w, N) 1 A exp { βnφ + (w) +...} for w > 2

124 3-rd Order Phase Transition Cumulative prob. of λ max : exp { βn 2 Φ (w) +... } for w < 2 P (λ max w, N) 1 A exp { βnφ + (w) +...} for w > 2 lim 1 N βn 2 ln [P (λ max w)] = 3-rd derivative discontinuous Φ (w) ( 2 w ) 3 as w 2 0 as w 2 +

125 3-rd Order Phase Transition Cumulative prob. of λ max : exp { βn 2 Φ (w) +... } for w < 2 P (λ max w, N) 1 A exp { βnφ + (w) +...} for w > 2 lim 1 N βn 2 ln [P (λ max w)] = 3-rd derivative discontinuous Φ (w) ( 2 w ) 3 as w 2 0 as w 2 + Left strong-coupling phase perturbative higher order corrections (1/N expansion) to free energy [Borot, Eynard, S.M., & Nadal 21] Right weak-coupling phase non-perturbative higher order corrrections [Nadal & S.M. 21, Borot & Nadal, 22]

126 3-rd order transition ubiquitous λ max for other matrix ensembles: Wishart: W = X X (N N) covariance matrix Typical: Tracy-Widom [Johansson 2000, Johnstone 20] Large deviations: Exact rate functions [Vivo, S.M., Bohigas 2007, S.M. & Vergassola 2009]

127 3-rd order transition ubiquitous λ max for other matrix ensembles: Wishart: W = X X (N N) covariance matrix Typical: Tracy-Widom [Johansson 2000, Johnstone 20] Large deviations: Exact rate functions [Vivo, S.M., Bohigas 2007, S.M. & Vergassola 2009] large N gauge theory in 2-d [Gross, Witten, Wadia 80, Douglas & Kazakov 93] Distribution of MIMO capacity [Kazakopoulos et. al. 20] Complexity in spin glass models [Fyodorov & Nadal 23] Random tilings [Colomo & Pronko, 23]

128 3-rd order transition ubiquitous λ max for other matrix ensembles: Wishart: W = X X (N N) covariance matrix Typical: Tracy-Widom [Johansson 2000, Johnstone 20] Large deviations: Exact rate functions [Vivo, S.M., Bohigas 2007, S.M. & Vergassola 2009] large N gauge theory in 2-d [Gross, Witten, Wadia 80, Douglas & Kazakov 93] Distribution of MIMO capacity [Kazakopoulos et. al. 20] Complexity in spin glass models [Fyodorov & Nadal 23] Random tilings [Colomo & Pronko, 23] Conductance and Shot Noise in Mesoscopic Cavities Entanglement entropy of a random pure state in a bipartite system Maximum displacement in Vicious walker problem Distribution of Andreev conductance... [ Bohigas, Comtet, Forrester, Nadal, Schehr, Texier, Vergassola, Vivo,..+S.M. ( ) ]

129 Basic mechanism for 3-rd order transition strong critical weak w w w gap Gap between the soft edge (square-root signularity) of the Coulomb droplet and the hard wall vanishes as a control parameter g goes through a critical value g c : gap 0 as g g c

130 Experimental Verification with Coupled Lasers

131 Experimental Verification with Coupled Lasers combined output power from fiber lasers λ max λ max top eigenvalue of the Wishart matrix W = X t X where X real symmetric Gaussian matrix (β = 1)

132 Generalization to Wishart Matrices W = X X (N N) square covariance matrix (Wishart, 1928) [ ] Entries of X Gaussian: Pr[X ] exp β 2 N Tr(X X ) β = 1 Real entries, β = 2 Complex

133 Generalization to Wishart Matrices W = X X (N N) square covariance matrix (Wishart, 1928) [ ] Entries of X Gaussian: Pr[X ] exp β 2 N Tr(X X ) β = 1 Real entries, β = 2 Complex All eigenvalues of W = X X are non-negative

134 Generalization to Wishart Matrices W = X X (N N) square covariance matrix (Wishart, 1928) [ ] Entries of X Gaussian: Pr[X ] exp β 2 N Tr(X X ) β = 1 Real entries, β = 2 Complex All eigenvalues of W = X X are non-negative Average density of states for large N: Marcenko-Pastur (1967) ρ(λ, N) = 1 N δ(λ λ i ) N ρ(λ) = 1 4 λ N 2π λ i=1

135 Generalization to Wishart Matrices W = X X (N N) square covariance matrix (Wishart, 1928) [ ] Entries of X Gaussian: Pr[X ] exp β 2 N Tr(X X ) β = 1 Real entries, β = 2 Complex All eigenvalues of W = X X are non-negative Average density of states for large N: Marcenko-Pastur (1967) ρ(λ, N) = 1 N δ(λ λ i ) N ρ(λ) = 1 4 λ N 2π λ i=1 ρ(λ) MARCENKO PASTUR 0 4 λ

136 Distribution of λ max ρ(λ) MARCENKO PASTUR TRACY WIDOM λ max = 4 (as N ) LEFT N 2/3 RIGHT 0 4 λ

137 Distribution of λ max MARCENKO PASTUR ρ(λ) TRACY WIDOM N 2/3 LEFT RIGHT 0 4 λ λ max = 4 (as N ) typical fluctuations: λ max 4 O(N 2/3 ) distributed via Tracy-Widom (Johansson 2000, Johnstone 20)

138 Distribution of λ max MARCENKO PASTUR ρ(λ) TRACY WIDOM N 2/3 LEFT RIGHT 0 4 λ λ max = 4 (as N ) typical fluctuations: λ max 4 O(N 2/3 ) distributed via Tracy-Widom (Johansson 2000, Johnstone 20) For large deviations: λ max 4 O(1) exp { βn 2 Ψ (w) } for w < 4 P (λ max = w, N) exp { βnψ + (w)} for w > 4

139 Distribution of λ max MARCENKO PASTUR ρ(λ) TRACY WIDOM N 2/3 LEFT RIGHT 0 4 λ λ max = 4 (as N ) typical fluctuations: λ max 4 O(N 2/3 ) distributed via Tracy-Widom (Johansson 2000, Johnstone 20) For large deviations: λ max 4 O(1) exp { βn 2 Ψ (w) } for w < 4 P (λ max = w, N) exp { βnψ + (w)} for w > 4 Ψ (w) and Ψ + (w) computed exactly (Vivo, S.M. & Bohigas 2007, S.M. & Vergassola 2009)

140 Exact Left and Right Large Deviation Functions Using Coulomb gas + Saddle point method for large N: Left large deviation function: [ ] 2 Ψ (w) = ln w 4 (w 4)2 w 8 64 w 4 (Vivo, S.M. and Bohigas, 2007)

141 Exact Left and Right Large Deviation Functions Using Coulomb gas + Saddle point method for large N: Left large deviation function: [ ] 2 Ψ (w) = ln w 4 (w 4)2 w 8 64 Right large deviation function: [ w(w 4) w 2 ] w(w 4) Ψ + (w) = + ln 4 2 w 4 (Vivo, S.M. and Bohigas, 2007) w 4 (S.M. and Vergassola, 2009)

142 Experimental Verification with Coupled Lasers

143 Experimental Verification with coupled lasers

144 Tracy-Widom density with β = 1

145 Collaborators Students: C. Nadal (Oxford Univ., UK) J. Randon-Furling (Univ. Paris-1, France) Postdoc: D. Villamaina (ENS, Paris, France) Collaborators: O. Bohigas, A. Comtet, G. Schehr, C. Texier, P. Vivo (LPTMS, Orsay, France) R. Allez (Paris-Dauphin, France) G. Borot (Geneva, Switzerland) J.-P. Bouchaud (CFM, Ecole Polytechnique, Paris, France) B. Eynard (Saclay, France) K. Damle, V. Tripathi (Tata Institute, Bombay, India) D.S. Dean (Bordeaux, France) P. J. Forrester (Melbourne, Australia) A. Lakshminarayan (IIT Madras, India) A. Scardichio (ICTP, Trieste, Italy) S. Tomsovic (Washington State Univ., USA) M. Vergassola (Institut Pasteur, Paris, France)

146 Selected References; D.S. Dean,, Large Deviations of Extreme Eigenvalues of Random Matrices, Phys. Rev. Lett., 97, 1602 (2006); Phys. Rev. E., 77, (2008), M. Vergassola, Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices, Phys. Rev. Lett. 102, 0606 (2009) P. Vivo,, O. Bohigas, Distributions of Conductance and Shot Noise and associated Phase Transitions, Phys. Rev. Lett. 1, (2008), Phys. Rev. B 81, (20) C. Nadal,, M. Vergassola Phase Transitions in the Distribution of Bipartite Entanglement of a Random Pure State, Phys. Rev. Lett. 104, 1105 (20); J. Stat. Phys. 142, 403 (21), C. Nadal, A. Scardichio, P. Vivo The Index Distribution of Gaussian Random Matrices, Phys. Rev. Lett. 103, (2009); Phys. Rev. E 83, (21) G. Schehr,, A. Comtet, J. Randon-Furling, Exact distribution of the maximal height of p vicious walkers, Phys. Rev. Lett. 1, 1506 (2008) P.J. Forrester,, G. Schehr, Non-intersecting Brownian walkers and Yang-Mills theory on the sphere, Nucl. Phys. B., 844, 500 (21) K. Damle,, V. Tripathi, P. Vivo, Phase Transitions in the Distribution of the Andreev Conductance of Superconductor-Metal Junctions with Many Transverse Modes, Phys. Rev. Lett. 107, (21) C. Nadal and, A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix, J. Stat. Mech., P040 (21) G. Borot, B. Eynard, and C. Nadal, Large Deviations of the Maximal Eigenvalue of Random Matrices, J. Stat. Mech., P11024 (21) C. Texier and S. N. Majumdar, Wigner time-delay distribution in chaotic cavities and freezing transition, Phys. Rev. Lett. 110, (23).

147 Left Large Deviation: Beyond the Leading Order On the left side: λ max < 2

148 Left Large Deviation: Beyond the Leading Order On the left side: λ max < 2 Adapting loop (Pastur) equations approach developed by Chekov, Eynard and collaborators:

149 Left Large Deviation: Beyond the Leading Order On the left side: λ max < 2 Adapting loop (Pastur) equations approach developed by Chekov, Eynard and collaborators: ln[prob(λ max = w, N)] = β Φ (w) N 2 + (β 2)Φ 1 (w) N + + φ β ln N + Φ 2 (β, w) + O(1/N) where explicit expressions for Φ 1 (w), φ β and Φ 2 (β, w) were obtained recently (Borot, Eynard, S.M., & Nadal, JSTAT, P11024 (21))

150 Left Large Deviation: Beyond the Leading Order On the left side: λ max < 2 Adapting loop (Pastur) equations approach developed by Chekov, Eynard and collaborators: ln[prob(λ max = w, N)] = β Φ (w) N 2 + (β 2)Φ 1 (w) N + + φ β ln N + Φ 2 (β, w) + O(1/N) where explicit expressions for Φ 1 (w), φ β and Φ 2 (β, w) were obtained recently (Borot, Eynard, S.M., & Nadal, JSTAT, P11024 (21)) Setting w = /2 N 2/3 x (with x < 0) gives the left tail (x ) estimate of the TW density for all β

151 Left Large Deviation: Beyond the Leading Order On the left side: λ max < 2 Adapting loop (Pastur) equations approach developed by Chekov, Eynard and collaborators: ln[prob(λ max = w, N)] = β Φ (w) N 2 + (β 2)Φ 1 (w) N + + φ β ln N + Φ 2 (β, w) + O(1/N) where explicit expressions for Φ 1 (w), φ β and Φ 2 (β, w) were obtained recently (Borot, Eynard, S.M., & Nadal, JSTAT, P11024 (21)) Setting w = /2 N 2/3 x (with x < 0) gives the left tail (x ) estimate of the TW density for all β Prob. [ λ max < /2 N 2/3 x ] ] τ β x (β2 +4 6β)/2β exp [ β x (β 2) 6 x 3/2

152 Left Large Deviation: Beyond the Leading Order On the left side: λ max < 2 Adapting loop (Pastur) equations approach developed by Chekov, Eynard and collaborators: ln[prob(λ max = w, N)] = β Φ (w) N 2 + (β 2)Φ 1 (w) N + + φ β ln N + Φ 2 (β, w) + O(1/N) where explicit expressions for Φ 1 (w), φ β and Φ 2 (β, w) were obtained recently (Borot, Eynard, S.M., & Nadal, JSTAT, P11024 (21)) Setting w = /2 N 2/3 x (with x < 0) gives the left tail (x ) estimate of the TW density for all β Prob. [ λ max < /2 N 2/3 x ] ] τ β x (β2 +4 6β)/2β exp [ β x (β 2) 6 x 3/2 where the constant τ β is

153 The constant τ β ( 17 ln τ β = 8 25 ( β 2 )) + 4 ln(2) 1 ( ) πβ β 4 ln β ( ) ζ ( 1) + γ E 6β + [ ] 6 x coth(x/2) 12 x 2 + dx 12x 2 (e β x/2 1) 0 (Borot, Eynard, S.M., & Nadal, 21)

154 The constant τ β ( 17 ln τ β = 8 25 ( β 2 )) + 4 ln(2) 1 ( ) πβ β 4 ln β ( ) ζ ( 1) + γ E 6β + [ ] 6 x coth(x/2) 12 x 2 + dx 12x 2 (e β x/2 1) For β = 1, 2 and 4 0 (Borot, Eynard, S.M., & Nadal, 21) agrees with Baik, Buckingham and DiFranco (2008)

155 Right Large Deviation: Beyond the Leading Order On the right side: λ max > 2

156 Right Large Deviation: Beyond the Leading Order On the right side: λ max > 2 Using an orthogonal polynomial (with an upper cut-off) method (for β = 2) and adapting the techniques used by Gross and Matytsin, 94 in the context of two-dimensional Yang-Mills theory

157 Right Large Deviation: Beyond the Leading Order On the right side: λ max > 2 Using an orthogonal polynomial (with an upper cut-off) method (for β = 2) and adapting the techniques used by Gross and Matytsin, 94 in the context of two-dimensional Yang-Mills theory Prob(λ max = w, N) 1 2π 2 2 N Φ+(w) e (w 2 2)

158 Right Large Deviation: Beyond the Leading Order On the right side: λ max > 2 Using an orthogonal polynomial (with an upper cut-off) method (for β = 2) and adapting the techniques used by Gross and Matytsin, 94 in the context of two-dimensional Yang-Mills theory Prob(λ max = w, N) 1 2π 2 2 N Φ+(w) e (w 2 2) [ where Φ + (w) = 1 2 w w 2 w 2 + ln ] w (C. Nadal and S.M., J. Stat. Mech., P040, 21)

159 Right Large Deviation: Beyond the Leading Order On the right side: λ max > 2 Using an orthogonal polynomial (with an upper cut-off) method (for β = 2) and adapting the techniques used by Gross and Matytsin, 94 in the context of two-dimensional Yang-Mills theory Prob(λ max = w, N) 1 2π 2 2 N Φ+(w) e (w 2 2) [ where Φ + (w) = 1 2 w w 2 w 2 + ln ] w (C. Nadal and S.M., J. Stat. Mech., P040, 21) Close to w 2 +, this gives Prob. [ λ max < /2 N 2/3 x ] 1 3/2 (4/3) x e 16 π x 3/2 precise asymptotics of the right tail of TW for β = 2 (Baik, 2006)

160 Right Large Deviation: Beyond the Leading Order On the right side: λ max > 2 Using an orthogonal polynomial (with an upper cut-off) method (for β = 2) and adapting the techniques used by Gross and Matytsin, 94 in the context of two-dimensional Yang-Mills theory Prob(λ max = w, N) 1 2π 2 2 N Φ+(w) e (w 2 2) [ where Φ + (w) = 1 2 w w 2 w 2 + ln ] w (C. Nadal and S.M., J. Stat. Mech., P040, 21) Close to w 2 +, this gives Prob. [ λ max < /2 N 2/3 x ] 1 3/2 (4/3) x e 16 π x 3/2 precise asymptotics of the right tail of TW for β = 2 (Baik, 2006) For general β, precise right tail of TW obtained recently (Dumaz and Virag, 21)

161 Right Large Deviation: Beyond the Leading Order On the right side: λ max > 2 Using an orthogonal polynomial (with an upper cut-off) method (for β = 2) and adapting the techniques used by Gross and Matytsin, 94 in the context of two-dimensional Yang-Mills theory Prob(λ max = w, N) 1 2π 2 2 N Φ+(w) e (w 2 2) [ where Φ + (w) = 1 2 w w 2 w 2 + ln ] w (C. Nadal and S.M., J. Stat. Mech., P040, 21) Close to w 2 +, this gives Prob. [ λ max < /2 N 2/3 x ] 1 3/2 (4/3) x e 16 π x 3/2 precise asymptotics of the right tail of TW for β = 2 (Baik, 2006) For general β, precise right tail of TW obtained recently (Dumaz and Virag, 21) As a bonus, our method also provides a simpler derivation of TW distribution for β = 2 (Nadal and S.M., 21)

162 A simple example of large deviation tails Let M no. of heads in N tosses of an unbiased coin

163 A simple example of large deviation tails Let M no. of heads in N tosses of an unbiased coin Clearly P(M, N) = ( N M) 2 N (M = 0, 1,..., N) binomial distribution with mean= M = N 2 and variance=σ2 = ( M N 2 ) 2 = N 4

164 A simple example of large deviation tails Let M no. of heads in N tosses of an unbiased coin Clearly P(M, N) = ( N M) 2 N (M = 0, 1,..., N) binomial distribution with mean= M = N 2 and variance=σ2 = ( M N 2 ) 2 = N 4 typical fluctuations M N 2 O( N) are [ well described ( ) by the Gaussian form: P(M, N) exp 2 N M N 2 ] 2