Large deviations of the top eigenvalue of random matrices and applications in statistical physics

Transcription

1 Large deviations of the top eigenvalue of random matrices and applications in statistical physics Grégory Schehr LPTMS, CNRS-Université Paris-Sud XI Journées de Physique Statistique Paris, January 29-30, 2015 Collaborators: Alain Comtet (LPTMS, Orsay) Peter J. Forrester (Math. Dept., Univ. of Melbourne) Satya N. Majumdar (LPTMS, Orsay) S. N. Majumdar, G. S., J. Stat. Mech. P01012 (2014), arxiv:

2 Large spectrum of applications of random matrix theory «The Oxford handbook of random matrix theory», Ed. by G. Akemann, J. Baik and P. Di Francesco (2011) Physics: nuclear physics, quantum chaos, disordered systems, mesoscopic transport, quantum entanglement, neural networks, gauge theory, string theory, cosmology, statistical physics (growth models, interface, directed polymers),... Mathematics: number theory, combinatorics, knot theory, determinantal point processes, integrable systems, free probability,... Statistics: multivariate statistics, principal component analysis (PCA), image processing, data compression, Bayesian model selection,... Information theory: signal processing, wireless communications,... Biology: sequence matching, RNA folding, gene expression networks,... Economy and finance: time series and big data analysis,...

3 Spectral statistics in random matrix theory (RMT) Basic model: real, symmetric, Gaussian random matrix P (M) / exp / exp 2 4 N 2 apple X i,j M 2 i,j N 2 Tr(M 2 ) 3 5 Invariant under rotation Gaussian orthogonal ensemble (GOE) The matrix has real eigenvalues which are strongly correlated Spectral statistics in RMT: statistics of

4 Largest (top) eigenvalue of random matrices Density of eigenvalues for N 1 Wigner sea p 2 + p 2 λ

5 Largest (top) eigenvalue of random matrices Recent excitements in statistical physics and mathematics on largest eigenvalue WIGNER SEMI CIRCLE ρ (λ, Ν) LEFT LARGE DEVIATION TRACY WIDOM 2/3 N Typical fluctuations (small): Tracy-Widom distribution λ RIGHT LARGE DEVIATION

6 Tracy-Widom distribution exp 2 3 x3/2 log F 0 1(x) exp 1 24 x x

7 Largest (top) eigenvalue of random matrices Recent excitements in statistical physics and mathematics on largest eigenvalue WIGNER SEMI CIRCLE ρ (λ, Ν) LEFT LARGE DEVIATION TRACY WIDOM 2/3 N Typical fluctuations (small): Tracy-Widom distribution ubiquitous λ largest eigenvalue of RIGHT correlation matrices (Wishart-Laguerre) LARGE DEVIATION longest increasing subsequence of random permutations directed polymers and growth models in the KPZ universality class continuum KPZ equation sequence alignment problems mesoscopic fluctuations in quantum dots high-energy physics (Yang-Mills theory)...

8 Ubiquity of Tracy-Widom distributions Experimental observation of TW distributions for GOE and GUE in liquid crystals experiments (Carr-Helfrich instability) Takeuchi & Sano 10 Takeuchi, Sano, Sasamoto & Spohn 11 from Takeuchi, Sano, Sasamoto & Spohn, Sci. Rep. (Nature) 1, 34 (2011) Q: universality of the Tracy-Widom distributions?

9 Largest (top) eigenvalue of random matrices Recent excitements in statistical physics and mathematics on largest eigenvalue WIGNER SEMI CIRCLE 0 ρ (λ, Ν) LEFT LARGE DEVIATION 2 2 λ TRACY WIDOM 2/3 N RIGHT LARGE DEVIATION Typical fluctuations (small): Tracy-Widom distribution ubiquitous Q: universality of the Tracy-Widom distributions? In this talk: atypical and large fluctuations of Large deviation functions Third order phase transition

10 Stability of a large complex system Stable non-interacting population of densites Slightly perturbed densities species with equilibrium evolve via (assuming identical damping times) Switch on interactions between the species coupling strength random interaction matrix Q: what is the proba. that the system remains stable once the interactions are switched on?

11 Linear stability criterion Linear dynamical system Eigenvalues of : The system is stable iff i.e. iff Proba. that the system is stable

12 Stable/Unstable transition for large systems Assuming that the interaction matrix is real, symmetric and Gaussian May observed a sharp transition in the limit : stable, weakly interacting phase : unstable, strongly interacting phase P stable (α,n)=prob.[λ max w =1/α] 1 STABLE WEAK COUPLING 0 UNSTABLE STRONG COUPLING w c = 2 w =1/α May 72

13 Stable/Unstable transition for large systems What happens for finite but large systems, of size? P stable (α,n)=prob.[λ max w =1/α] 1 LEFT TAIL STABLE WEAK COUPLING O(N 2/3 ) RIGHT TAIL 0 UNSTABLE STRONG COUPLING w c = 2 w =1/α Is there any thermodynamic sense to this transition? What is the analogue of the free energy? What is the order of this transition?

14 Coulomb Gas approach

15 Gaussian random matrix models random matrix : Standard Dyson s ensembles: Orthogonal, Unitary, Symplectic (GOE) (GUE) (GSE) Gaussian probability measure Joint PDF of the real eigenvalues Wigner 51 Partition function with (GOE), (GUE) and (GSE)

16 Coulomb Gas picture and Wigner semi-circle Rewrite the partition function as 2-d Coulomb gas confined to a line, with the inverse temperature Typical scale of the eigenvalues: Mean density of eigenvalues Wigner sea p 2 + p 2 λ

17 Coulomb Gas with a wall

18 Cumulative distribution function of wall eigenvalues What happens when the wall is moved?

19 Pushed vs. pulled Coulomb gas Saddle point analysis Dean & Majumdar 06, 08 Mean density of eigenvalues in presence of the wall w w w ρ w(λ) ρ w (λ) ρ w(λ) λ λ L(w) w< 2 w = 2 w> 2 PUSHED CRITICAL PULLED λ

20 Left large deviation function F (w, N) =Pr.[ max apple w] = Z N (w) Z N (w!1) exp[ N 2 (w)],w< p 2 i.e. lim N!1 1 F (w, N) = N 2 (w) Physically, N 2 (w) is the energy to push the Coulomb gas Left deviation function when Dean & Majumdar 06, 08

21 Right large deviation function w w w ρ w(λ) ρ w (λ) ρ w(λ) λ λ L(w) w< 2 w = 2 w> 2 PUSHED CRITICAL PULLED λ The saddle point equation yields a trivial result for Non trivial corrections requires a different approach

22 Right tail: pulled Coulomb gas : energy to pull a single charge out of the Wigner sea WIGNER SEMI CIRCLE 2 2 λ λ max = w Majumdar & Vergassola 09 when

23 Third order phase transition unstable This implies the behavior of the free energy stable The third derivative of the free energy is discontinuous Third order phase transition The crossover, for finite, between the two phases is described by the Tracy-Widom distribution S. N. Majumdar, G. S., J. Stat. Mech. P01012 (2014)

24 Third order phase transition 1 N crossover STABLE ( weakly interacting ) UNSTABLE ( strongly interacting ) 0 α = 1 w 1 2 S. N. Majumdar, G. S., J. Stat. Mech. P01012 (2014)

25 Similar third order phase transition in gauge theory

26 Similar third order phase transition in gauge theory 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 α = 1 w 1 2 Similar transition in lattice gauge theory Unstable phase = strong coupling phase of Yang-Mills gauge theory Stable phase = weak coupling phase of Yang-Mills gauge theory Tracy-Widom ditribution describes the crossover between the two regimes (at finite but large ): double scaling regime

27 Conclusion Largest eigenvalue of a Gaussian random matrix Application to the stability of large complex system Proba. distrib. func. (PDF) of : Coulomb gas (CG) with a wall Tracy-Widom distribution Physics of large deviation tails Left tail: pushed CG Right tail: unpushed CG Third order phase transition pushed/unpushed unstable/stable Similar third order transition in Yang-Mills gauge theories and other systems (conductance fluctuations, complexity in spin glasses, non-intersecting Brownian motions...)