Large deviations of the top eigenvalue of random matrices and - PowerPoint PPT Presentation

Large deviations of the top eigenvalue of random matrices and applications in statistical physics Grégory Schehr LPTMS, CNRS-Université Paris-Sud XI Analytical Results in Statistical Physics, in memory of Bernard Jancovici

Large deviations of the top eigenvalue of random matrices and applications in statistical physics Grégory Schehr LPTMS, CNRS-Université Paris-Sud XI Analytical Results in Statistical Physics, in memory of Bernard Jancovici 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:1311.0580

Large spectrum of applications of random matrix theory

Large spectrum of applications of random matrix theory 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),...

Large spectrum of applications of random matrix theory 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,...

Large spectrum of applications of random matrix theory 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,...

Large spectrum of applications of random matrix theory 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,...

Large spectrum of applications of random matrix theory 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,...

Large spectrum of applications of random matrix theory 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 fi nance: time series and big data analysis,...

Large spectrum of applications of random matrix theory 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 fi nance: time series and big data analysis,... «The Oxford handbook of random matrix theory», Ed. by G. Akemann, J. Baik and P. Di Francesco (2011)

Spectral statistics in random matrix theory (RMT) Basic model: real, symmetric, Gaussian random matrix    − N � M 2 P ( M ) ∝ exp i,j  2 i,j � � − N 2 Tr( M 2 ) ∝ exp

Spectral statistics in random matrix theory (RMT) Basic model: real, symmetric, Gaussian random matrix    − N � M 2 P ( M ) ∝ exp i,j  2 i,j � � − N 2 Tr( M 2 ) ∝ exp Invariant under rotation Gaussian orthogonal ensemble (GOE)

Spectral statistics in random matrix theory (RMT) Basic model: real, symmetric, Gaussian random matrix    − N � M 2 P ( M ) ∝ exp i,j  2 i,j � � − N 2 Tr( M 2 ) ∝ exp Invariant under rotation Gaussian orthogonal ensemble (GOE) The matrix has real eigenvalues which are strongly correlated

Spectral statistics in random matrix theory (RMT) Basic model: real, symmetric, Gaussian random matrix    − N � M 2 P ( M ) ∝ exp i,j  2 i,j � � − N 2 Tr( M 2 ) ∝ exp Invariant under rotation Gaussian orthogonal ensemble (GOE) The matrix has real eigenvalues which are strongly correlated Spectral statistics in RMT: statistics of

Largest (top) eigenvalue of random matrices Density of eigenvalues for N ≫ 1 Wigner sea √ √ λ − 2 + 2

Largest (top) eigenvalue of random matrices Recent excitements in statistical physics and mathematics on largest eigenvalue TRACY−WIDOM ρ (λ, Ν) WIGNER SEMI−CIRCLE −2/3 N LEFT LARGE DEVIATION − 2 0 2 λ RIGHT LARGE DEVIATION

Largest (top) eigenvalue of random matrices Recent excitements in statistical physics and mathematics on largest eigenvalue Typical fl uctuations (small): TRACY−WIDOM ρ (λ, Ν) WIGNER SEMI−CIRCLE −2/3 Tracy-Widom distribution N LEFT LARGE DEVIATION − 2 0 2 λ RIGHT LARGE DEVIATION

Tracy-Widom distribution 10 0 10 -50 � � − 2 3 x 3 / 2 ∼ exp 10 -100 1 ( x ) 10 -150 log F ′ 10 -200 � − 1 � 24 | x | 3 ∼ exp 10 -250 10 -300 10 0 -20 0 20 40 60 x

Largest (top) eigenvalue of random matrices Recent excitements in statistical physics and mathematics on largest eigenvalue Typical fl uctuations (small): TRACY−WIDOM ρ (λ, Ν) WIGNER SEMI−CIRCLE −2/3 Tracy-Widom distribution N LEFT ubiquitous LARGE DEVIATION − 2 0 2 λ RIGHT LARGE DEVIATION

Largest (top) eigenvalue of random matrices Recent excitements in statistical physics and mathematics on largest eigenvalue Typical fl uctuations (small): TRACY−WIDOM ρ (λ, Ν) WIGNER SEMI−CIRCLE −2/3 N Tracy-Widom distribution LEFT LARGE DEVIATION ubiquitous − 2 0 2 λ RIGHT LARGE DEVIATION largest eigenvalue of correlation matrices (Wishart-Laguerre) longest increasing subsequence of random permutations directed polymers and growth models in the KPZ universality class continuum KPZ equation sequence alignment problems mesoscopic fl uctuations in quantum dots high-energy physics (Yang-Mills theory)...

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 ?

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

Stability of a large complex system

Stability of a large complex system Stable non-interacting population of species with equilibrium densites

Stability of a large complex system Stable non-interacting population of species with equilibrium densites Slightly perturbed densities evolve via (assuming identical damping times)

Stability of a large complex system Stable non-interacting population of species with equilibrium densites Slightly perturbed densities evolve via (assuming identical damping times) Switch on interactions between the species random interaction coupling strength matrix