Extreme value problems in Random Matrix Theory and other disordered systems

Abstract

We review some applications of central limit theorems and extreme values statistics in the context of disordered systems. We discuss several problems, in particular concerning Random Matrix Theory and the generalisation of the Tracy-Widom distribution when the disorder has ``fat tails''. We underline the relevance of power-law tails for Directed Polymers and mean-field Spin Glasses, and we point out various open problems and conjectures on these matters. We find that in many instances the assumption of Gaussian disorder cannot be taken for granted.

to appear in JSTAT, proceedings of `Principles of Dynamics of Nonequilibrium Systems', ISI Cambridge 2006

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• The intermediate disorder regime for directed polymers in dimension $1+1$
• Random Convex Hulls and Extreme Value Statistics
• Freezing Transitions and Extreme Values: Random Matrix Theory, $\zeta (1/2+it)$, and Disordered Landscapes
• Multifractality of eigenstates in the delocalized non-ergodic phase of some random matrix models : Wigner-Weisskopf approach
• Large deviations of the maximum of independent and identically distributed random variables
• Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble
• Asymptotics for the Expected Maximum of Random Walks and Lévy Flights with a Constant Drift
• Time to reach the maximum for a stationary stochastic process
• Discrete Sampling of Extreme Events Modifies Their Statistics
• Extreme value statistics and return intervals in long-range correlated uniform deviates
• The phase diagram of Lévy spin glasses
• On integrable directed polymer models on the square lattice
• Revisiting Directed Polymers with heavy-tailed disorder
• Maximum relative height of elastic interfaces in random media
• Almost-conserved operators in nearly many-body localized systems
• Probability distributions for directed polymers in random media with correlated noise
• Matching between typical fluctuations and large deviations in disordered systems : application to the statistics of the ground state energy in the SK spin-glass model
• Localization transition in random Levy matrices : multifractality of eigenvectors in the localized phase and at criticality
• Role of the Tracy-Widom distribution in the finite-size fluctuations of the critical temperature of the Sherrington-Kirkpatrick spin glass
• An exact relation between free energy fluctuations and bond chaos in the Sherrington-Kirkpatrick model
• Spectral densities of Wishart-Levy free stable random matrices: Analytical results and Monte Carlo validation
• Gumbel Central Limit Theorem for Max-Min and Min-Max
• Breaking the chains: extreme value statistics and localization in random spin chains
• Occupancy correlations in the asymmetric simple inclusion process
• Maxima of log-correlated fields: some recent developments
• Large Deviations of the Finite-Time Magnetization of the Curie-Weiss Random Field Ising Model
• Extreme matrices or how an exponential map links classical and free extreme laws
• Sample-to-sample fluctuations and bond chaos in the $m$-component spin glass
• Asymmetric scaling in large deviations for rare values bigger or smaller than the typical value
• Ordered spectral statistics in 1D disordered supersymmetric quantum mechanics and Sinai diffusion with dilute absorbers