Satya Majumdar (LPTMS Orsay)

Extreme statistics for correlated variables (11.5 h)

1. Introduction to the extreme value statistics (EVS): statistics of the maximum (or minimum) of a set of random variables

EVS for uncorrelated random variables: the three limiting distributions (Gumbel, Frechet and Weibull). Simple examples of EVS for correlated random variables: (i) Weakly Correlated: Ornstein-Uhlenbeck process (particle moving in a harmonic potential), limiting distribution of the maximum is the same as that of uncorrelated variables; (ii) Strongly Correlated: Brownian motion - the limiting distribution is half-Gaussian: different from that of the uncorrelated case.

2. EVS for two strongly correlated systems with many applications

Variety of constrained Brownian motions. Random Matrices.

3. Distribution of maximum for constrained Brownian motions:

Path-integral (Feynman-Kac) computation of the maximum distribution. Application to: (i) Brownian Bridge; (ii) Brownian Excursion; (iii) Brownian Meander; (iv) Fluctuating (1+1)-dimensional Interfaces: Airy distribution function (applications in computer science).

4. Large deviations and rare events in random matrices

Large deviations of the maximum eigenvalue: probability of rare fluctuations. Wishart radom matrices: average density of states (Marcenko-Pastur law), maximum eigenvalue (Tracy-Widom again), minimum eigenvalue (when different from Tracy-Widom), application to Quantum Entanglement problem.

5. Ubiquity of the Tracy-Widom law:

Longest increasing subsequence problem. Hamersley process - interacting particle systems. Directed polymer. (2+1)-dimensional directed percolation. Sequence matching problem. Maximum Agreement SubTree problem in Phylogeny. Etc.