Extreme statistics in stochastic processes.

Wednesday 25 January 2017, IHP

Moderator: Satya N. Majumdar (Orsay)

David Dean (Bordeaux), Bernard Derrida (Paris), Jon Keating (Bristol), Pierre Le Doussal (Paris), Sandrine Péché (Paris), Alberto Rosso (Orsay), Grégory Schehr (Orsay), Zhan Shi (Paris)

Satya N. Majumdar (Orsay): Introduction. pdf

Extreme value statistics (EVS) concerns the statistics of the maximum (or minimum) of a set of random variables. This is a relevant problem in diverse fields in physics, mathematics, statistics, biology and computer science. EVS is well understood when the variables are uncorrelated or weakly correlated. In contrast, very few results exist when the variables are strongly correlated. EVS of strongly correlated variables is important in many contexts, e.g., the largest eigenvalue of a random matrix, the ground state energy of a disordered system (such as spin glasses or directed polymers), the maximum of log-correlated Gaussian processes, the maximum of a branching Brownian motion, amongst many others.

David Dean (Bordeaux): Extreme value statistics for free fermions.

I will discuss the problem of N non-interacting fermions in an isotropic d-dimensional harmonic trap and show how one can compute the cumulative distribution of the maximal radial distance of the fermions from the trap center at zero temperature. While in d=1 the limiting distribution (in the large N limit), properly centered and scaled, converges to the squared Tracy-Widom distribution of the Gaussian Unitary Ensemble in Random Matrix Theory, for all d>1, the limiting distribution converges to the Gumbel distribution.

Bernard Derrida (Paris): Genealogy and large deviations of the branching Brownian motion.

This talk will review some recents results on the genealogy and on the large deviation function of the position of the rightmost particles of a branching Brownian motion. Our results on the genealogy can be interpreted as a finite size correction to the overlap function of the mean field directed polymer, in the broken replica symmetry phase. The large deviation function of the position exhibits three regimes, with corrections one can estimate when one introduces selection, as in the L-BBM or the N-BBM.
Derrida, B., Mottishaw, P. (2016). On the genealogy of branching random walks and of directed polymers. EPL (Europhysics Letters), 115(4), 40005.
Derrida, B., Shi, Z. (2016). Large deviations for the branching Brownian motion in presence of selection or coalescence. Journal of Statistical Physics, 163(6), 1285-1311.
Derrida, B., Shi, Z. (2016). Large deviations for the rightmost position in the branching Brownian motion. Preprint (2016) and work in preparation.
Derrida, B., Meerson, B., Sasorov, P. V. (2016). Large-displacement statistics of the rightmost particle of the one-dimensional branching Brownian motion. Physical Review E, 93(4), 042139.

Jon Keating (Bristol): Extreme value statistics: from random matrices to number theory.

I will review some recent developments concerning the extreme value statistics of the characteristic polynomials of random matrices and of the Riemann zeta function, focusing in particular on connections with the extreme value statistics of log-correlated Gaussian fields.

Pierre Le Doussal (Paris): Exact results for freezing and extreme value statistics of log-correlated fields. pdf

Extreme value statistics of log-correlated fields, such as the 2D Gaussian free field, can be studied using methods from statistical mechanics. This leads to so-called log-REM models, i.e. correlated versions of Derrida Random Energy Model, which exhibit an ubiquitous freezing transition into a glass phase dominated by the extrema. We will review the topic and discuss recent exact results for values and positions of extrema. The tools are replica methods, integrability, and, in 2D, relations to Liouville field theory. Universality of the results for a large class of log-REM's will be discussed.
Collaborators: Xiangyu Cao, Yan Fyodorov, A. Rosso, R. Santachiara.

Sandrine Péché (Paris): Some results on delocalization and localization of eigenvectors of random matrices.

We will discuss what can impact on the delocalization of eigenvectors of random matrices. The talk will make a review of some recent results for Hermitian random matrices.

Alberto Rosso (Orsay): Extreme statistics and Gaussian free fields: effect of bounded domains and Liouville field theory.

I will present some recent results on the extreme statistics of 2D Gaussian free field. In particular I will discuss the exact mapping between the Liouville field theory and the Gibbs measure statistics of a thermal particle in a 2D Gaussian Free Field plus a logarithmic confining potential.
X. Cao, P. Le Doussal, A. Rosso, R. Santachiara, arXiv:1611.02193

Grégory Schehr (Orsay): Extremes and order statistics of one-dimensional branching Brownian motion. pdf

I will review some recent results for the extremes and order statistics of one-dimensional branching Brownian motion (BBM) in which particles either diffuse, die (with rate d) or split into two particles (with rate b). In particular, I will present exact results for the probability distributions of the k-th gap between two successive particles, conditioned on the event that there are exactly n particles in the system at a given time t. In particular, I will show that these distributions become stationary, at late times. I will also discuss applications of extreme value questions to the study of the spatial extent of BBM.

Zhan Shi (Paris): An elementary model of percolation on trees.

I am going to discuss a few questions, with but more often without answers, about the free energy in a simplified model of depinning transition in the limit of strong disorder. The study, initiated by Derrida and Retaux in 2014, can also be formulated for an elementary percolation model on trees.
Joint work with Xinxing Chen, Bernard Derrida, Yueyun Hu and Mikhail Lifshits.