Morning Session Chair: Eugene Kanzieper
Afternoon Session Chair: Uzy Smilansky
|Finite temperature free fermions and the Kardar-Parisi-Zhang equation at finite time
|I will consider a system of $N$ one-dimensional free fermions confined by a harmonic well. At zero temperature ($T=0$), it is well known that this system is intimately connected to random matrices belonging to the Gaussian Unitary Ensemble. In particular, the density of fermions has, for large $N$, a finite support and it is given by the Wigner semi-circular law. Besides, close to the edges of the support, the quantum fluctuations are described by the so-called Airy-Kernel (which plays an important role in random matrix theory). What happens at finite temperature $T$? I will show that at finite but low temperature, the fluctuations close to the edge, are described by a generalization of the Airy kernel, which depends continuously on temperature. Remarkably, exactly the same kernel arises in the exact solution of the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimensions at finite time. I will also discuss recent results for fermions in higher dimensions.
|Variants of geometric RSK correspondence and the multipoint distribution of the log-gamma polymer
|We extend Kirillov’s geometric Robinson-Schested-Knuth correspondence (gRSK) in two ways: one that replaces the input matrix by a general polygonal array and one which provides the geometric version of the polynuclear growth process (PNG). These variants of gRSK allow to write explicit integral formulae for the joint Laplace transform of point-to-point partition functions for the directed polymer model with log-gamma disorder. Under an assumption of convergence of certain series, we show that the joint law of the two point-to-point partition functions converges to the two point function of the Airy process. This is a joint work with Vu Lan Nguyen (Paris 7).
| Coffee in the Common Room
|Spectral properties of random fermionic models
|I will discuss our recent work on fermionic $XY$ models with interactions that are taken to be i.i.d. Gaussian random variables. We focus on the ground state energy gap, density of states, and the local eigenvalue statistics in the case of all-to-all interactions. We are also examine what happens as the number of interacting neighbors
changes. This is based on a joint work with F. Mezzadri and F. Cunden.
|Global fluctuations for non-colliding process
|In this talk I will discuss a recent approach for studying the global fluctuations for a class of non-colliding processes with determinantal correlations. The method is based on recurrence relations for the associated biorthogonal family of functions. The main results are Central Limit Theorems for multi-time linear statistics. The results show the universality of the Gaussian Free Field appearing in the global fluctuations for these models. Special attention will be given to Dyson’s Brownian motion with one initial point and several endpoints. In that case, the underlying family is then given by multiple Hermite polynomials.
| Lunch in the Coffee Room
|Diffusion of characteristic polynomials and edge universality in non-hermitian random matrix models.
|We propose the mathematical framework for studying the evolution of non-hermitian random matrix ensembles, involving a new, hidden variable, which turns out to be crucial to unravel the dynamics of co-evolving eigenvalues and eigenvectors. We demonstrate this phenomenon on the example of complex Ornstein-Uhlenbeck process.
 Z. Burda, J. Grela, M. A. Nowak, W. Tarnowski and P. Warchoł, Phys. Rev. Lett. 114 (2014) 104102; Nucl. Phys. B897 (2015) 421.
 J.-P. Blaizot, J. Grela, M. A. Nowak, W. Tarnowski and P. Warchoł, arXiv:1512.06599v2 [math-ph].
 Y. Liu, M. A. Nowak and I. Zahed, arXiv:1602.02578v1 [hep-lat].
|Tea in the Common Room
Much work has been devoted to the understanding of the motion of eigenvalues in response to randomness. The folklore of random matrix analysis, especially in the case of Hermitian matrices, suggests that the eigenvalues of a perturbed matrix repel. We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying real matrix attract. We offer a dynamical perspective on the motion and interaction of the eigenvalues in the complex plane, derive their governing equations and discuss applications. C.c. pairs closest to the real axis, or those that are ill-conditioned, attract most strongly and can collide to become exactly real. We apply the results to the Hatano-Nelson model, random perturbations of a fixed matrix, real sto-chastic processes with zero-mean and independent intervals and discuss open problems.
Reference: Journal of Statistical Physics February 2016, Volume 162, Issue 3, pp 615-643
|Dinner in the Common Room