Thimothée Thiery (KULeuven) : Random-walks in time-dependent random environment and the KPZ universality class: the Beta polymer.

Wednesday 22 Feb 2017, 15:00 → 17:00 Europe/Brussels

CYCL09B (MdeHemptinne)

The KPZ universality class in 1+1D is a remarkable example of universality in out-of-equilibrium statistical mechanics. 'KPZ-type’ scaling and fluctuations are indeed observed in a variety of statistical mechanics model, ranging e.g. from out-of-equilibrium growth to interacting particle systems, but also in some modern experiments. The directed polymer (DP) problem, that is the equilibrium statistical mechanics of directed paths in a random environment, is also believed to be in the KPZ universality class for a large choice of random environments. In a recent paper [1] Barraquand and Corwin introduced the Beta polymer, a Bethe ansatz exactly solvable model of DP on the square lattice which has the peculiarity of being equivalent to a model of random walk in a one-dimensional dynamic random environment (TD-RWRE). This is due to a special choice of short-range correlations of the random environment that encodes an additional conservation law (compared to the usual DP problem): the conservation of probability. Interestingly, the analysis of the statistical properties of the TD-RWRE cumulative distribution function (in the original BC paper [1]), and transition probability (in a recent paper by myself and Pierre Le Doussal [2]), suggest that: (i) the TD-RWRE nature of the Beta polymer breaks KPZ universality in the diffusive regime of the random walk; (ii) the DP nature of the TD-RWRE can still be observed and KPZ universality is found in the large deviations regime of the random walk. The Beta polymer thus shed a new light on both the DP and TD-RWRE problems. In this talk I will present the known results on and around the Beta polymer, focusing mostly on the techniques used in [2]. [1] G. Barraquand and I. Corwin, arXiv:1503:04117, Probab. Theory Relat. Fields (2016). [2] T. Thiery and P. Le Doussal, arXiv:1605.07538, JPhysA (2016).