Description: tba

Description: Random walk in random potential has been studied as a stochastic process related to the Anderson Hamiltonian since 1970’s. The aim of this course is to explain some recent progress in the case of hard core potentials. In this case, the annealed law of the random walk can be regarded as a model of a polymer with self-attractive interaction. In 1990’s, it was proved that the polymer is localized (or folded) in a ball at all temperature. But the structure inside that ball remained unclear in dimensions three and higher. Recently, it has been proved that the ball is filled by the polymer, which implies that the polymer macroscopically looks like a solid ball. In addition, a partial result on the boundary fluctuation has been obtained. I will explain the basic ideas of the proofs of the localization, ball covering, and boundary fluctuation. Main part of this course will be based on a joint work with Jian Ding, Rongfeng Sun, and Changji Xu. ​

Description: We will describe and study the Interacting Partially-Directed Self-Avoiding Walk (IPDSAW), a model initially introduced in 1968 by Zwanzig and Lauritzen to investigate the collapse transition of an homopolymer dipped in a poor solvent. In size L, the allowed configurations of the model are given by the L-step trajectories of a self-avoiding walk on Z2 that takes unitary steps up, down and to the right. The coupling parameter is β≥0 and the Hamiltonian associated with every configuration is β times the number of « self-touching » of that configuration. This model was known to undergo a collapse transition at some critical parameter βc>0 (see e.g. [Brak, Guttmann & Whittington 92]) between an extended phase {β<βc} and a collapsed phase {β≥βc}. 
We will begin by displaying a new probabilistic representation of the partition function, based on an auxiliary random walk, which relates some geometric features of IPDSAW trajectories to that of a particular random walk conditioned on enclosing a prescribed geometric area. This method allowed us to push the mathematical understanding of IPDSAW some steps further in the last years. We will illustrate this new family of results by providing a sharp geometric description of a typical configuration of IPDSAW both inside the collapse phase {β>βc}, the extended phase {β<βc} and also at criticality {β=βc}. 
Finally, we will consider the 2-dimensional Interacting Prudent Walk (IPRW), an extension of the IPDSAW that is not directed. Using some particular decomposition of prudent paths into partially directed paths, we will prove that the IPRW undergoes a collapse transition as well. ​