Abstract:
We investigate the 1 + 1 dimensional self-interacting and partially directed self-avoiding walk, usually referred to by the acronym IPDSAW. The IPDSAW is known to undergo a extended-collapsed transition at a critical point \beta_c. We present here a new method that provides a probabilistic representation of the partition function, from which we derive a variational formula for the free energy. This variational formula allows us to prove the existence of the collapse transition and to identify the critical point in a simple way. We also provide the precise asymptotic of the free energy close to criticality and establish some path properties of the random walk inside the collapsed phase. This is a joint work with Nicolas Pétrélis and Philippe Carmona (University of Nantes, France).