Collapse transition of the interacting prudent walk

This article is dedicated to the study of the 2-dimensional interacting prudent self-avoiding walk (referred to by the acronym IPSAW) and in particular to its collapse transition. The interaction intensity is denoted by $\beta>0$ and the set of trajectories consists of those self-avoiding paths respecting the prudent condition, which means that they do not take a step towards a previously visited lattice site. The IPSAW interpolates between the interacting partially directed self-avoiding walk (IPDSAW) that was analyzed in details in, e.g., [Zwanzig and Lauritzen, 1968], [Brak et al., 1992], [Carmona, Nguyen and P\'etr\'elis, 2013-2016], and the interacting self-avoiding walk (ISAW) for which the collapse transition was conjectured in [Saleur, 1986]. Three main theorems are proven. We show first that IPSAW undergoes a collapse transition at finite temperature and, up to our knowledge, there was so far no proof in the literature of the existence of a collapse transition for a non-directed model built with self-avoiding path. We also prove that the free energy of IPSAW is equal to that of a restricted version of IPSAW, i.e., the interacting two-sided prudent walk. Such free energy is computed by considering only those prudent path with a general north-east orientation. As a by-product of this result we obtain that the exponential growth rate of generic prudent paths equals that of two-sided prudent paths and this answers an open problem raised in e.g., [Bousquet-M\'elou, 2010] or [Dethridge and Guttmann, 2008]. Finally we show that, for every $\beta>0$, the free energy of ISAW itself is always larger than $\beta$ and this rules out a possible self-touching saturation of ISAW in its conjectured collapsed phase.