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# Annales de l’Institut Henri Poincaré D

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**Volume 5, Issue 3, 2018, pp. 387–435**

**DOI: 10.4171/AIHPD/58**

Published online: 2018-07-25

Collapse transition of the interacting prudent walk

Nicolas Pétrélis^{[1]}and Niccolò Torri

^{[2]}(1) Université de Nantes, France

(2) Université de Nantes, France

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., [16], [4], [5] and [10], and the interacting self-avoiding walk (ISAW) for which the collapse transition was conjectured in [11].

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., [3] or [8]. 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.

*Keywords: *Polymer collapse, phase transition, prudent walk, self-avoiding random walk, free energy

Pétrélis Nicolas, Torri Niccolò: Collapse transition of the interacting prudent walk. *Ann. Inst. Henri Poincaré Comb. Phys. Interact.* 5 (2018), 387-435. doi: 10.4171/AIHPD/58