Collapse transition of the interacting prudent walk

Abstract

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 β > 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), and Nguyen and Pétrélis (2013), 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-Mélou (2010) or Dethridge and Guttmann (2008). Finally we show that, for every β > 0, the free energy of ISAW itself is always larger than β and this rules out a possible self-touching saturation of ISAW in its conjectured collapsed phase.2010 Mathematics Subject Classification. 82B26, 60K35, 82B41, 60K15.

Interacting partially directed self-avoiding walk: a probabilistic perspective

“…The prudent model is, to our knowledge, the only non-directed model of a 2-dimensional interacting self-avoiding walk for which the existence of a collapse transition has been proven rigorously. This is the main result in Pétrélis and Torri (2016+) The proof of Theorem 5.4 is purely combinatorial. It consists in building a sequence of path transformations (M L ) L∈ such that for every L ∈ , M L maps Ω PSAW L onto Ω NE L and satisfies the following properties:…”

“…Thus, one could consider a model that interpolates between IPDSAW and ISAW, in the sense that its allowed configurations are not directed anymore and have a connective constant strictly between that of IPDSAW trajectories and that of ISAW trajectories. This is the case for the Interacting Prudent Self-Avoiding Walk that has been investigated recently in Pétrélis and Torri (2016+) and will be discussed further in Section 5 below.…”

“…In Sections 2 and 3, we will use the second definition since it fits with the probabilistic approach that we wish to display. However, in Section 5 we will come back to the original definition to present the IPSAW, a non-directed extension of IPDSAW that has been investigated in Pétrélis and Torri (2016+).…”

“…In the spirit of the second class of open problems mentioned in Section 4, the Interacting Prudent Self-Avoiding Walk (IPSAW) is studied in Pétrélis and Torri (2016+). In size L ∈ , the set of configurations consists of the L-step prudent paths introduced in Turban and Debierre (1987), i.e.,…”

“…A new probabilistic approach of IPDSAW has been introduced in Nguyen and Pétrélis (2013) which turned out to strongly simplify its investigation. In the present paper we review the results obtained using this new framework concerning the analytic properties of free energy in Carmona, Nguyen and Pétrélis (2016) and Pétrélis and Torri (2016+) and also concerning the path properties of IPDSAW in and Carmona and Pétrélis (2017b). For every result that we state here, we provide a sketch of its rigorous proof.…”