Scaling limit of the uniform prudent walk

We study the 2-dimensional uniform prudent self-avoiding walk, which assigns equal probability to all nearest-neighbor self-avoiding paths of a fixed length that respect the prudent condition, namely, the path cannot take any step in the direction of a previously visited site. The uniform prudent walk has been investigated with combinatorial techniques in [Bousquet-M\'elou, 2010], while another variant, the kinetic prudent walk has been analyzed in detail in [Beffara, Friedli and Velenik, 2010]. In this paper, we prove that the $2$-dimensional uniform prudent walk is ballistic and follows one of the $4$ diagonals with equal probability. We also establish a functional central limit theorem for the fluctuations of the path around the diagonal.Comment: 16 pages, 5 figure

Collapse transition of the interacting prudent walk

33 pages, 13 figuresInternational audienceThis 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), Carmona et al. (2016) 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 northeast 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

One-dimensional polymers in random environments: stretching vs. folding

In this article we study a \emph{non-directed polymer model} on $\mathbb Z$, that is a one-dimensional simple random walk placed in a random environment. More precisely, the law of the random walk is modified by the exponential of the sum of ``rewards'' (or penalities) $\beta \omega_x -h$ sitting on the range of the random walk, where $(\omega_x)_{x\in \mathbb Z}$ are i.i.d.\ random variables (the disorder), and where $\beta\geq 0$ (disorder strength) and $h\in \mathbb{R}$ (external field) are two parameters. When $\beta=0,h>0$, this corresponds to a random walk penalized by its range; when $\beta>0, h=0$, this corresponds to the ``standard'' polymer model in random environment, except that it is non-directed. In this work, we allow the parameters $\beta,h$ to vary according to the length of the random walk, and we study in detail the competition between the \emph{stretching effect} of the disorder, the \emph{folding effect} of the external field (if $h\ge 0$), and the \emph{entropy cost} of atypical trajectories. We prove a complete description of the (rich) phase diagram. For instance, in the case $\beta>0, h=0$ of the non-directed polymer, if \go_x ha a finite second moment, we find a transversal fluctuation exponent $\xi=2/3$, and we identify the limiting distribution of the rescaled log-partition function