Interacting partially directed self-avoiding walk: a probabilistic perspective

Abstract

We review some recent results obtained in the framework of the 2-dimensional Interacting Self-Avoiding Walk (ISAW). After a brief presentation of the rigorous results that have been obtained so far for ISAW we focus on the Interacting Partially Directed Self-Avoiding Walk (IPDSAW), a model introduced in Zwanzig and Lauritzen (1968) to decrease the mathematical complexity of ISAW.In the first part of the paper, we discuss how a new probabilistic approach based on a random walk representation (see Nguyen and Pétrélis (2013)) allowed for a sharp determination of the asymptotics of the free energy close to criticality (see Carmona, Nguyen and Pétrélis (2016)). Some scaling limits of IPDSAW were conjectured in the physics literature (see e.g. Brak et al. (1993)). We discuss here the fact that all limits are now proven rigorously, i.e., for the extended regime in Carmona and Pétrélis (2016), for the collapsed regime in Carmona, Nguyen and Pétrélis (2016) and at criticality in Carmona and Pétrélis (2017a).The second part of the paper starts with the description of four open questions related to physically relevant extensions of IPDSAW. Among such extensions is the Interacting Prudent Self-Avoiding Walk (IPSAW) whose configurations are those of the 2-dimensional prudent walk. We discuss the main results obtained in Pétrélis and Torri (2016+) about IPSAW and in particular the fact that its collapse transition is proven to exist rigorously.MSC 2010 subject classifications: Primary 60K35; Secondary 82B26, 82B41.

Localization of a one-dimensional simple random walk among power-law renewal obstacles

“…Step 3. Let us now prove that ℓ 0 is uniquely defined and actually coincides with ℓ 0 for n large enough, with probability larger than 1 − p. To this end, let us first notice that for every η ∈ (0, 1), every ℓ that minimizes G n , on the intersection of (B.26) and A (10) n (ε 0 , η), and for n large enough (so that, in particular, A…”

“…This phenomenon can be observed in various models and under different forms. Let us mention for instance the collapse transition of a polymer in a poor solvent [10], the pinning of a polymer on a defect line [22,23], Anderson localization [27], localization of a polymer in a heterogeneous medium [12], confinement of random walks among obstacles [41]. Such models, which are often motivated by Biology, Chemistry or Physics, offer challenging mathematical problems and have been an active field of research.…”

A sharp asymptotics of the partition function for the collapsed interacting partially directed self-avoiding walk

“…We also recall from (2.6) that A N defines the algebraic area enclosed in-between a random walk trajectory and the x-axis after N steps. Recall (3.1) and (3.2), and let us now briefly remind the transformation that allows us to give a probabilistic representation of Z • L,β (we refer to [3] for a review on the recent progress made on IPDSAW by using probabilistic tools). First, note that for x, y ∈ Z one can write x ∧ y = 1 2 (|x| + |y| − |x + y|).…”

“…Appendix A. Local limit estimates, proof of Lemma 5.12 We will prove Lemma 5.12 subject to Proposition A.1 and Lemma A.2 that are stated below and which were proven in [3,Section 6]. To that aim, we recall (5.5-5.8) and we set B(h) = Hess L Λ (h), h ∈ D, (A.1) and…”

Limit theorems for random walk excursion conditioned to enclose a typical area

“…We refer to [5] for a recent topical review about IPDSAW, but let us introduce the model in a few words. The IPDSAW is a polymer model whose configurations (of size L ∈ N) are given by the L-step trajectories of a self-avoiding walk taking unit steps up, down, and right.…”