Indexed on: 23 Feb '18Published on: 23 Feb '18Published in: arXiv - Mathematics - Probability
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 P\'etr\'elis (2013)) allowed for a sharp determination of the asymptotics of the free energy close to criticality (see Carmona, Nguyen and P\'etr\'elis (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 P\'etr\'elis (2016), for the collapsed regime in Carmona, Nguyen and P\'etr\'elis (2016) and at criticality in Carmona and P\'etr\'elis (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 P\'etr\'elis and Torri (2016+) about IPSAW and in particular the fact that its collapse transition is proven to exist rigorously.