The collapse transition of self-interacting random walks is a challenging issue, arising in the study of an homopolymer dipped in a repulsive solvent. Different mathematical models have been built by physicists to try and improve their understanding of this phenomenon. For such models, the possible spatial configurations of the polymer are provided by self-avoiding walk trajectories. However, self avoiding walks, especially in dimension 2 and 3, are complicated objects.
This is the reason why, in the mathematical literature, collapse transition models were rather built by either relaxing the self-avoiding feature of the walk or by considering partially directed path. In our main result, we prove the existence of a collapse transition for a non-directed model built with prudent paths, i.e., non-directed self-avoiding paths which can not take a step towards a previously visited lattice site. The prudent self-avoiding walk has attracted the attention of the combinatorics community over recent years and an open problem is to find out the exponential growth rate of the set of prudent paths. As a by product of our results we are able to solve this problem.
In the first part of the talk we discuss the main models already studied in the literature, while in the second part of the talk we present our main results.
The presentation will be kept at an elementary level: only a basic probabilistic background is assumed.
(Joint work with Nicolas Pétrélis)