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

TL;DR

This paper rigorously derives sharp asymptotics for the partition function of the collapsed phase of the interacting partially-directed self-avoiding walk, confirming a longstanding conjecture and describing the typical structure of trajectories.

Contribution

It provides the first rigorous proof of the asymptotic behavior of the partition function in the collapsed phase of IPDSAW, confirming previous conjectures.

Findings

Sharp asymptotics of the partition function inside the collapsed phase

Confirmation of a conjecture from Guttmann (2015) and Owczarek et al. (1993)

Typical IPDSAW trajectory consists of a single macroscopic bead with long vertical stretches

Abstract

In the present paper, we investigate the collapsed phase of the interacting partially-directed self-avoiding walk (IPDSAW) that was introduced in Zwanzig and Lauritzen (1968). We provide sharp asymptotics of the partition function inside the collapsed phase, proving rigorously a conjecture formulated in Guttmann (2015) and Owczarek et al. (1993). As a by-product of our result, we obtain that, inside the collapsed phase, a typical IPDSAW trajectory is made of a unique macroscopic bead, consisting of a concatenation of long vertical stretches of alternating signs, outside which only finitely many monomers are lying.