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

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. NotationsLet (a L ) L≤1 and (b L ) L≤1 be two sequences of positive numbers. We will writeWe will also write (const.) to denote generic positive constant whose value may change from line to line.