Interacting partially directed self avoiding walk: scaling limits

Abstract

This paper is dedicated to the investigation of a $1+1$ dimensional self-interacting and partially directed self-avoiding walk. The intensity of the interaction between monomers is denoted by $\beta \in (0,\infty )$ and there exists a critical threshold $\beta _c$ which determines the three regimes displayed by the model, i.e., extended for $\beta <\beta _c$, critical for $\beta =\beta _c$ and collapsed for $\beta >\beta _c$.

In [4], physicists displayed some numerical results concerning the typical growth rate of some geometric features of the path as its length $L$ diverges. From this perspective the quantities of interest are the horizontal extension of the path and its lower and upper envelopes.

With the help of a new random walk representation, we proved in [10] that the path grows horizontally like $\sqrt{L} $ in its collapsed regime and that, once rescaled by $\sqrt{L} $ vertically and horizontally, its upper and lower envelopes converge towards some deterministic Wulff shapes.

In the present paper, we bring the geometric investigation of the path several steps further. In the collapsed regime, we identify the joint limiting distribution of the fluctuations of the upper and lower envelopes around their associated limiting Wulff shapes, rescaled in time by $\sqrt{L} $ and in space by $L^{1/4}$. In the critical regime we identify the limiting distribution of the horizontal extension rescaled by $L^{2/3}$ and we show that the excess partition function decays as $L^{2/3}$ with an explicit prefactor. In the extended regime, we prove a law of large number for the horizontal extension of the polymer rescaled by its total length $L$, we provide a precise asymptotics of the partition function and we show that its lower and upper envelopes, once rescaled in time by $L$ and in space by $\sqrt{L} $, converge towards the same Brownian motion.