Interacting partially directed self avoiding walk : scaling limits

TL;DR

This paper studies the geometric and scaling properties of a 1+1 dimensional self-interacting, partially directed self-avoiding walk, revealing new limit theorems and asymptotics across different regimes including collapsed, extended, and critical phases.

Contribution

It introduces a new random walk representation and provides detailed asymptotic results and limit theorems for the path's geometric features in various regimes.

Findings

Horizontal extension grows like  in the collapsed regime

Rescaled envelopes converge to Brownian motion in the extended regime

At criticality, the horizontal extension scaled by L^{2/3} converges in distribution

Abstract

This paper is dedicated to the investigation of a $1+1$ dimensional self-interacting and partially directed self-avoiding walk, usually referred to by the acronym IPDSAW and introduced in \cite{ZL68} by Zwanzig and Lauritzen to study the collapse transition of an homopolymer dipped in a poor solvant. In \cite{POBG93}, physicists displayed 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 projections of the path onto the horizontal axis (also called horizontal extension) and onto the vertical axis for which it is useful to define the lower and the upper envelopes of the path. With the help of a new random walk representation, we proved in \cite{CNGP13} 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 to some deterministic Wulff shapes. In the present paper, we bring the geometric investigation of the path several steps further. 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 to the same Brownian motion. At criticality, 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 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 $L^{1/4}$.