Exact solution of two friendly walks above a sticky wall with single and double interactions

TL;DR

This paper provides an exact analytical solution for a model of two directed polymer-like walks interacting with a wall, revealing a complex phase diagram with multiple phase transitions.

Contribution

It introduces a solvable model of two friendly walks with separate wall interaction parameters, analyzing the phase diagram and transition types.

Findings

Exact generating function solution obtained

Three distinct phases identified with different transition types

Two phase transitions predicted as temperature varies

Abstract

We find, and analyse, the exact solution of two friendly directed walks, modelling polymers, which interact with a wall via contact interactions. We specifically consider two walks that begin and end together so as to imitate a polygon. We examine a general model in which a separate interaction parameter is assigned to configurations where both polymers touch the wall simultaneously, and investigate the effect this parameter has on the integrability of the problem. We find an exact solution of the generating function of the model, and provide a full analysis of the phase diagram that admits three phases with one first-order and two second-order transition lines between these phases. We argue that one physically realisable model would see two phase transitions as the temperature is lowered.