Exact solution of weighted partially directed walks crossing a square

TL;DR

This paper provides an exact analytical solution for weighted partially directed walks crossing a square, revealing a phase transition with distinct scaling behaviors of the partition function depending on the fugacity parameter.

Contribution

It introduces an exact solution for the model in different regimes of the fugacity, detailing the asymptotic behavior and phase transition characteristics.

Findings

Identifies three scaling regimes for the partition function based on fugacity value.

Derives explicit asymptotic expressions for each phase.

Shows a phase transition from dilute to dense phases at critical fugacity.

Abstract

We consider partially directed walks crossing a $L\times L$ square weighted according to their length by a fugacity $t$. The exact solution of this model is computed in three different ways, depending on whether $t$ is less than, equal to or greater than 1. In all cases a complete expression for the dominant asymptotic behaviour of the partition function is calculated. The model admits a dilute to dense phase transition, where for $0 < t < 1$ the partition function scales exponentially in $L$ whereas for $t>1$ the partition function scales exponentially in $L^2$, and when $t=1$ there is an intermediate scaling which is exponential in $L \log{L}$.