Exact solution of asymmetric gelation between three walks on the square lattice

TL;DR

This paper provides an exact solution for a model of three asymmetric polymers on a square lattice, revealing a four-phase diagram and detailed phase transition analysis using functional equations.

Contribution

It introduces an exact analytical solution for asymmetric contact interactions among three directed walks, extending previous symmetric models.

Findings

Four distinct phases identified in the phase diagram.

Explicit calculation of entropic exponents for each phase.

Determination of the order of phase transitions.

Abstract

We find and analyse the exact solution of a model of three different polymers with asymmetric contact interactions in two dimensions, modelling a scenario where there are different types of polymers involved. In particular, we find the generating function of three directed osculating walks in star configurations on the square lattice with two interaction Boltzmann weights, so that there is one type of contact interaction between the top pair of walks and a different interaction between the bottom pair of walks. These osculating stars are found to be the most amenable to exact solution using functional equation techniques in comparison to the symmetric case where three friendly walks in watermelon configurations were successfully solved with the same techniques. We elucidate the phase diagram, which has four phases, and find the order of all the phase transitions between them. We also calculate the entropic exponents in each phase.