A variational formula for the free energy of the partially directed polymer collapse

TL;DR

This paper introduces a new variational approach to analyze the free energy of a partially directed polymer model, proving the collapse transition's existence, identifying the critical temperature, and determining its order.

Contribution

It develops a novel variational formula for the free energy of the model, enabling a rigorous proof of the collapse transition and its critical properties.

Findings

Proves the existence of the collapse transition.

Identifies the critical temperature for the transition.

Determines the transition order as 3/2.

Abstract

Long linear polymers in dilute solutions are known to undergo a collapse transition from a random coil (expand itself) to a compact ball (fold itself up) when the temperature is lowered, or the solvent quality deteriorates. A natural model for this phenomenon is a 1+1 dimensional self-interacting and partially directed self-avoiding walk. In this paper, we develop a new method to study the partition function of this model, from which we derive a variational formula for the free energy. This variational formula allows us to prove the existence of the collapse transition and to identify the critical temperature in a simple way. We also prove that the order of the collapse transition is 3/2.