%0 Journal Article
%T A variational formula for the free energy of the partially directed polymer collapse
%A Gia Bao Nguyen
%A Nicolas Petrelis
%J Mathematics
%D 2012
%I arXiv
%R 10.1007/s10955-013-0748-2
%X 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.
%U http://arxiv.org/abs/1211.0925v1