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

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