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A variational formula for the free energy of the partially directed polymer collapse. (English) Zbl 1273.82079

Summary: 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 provide a probabilistic proof of the fact that the collapse transition is of second order with critical exponent 3/2.

MSC:

82D60 Statistical mechanics of polymers
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
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