A Variational Formula for the Free Energy of the Partially Directed Polymer Collapse |
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Authors: | Gia Bao Nguyen Nicolas Pétrélis |
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Institution: | 1. Laboratoire de Mathématiques Jean Leray UMR 6629, Université de Nantes, 2 Rue de la Houssinière, BP 92208, 44322, Nantes Cedex 03, France
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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 provide a probabilistic proof of the fact that the collapse transition is of second order with critical exponent 3/2. |
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