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The authors consider a directed walk model of linear polymers in dilute solution, with an energy associated with the number of near-neighbour contacts in the walk. For this model they can derive an exact expression for the generating function in two variables conjugate to the number of steps and the number of contacts. They discuss the analytic structure of this generating function and identify the transition corresponding to collapse.
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Brak et al. (1992) studied this question.
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