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Random Matrix Solution of a Polymer Collapse Model

A polymer folding model on the square lattice is constructed with attractive contact interactions of strength 1/c^2, 0<c<1. The corresponding model on a dynamical random lattice, with freely fluctuating co-ordination number at each vertex, is formulated as a random two-matrix model and an expression for the partition function of a length-L chain is derived. Numerical estimates and analytical evaluation for L \to \infty shows a third-order collapse transition at c=\sqrt{2}-1. Geometrical critical exponents are computed in each phase and interpreted. The Knizhnik-Polyakov-Zamolodchikov 2D quantum gravity scaling relations are used to predict the corresponding behaviour on the regular lattice, which lies in a different universality class from the percolation Theta-point of Duplantier and Saleur.