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Contact matrices provide a coarse grained description of the configuration omega of a linear chain (polymer or random walk) on Z^n: C_ij(omega)=1 when the distance between the position of the i-th and j-th step are less than or equal to some distance "a" and C_ij(omega)=0 otherwise. We consider models in which polymers of length N have weights corresponding to simple and self-avoiding random walks, SRW and SAW, with "a" the minimal permissible distance. We prove that to leading order in N, thedoi:10.1088/0305-4470/36/21/303 fatcat:rkfsrnbnuvgs7g3pgrc4kamzya