Self-avoiding random walks: some exactly soluble cases

Abstract

We use the exact renormalization group equations to determine the asymptotic behavior of long self-avoiding random walks on some pseudolattices. The lattices considered are the truncated 3-simplex, the truncated 4-simplex, and the modified rectangular lattices. The total number of random walks C_n, the number of polygons P_n of perimeter n, and the mean square end to end distance <R²_n> are assumed to be asymptotically proportional to μⁿn^γ-1, μⁿn^α-3, and n^2ν respectively for large n, where n is the total length of the walk. The exact values of the connectivity constant μ, and the critical exponents λ, α, ν are determined for the three lattices. We give an example of two lattice systems that have the same effective nonintegral dimensionality 3/2 but different values of the critical exponents γ, α, and ν.