Exact critical properties of two-dimensional dense self-avoiding walks

Abstract

We consider two-dimensional dense self-avoiding walks (SAW). They are obtained by putting a chain, and more generally a network G of fixed topology, made of N polymers of equal length l, on a finite lattice of Λ sites, and taking the limit l, Λ → ∞ with a finite density Nl/Λ → ƒ > 0 . We show how this system is described by the low-temperature phase of the n = 0 vector model. The entropy defines a connectivity constant μ D (ƒ) which we determine numerically using transfer matrix calculations. The exponent νD is simply 1 2 but the determination of gamma exponents is more delicate. Since dense polymers are sensitive to boundary conditions we argue that γ G D which gives the ratio of the number of configurations of G and the number of configurations of one loop of the same lenght ω G,l D /ω 0, Nl D ∼l γ G D , l ⪢ 1 should be universal and we give its exact value using Coulomb gas methods: γ G D = Σ L⩾1 1 32 n L(2 − L)(L + 18), n L being the number of L-leg vertices in G . The values of the contact exponents for a dense SAW are derived from this result. The asymptotic behaviour of ω G l D itself depends on boundary conditions, and in the case of periodic ones we propose ω G,l D ∼ [μ D (ƒ)] Nl l γ G−1 , l ⪢ 1 with γ G γ G D + 1 . The values for the simplest networks are tested using a Monte Carlo simulation. We also consider free boundary conditions. The related surface exponents for dense SAW are then given. We discuss the relations with conformal invariance formalism. Finally we calculate exactly the continuum limit of some partition functions for dense SAW on a torus. In particular we recover an old result obtained by Kasteleyn for the Manhattan lattice.