Two mutually repelling self-avoiding walks: a Monte Carlo study

Abstract

We consider two self-avoiding walks (SAWs) on a square lattice, beginning simultaneously from two arbitrary nearest neighbours. The walks cross neither themselves nor each other, and grow simultaneously. Using Monte Carlo technique we study the variation, with the length N of the walks, of the average end-to-end distance < R_N> of each walk and of their average separation <S_N>. We find <R_N²> ∞ N^2ν, < S_N²> ∞ N^2λ, where ν ≈ λ ≈ 0.75 for two ordinary SAWs and ν ≈ λ ≈ 0.67 for two growing SAWs in two dimensions. For two directed SAWs, we find < R_Nǁ_/⊥² > ∞ N^2νǁ_/⊥ and < S_Nǁ/² > ∞ N^2λǁ_/⊥, where νǁ ≈ 1.00, ν_⊥ ≈ 0.56, λ_⊥≈ 0.50 and λ_⊥ ≈ 0.59 in two dimensions. We thus find an indirect excluded volume effect on one directed SAW, due to the other.