Quenched averages for self-avoiding walks and polygons on deterministic fractals

Abstract

We study rooted self avoiding polygons and self avoiding walks on deterministic fractal lattices of finite ramification index. Different sites on such lattices are not equivalent, and the number of rooted open walks W_n(S), and rooted self-avoiding polygons P_n(S) of n steps depend on the root S. We use exact recursion equations on the fractal to determine the generating functions for P_n(S), and W_n(S) for an arbitrary point S on the lattice. These are used to compute the averages ⟨P_n(S)⟩,⟨W_n(S)⟩,⟨logPn(S)⟩ and ⟨logW_n(S)⟩ over different positions of S. We find that the connectivity constant μ, and the radius of gyration exponent ν are the same for the annealed and quenched averages. However, ⟨logP_n(S)⟩≃nlogμ+(α_q−2)logn, and ⟨logW_n(S)⟩≃nlogμ+(γ_q−1)logn, where the exponents α_q and γ_q, take values different from the annealed case. These are expressed as the Lyapunov exponents of random product of finite-dimensional matrices. For the 3-simplex lattice, our numerical estimation gives α_q≃0.72837±0.00001; and γ_q≃1.37501±0.00003, to be compared with the known annealed values α_a=0.73421 and γ_q=1.37522.