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 $\langle P_{n}(S) \rangle$,$\langle W_{n}(S) \rangle$,$\langle \log P_{n}(S) \rangle$ and $\langle \log W_{n}(S) \rangle$ over different positions of S. We find that the connectivity constant μ, and the radius of gyration exponent $\nu$ are the same for the annealed and quenched averages. However, $\langle \log P_{n}(S) \rangle \simeq n \log \mu + (\alpha_q - 2)\log n$, and $\langle \log W_{n}(S) \rangle \simeq n \log \mu + (\gamma_q-1) log{n}$, where the exponents $\alpha_q$ and $\gamma_q$, 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 $\alpha_q \simeq 0.72837 \pm 0.00001;$ and $\gamma_q \simeq 1.37501 \pm 0.00003$, to be compared with the known annealed values $\alpha_a = 0.73421$ and $\gamma_q = 1.37522$.