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

TL;DR

This paper investigates self-avoiding walks and polygons on deterministic fractals, deriving exact recursion relations to compute quenched averages and revealing differences in exponents compared to annealed averages, with implications for understanding fractal lattice properties.

Contribution

The paper introduces exact recursion equations for rooted self-avoiding walks and polygons on deterministic fractals, enabling computation of quenched averages and analysis of their exponents.

Findings

Quenched and annealed averages share the same connectivity constant and radius of gyration exponent.

Exponents for quenched averages differ from annealed values, indicating distinct scaling behaviors.

Numerical estimates for the 3-simplex lattice provide precise values for the exponents lpha_q and mma_q.

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 $< P_n(S)>, <W_n(S)>, <log P_n(S)>$ and $<log W_n(S)>$ over different positions of S. We find that the connectivity constant $\mu$, and the radius of gyration exponent $\nu$ are the same for the annealed and quenched averages. However, $<log P_n(S)> ~ n log \mu + (\alpha_q -2) log n$, and $<log W_n(S)> ~ n log \mu + (\gamma_q -1)log n$, 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 annealed values $\alpha_a = 0.73421$ and $\gamma_a = 1.37522$.