Directed Random Walk on the Lattices of Genus Two

TL;DR

This paper studies directed random walks on complex lattices derived from genus two Riemann surfaces, analyzing multifractal properties through numerical and statistical methods.

Contribution

It introduces a novel model of directed random walks on genus two lattice graphs and computes their multifractal scaling exponents.

Findings

Multifractal scaling exponents depend on lattice parameters.

Numerical and statistical methods agree on the multifractal behavior.

The model reveals complex scaling properties of genus two lattice graphs.

Abstract

The object of the present investigation is an ensemble of self-avoiding and directed graphs belonging to eight-branching Cayley tree (Bethe lattice) generated by the Fucsian group of a Riemann surface of genus two and embedded in the Pincar\'e unit disk. We consider two-parametric lattices and calculate the multifractal scaling exponents for the moments of the graph lengths distribution as functions of these parameters. We show the results of numerical and statistical computations, where the latter are based on a random walk model.