The calculation of multi-fractal properties of directed random walks on hierarchic trees with continuous branching

TL;DR

This paper investigates the multifractal properties of directed random walks on hierarchic trees with continuous branching, deriving formulas for moments and connecting reaction-diffusion equations with multi-scaling phenomena.

Contribution

It introduces a novel analysis of multifractal properties in hierarchic tree models with continuous branching, deriving new formulas and conjectures for partition function distributions.

Findings

Derived formulas for moments of the partition function.

Established multifractal properties of directed random walks.

Conjectured a power-law tail for the partition function distribution.

Abstract

We consider the hierarchic tree Random Energy Model with continuous branching and calculate the moments of the corresponding partition function. We establish the multifractal properties of those moments. We derive formulas for the normal distribution of random variables, as well as for the general case. We compare our results for the moments of partition function with corresponding results of logarithmic 1-d REM and conjecture a specific power-law tail for the partition function distribution in the high-temperature phase. Our results establish a connection between reaction-diffusion equations and multi-scaling.