Directed random walks on hierarchic trees with continuous branching: a renormalization group approach

TL;DR

This paper develops a renormalization group framework for analyzing directed random walks on hierarchic trees with continuous and random branching, deriving new reaction-diffusion equations for various probabilistic models.

Contribution

It introduces a novel renormalization group approach and derives new reaction-diffusion equations for random walks on hierarchic trees with different branching distributions.

Findings

Derived PDEs for binomial, Poisson, and compound Poisson branching models.

Established a new class of reaction-diffusion equations in 1D.

Analyzed random walks on both deterministic and stochastic hierarchic trees.

Abstract

We investigate the directed random walk on hierarchic trees. Two cases are investigated: random variables on deterministic trees with a continuous branching, and random variables on the trees constructed trough the random branching process. We derive renormalization group (partial differential) equations for the branching models with binomial, Poisson and compound Poisson distributions of random variables on the links of tree. These renormaliation group equations are new class of reaction-diffusion equations in 1-dimension.