SPHERICALLY SYMMETRIC RANDOM WALKS II. DIMENSIONALLY DEPENDENT CRITICAL BEHAVIOR

TL;DR

This paper extends a model of random walks on hyperspherical lattices to include creation and annihilation, revealing dimension-dependent critical behavior and a second-order phase transition across all positive dimensions.

Contribution

It introduces a generalized model with birth and death processes on hyperspherical lattices, demonstrating critical phenomena for non-integer dimensions.

Findings

Steady-state distributions are obtained for all D>0.

The model exhibits a second-order phase transition.

Critical exponents depend on the dimension D.

Abstract

A recently developed model of random walks on a $D$-dimensional hyperspherical lattice, where $D$ is {\sl not} restricted to integer values, is extended to include the possibility of creating and annihilating random walkers. Steady-state distributions of random walkers are obtained for all dimensions $D>0$ by solving a discrete eigenvalue problem. These distributions exhibit dimensionally dependent critical behavior as a function of the birth rate. This remarkably simple model exhibits a second-order phase transition with a nontrivial critical exponent for all dimensions $D>0$.