Universality in Random Walk Models with Birth and Death

TL;DR

This paper investigates a class of random walk models with birth and death processes, revealing universal critical behavior and phase transitions dependent on spatial dimension, with implications for polymer adsorption at curved interfaces.

Contribution

It introduces a new class of random walk models with uniform death rates, deriving exact critical behavior and demonstrating universality across different lattice types.

Findings

Exact second-order phase transition in hyperspherical lattices

Universal critical exponents across various lattice geometries

Implications for polymer adsorption at curved interfaces

Abstract

Models of random walks are considered in which walkers are born at one location and die at all other locations with uniform death rate. Steady-state distributions of random walkers exhibit dimensionally dependent critical behavior as a function of the birth rate. Exact analytical results for a hyperspherical lattice yield a second-order phase transition with a nontrivial critical exponent for all positive dimensions $D\neq 2,~4$. Numerical studies of hypercubic and fractal lattices indicate that these exact results are universal. Implications for the adsorption transition of polymers at curved interfaces are discussed.