One-dimensional random walks with self-blocking immigration

TL;DR

This paper studies a one-dimensional random walk system with self-blocking immigration, confirming a predicted growth rate of particles and describing the large-scale particle distribution.

Contribution

It introduces a model of self-blocking immigration in 1D random walks and rigorously confirms the predicted particle growth and macroscopic profile.

Findings

Total particles grow as c√t log t from empty state

Asymptotic particle distribution profile characterized

Poisson ansatz accurately predicts system behavior

Abstract

We consider a system of independent one-dimensional random walkers where new particles are added at the origin at fixed rate whenever there is no older particle present at the origin. A Poisson ansatz leads to a semi-linear lattice heat equation and predicts that starting from the empty configuration the total number of particles grows as $c \sqrt{t} \log t$. We confirm this prediction and also describe the asymptotic macroscopic profile of the particle configuration.