Diffusion-limited annihilating systems and the increasing convex order

TL;DR

This paper investigates how increasing the initial number or variability of particles in diffusion-limited annihilating systems affects the total occupation time, revealing a monotonic relationship across various graph structures.

Contribution

It establishes that augmenting initial particle configurations increases occupation time and lifespan, extending the understanding of these systems on broad graph families.

Findings

Increasing initial particle size raises occupation time.

Higher variability in initial placements increases total occupation.

Results apply to internal diffusion-limited aggregation lifespan.

Abstract

We consider diffusion-limited annihilating systems with mobile $A$-particles and stationary $B$-particles placed throughout a graph. Mutual annihilation occurs whenever an $A$-particle meets a $B$-particle. Such systems, when ran in discrete time, are also referred to as parking processes. We show for a broad family of graphs and random walk kernels that augmenting either the size or variability of the initial placements of particles increases the total occupation time by $A$-particles of a given subset of the graph. A corollary is that the same phenomenon occurs with the total lifespan of all particles in internal diffusion-limited aggregation.