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Dissipative particle systems on expanders

John Haslegrave, Peter Keevash

TL;DR

This work develops a general dissipative framework for multi-type interacting particle systems on graphs and analyzes the equilibrium time—the time until no further interactions can occur. Central contributions include a Moving Target Lemma, Poissonised lonely-walker analysis, and abelian-sandpile toppling techniques, enabling tight $\Theta(n\log n)$ bounds on the equilibrium time for broad classes of models, especially on expander graphs with fixed speeds. The authors treat both general agential dissipative systems and the harder balanced two-type annihilation, deriving unified upper bounds via rounds, Hall matching, and careful mixing/hitting arguments, and matching lower bounds through trajectory-reversal, occupancy, and probabilistic combinatorics. The results extend prior mean-field and lattice-based analyses to general graphs and variable speeds, with implications for chemical-reaction-like dynamics, infection models, predator-prey interactions, and diffusion on networks. The work introduces new combinatorial tools that can handle non-monotone interactions and non-integrable dynamics, offering a robust framework for finite-graph particle systems with dissipative interactions.

Abstract

We consider a general framework for multi-type interacting particle systems on graphs, where particles move one at a time by random walk steps, different types may have different speeds, and may interact, possibly randomly, when they meet. We study the equilibrium time of the process, by which we mean the number of steps taken until no further interactions can occur. Under a rather general framework, we obtain high probability upper and lower bounds on the equilibrium time that match up to a constant factor and are of order $n\log n$ if there are order $n$ vertices and particles. We also obtain similar results for the balanced two-type annihilation model of chemical reactions; here, the balanced case (equal density of types) does not fit into our general framework and makes the analysis considerably more difficult. Our models do not admit any exact solution as for integrable systems or the duality approach available for some other particle systems, so we develop a variety of combinatorial tools for comparing processes in the absence of monotonicity.

Dissipative particle systems on expanders

TL;DR

This work develops a general dissipative framework for multi-type interacting particle systems on graphs and analyzes the equilibrium time—the time until no further interactions can occur. Central contributions include a Moving Target Lemma, Poissonised lonely-walker analysis, and abelian-sandpile toppling techniques, enabling tight bounds on the equilibrium time for broad classes of models, especially on expander graphs with fixed speeds. The authors treat both general agential dissipative systems and the harder balanced two-type annihilation, deriving unified upper bounds via rounds, Hall matching, and careful mixing/hitting arguments, and matching lower bounds through trajectory-reversal, occupancy, and probabilistic combinatorics. The results extend prior mean-field and lattice-based analyses to general graphs and variable speeds, with implications for chemical-reaction-like dynamics, infection models, predator-prey interactions, and diffusion on networks. The work introduces new combinatorial tools that can handle non-monotone interactions and non-integrable dynamics, offering a robust framework for finite-graph particle systems with dissipative interactions.

Abstract

We consider a general framework for multi-type interacting particle systems on graphs, where particles move one at a time by random walk steps, different types may have different speeds, and may interact, possibly randomly, when they meet. We study the equilibrium time of the process, by which we mean the number of steps taken until no further interactions can occur. Under a rather general framework, we obtain high probability upper and lower bounds on the equilibrium time that match up to a constant factor and are of order if there are order vertices and particles. We also obtain similar results for the balanced two-type annihilation model of chemical reactions; here, the balanced case (equal density of types) does not fit into our general framework and makes the analysis considerably more difficult. Our models do not admit any exact solution as for integrable systems or the duality approach available for some other particle systems, so we develop a variety of combinatorial tools for comparing processes in the absence of monotonicity.
Paper Structure (24 sections, 28 theorems, 28 equations)

This paper contains 24 sections, 28 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Balanced two-type annihilation
  3. Dissipative particle systems
  4. Simultaneous movement and the Big Bang
  5. Special models
  6. Comparison with previous results
  7. Methods and organisation
  8. Notation
  9. General models
  10. Hitting moving targets
  11. Poissonised lonely walkers
  12. Hitting times and toppling
  13. Few predators
  14. Annihilation
  15. Annihilation upper bound
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $G$ be a regular graph on $n$ vertices with spectral gap at least $0.425$. Consider balanced two-type annihilation on $G$ from any starting configuration and let $T$ be the extinction time. Then $cn\log n\leq T\leq Cn\log n$ with high probability and in expectation, where $c$ and $C$ are absolut

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark
  • Corollary 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Lemma 2.1
  • ...and 44 more