November 2023 Particle density in diffusion-limited annihilating systems
Tobias Johnson, Matthew Junge, Hanbaek Lyu, David Sivakoff
Author Affiliations +
Ann. Probab. 51(6): 2301-2344 (November 2023). DOI: 10.1214/23-AOP1653


Place an A-particle at each site of a graph independently with probability p, and otherwise place a B-particle. A- and B-particles perform independent continuous time random walks at rates λA and λB, respectively, and annihilate upon colliding with a particle of opposite type. Bramson and Lebowitz studied the setting λA=λB in the early 1990s. Despite recent progress, many basic questions remain unanswered when λAλB. For the critical case p=1/2 on low-dimensional integer lattices, we give a lower bound on the expected number of particles at the origin that matches physicists’ predictions. For the process with λB=0 on the integers and on the bidirected regular tree, we give sharp upper and lower bounds for the expected total occupation time of the root at and approaching criticality.

Funding Statement

Johnson was partially supported by NSF Grant DMS-1811952, Junge by NSF Grant-185551, and Lyu by NSF Grants DMS-2206296 and DMS-2010035. Lyu and Sivakoff were partially supported by NSF Grant CCF-1740761.


We thank Michael Damron for helpful feedback and his assistance with Section 4.2.


Download Citation

Tobias Johnson. Matthew Junge. Hanbaek Lyu. David Sivakoff. "Particle density in diffusion-limited annihilating systems." Ann. Probab. 51 (6) 2301 - 2344, November 2023.


Received: 1 July 2022; Revised: 1 July 2023; Published: November 2023
First available in Project Euclid: 12 November 2023

Digital Object Identifier: 10.1214/23-AOP1653

Primary: 60K35

Keywords: Critical behavior , Interacting particle system

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 6 • November 2023
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