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August 2020 Exact formulas for two interacting particles and applications in particle systems with duality
Gioia Carinci, Cristian Giardinà, Frank Redig
Ann. Appl. Probab. 30(4): 1934-1970 (August 2020). DOI: 10.1214/19-AAP1548

Abstract

We consider two particles performing continuous-time nearest neighbor random walk on $\mathbb{Z}$ and interacting with each other when they are at neighboring positions. The interaction is either repulsive (partial exclusion process) or attractive (inclusion process). We provide an exact formula for the Laplace–Fourier transform of the transition probabilities of the two-particle dynamics. From this we derive a general scaling limit result, which shows that the possible scaling limits are coalescing Brownian motions, reflected Brownian motions and sticky Brownian motions.

In particle systems with duality, the solution of the dynamics of two dual particles provides relevant information. We apply the exact formula to the the symmetric inclusion process, that is self-dual, in the condensation regime. We thus obtain two results. First, by computing the time-dependent covariance of the particle occupation number at two lattice sites we characterise the time-dependent coarsening in infinite volume when the process is started from a homogeneous product measure. Second, we identify the limiting variance of the density field in the diffusive scaling limit, relating it to the local time of sticky Brownian motion.

Citation

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Gioia Carinci. Cristian Giardinà. Frank Redig. "Exact formulas for two interacting particles and applications in particle systems with duality." Ann. Appl. Probab. 30 (4) 1934 - 1970, August 2020. https://doi.org/10.1214/19-AAP1548

Information

Received: 1 March 2018; Revised: 1 August 2019; Published: August 2020
First available in Project Euclid: 4 August 2020

MathSciNet: MR4133384
Digital Object Identifier: 10.1214/19-AAP1548

Subjects:
Primary: 60K35, 82C22

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 4 • August 2020
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