Abstract
We analyze the -mixing of a generalization of the averaging process introduced by Aldous (2011). The process takes place on a growing sequence of graphs which we assume to be finite-dimensional, in the sense that the random walk on those geometries satisfies a family of Nash inequalities. As a byproduct of our analysis, we provide a complete picture of the total variation mixing of a discrete dual of the averaging process, which we call binomial splitting process. A single particle of this process is essentially the random walk on the underlying graph. When several particles evolve together, they interact by synchronizing their jumps when placed on neighboring sites. We show that, given k the number of particles and n the (growing) size of the underlying graph, the system exhibits cutoff in total variation if and . Finally, we exploit the duality between the two processes to show that the binomial splitting process satisfies a version of Aldous’ spectral gap identity, namely, the relaxation time of the process is independent of the number of particles.
Funding Statement
M.Q. was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 945045, and by the NWO Gravitation project NETWORKS under grant no. 024.002.003. Part of this work was completed while being a member of GNAMPA-INdAM and of COST Action GAMENET, and receiving partial support by the GNAMPA-INdAM Project 2020 “Random walks on random games” and PRIN 2017 project ALGADIMAR.
F.S. gratefully acknowledges funding by the Lise Meitner fellowship, Austrian Science Fund (FWF): M3211. Part of this work was completed while funded by the European Union’s Horizon 2020 research and innovation programme under the Marie-Skłodowska-Curie grant agreement No. 754411.
Acknowledgments
The authors wish to thank Pietro Caputo and Jan Maas for several fruitful discussions.
Citation
Matteo Quattropani. Federico Sau. "Mixing of the averaging process and its discrete dual on finite-dimensional geometries." Ann. Appl. Probab. 33 (2) 1136 - 1171, April 2023. https://doi.org/10.1214/22-AAP1838
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