Does exclusion equilibrate as fast as independent particles?
Let $G = (V, E)$ be a connected finite graph with $n$ vertices and positive symmetric edge rates $(r_e)_{e \in E}$, and let $1 \le k \le n/2$. In the symmetric exclusion process $\mathrm{EX}(k)$, $k$ indistinguishable particles occupy distinct vertices, and each edge $e$ rings at rate $r_e$ and exchanges the occupations of its endpoints; its equilibrium is uniform on the $k$-subsets of $V$. In $\mathrm{RW}^{\otimes k}$, $k$ independent continuous-time random walks each jump across $e$ at rate $r_e$, and may share vertices; its equilibrium is uniform on $V^k$. Write $t_{\mathrm{mix}}$ for the total-variation mixing time at error $1/4$ from the worst starting state. Oliveira proved that $t_{\mathrm{mix}}(\mathrm{EX}(k)) \le C\, t_{\mathrm{mix}}(\mathrm{RW}^{\otimes 1}) \log(4n)$ for an absolute constant $C$. By the theorem of Caputo, Liggett and Richthammer, formerly Aldous' spectral-gap conjecture, $\mathrm{EX}(k)$ has the same spectral gap as a single walk. Hermon and Pymar proved the comparison below up to a constant factor for $d$-regular graphs with all rates $1/d$ in several regimes, among them $k$ of order $n^a$ for a fixed $0 < a < 1$.
Problem. Is there an absolute constant $C$ such that $$t_{\mathrm{mix}}(\mathrm{EX}(k)) \le C\, t_{\mathrm{mix}}(\mathrm{RW}^{\otimes k})$$ for all choices of $G$, $(r_e)$ and $k$?
Source: Oliveira (2013), as a heuristic assumption at every error level; stated as Oliveira's conjecture by Hermon and Pymar (2020), (2). Stated by AIM — Open Applied Problems at error 1/4.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- CO
- Keywords
- AIM Open Applied Problems, exclusion processes, interacting particle systems, mixing times
References
@article{oliveira2013,
author = {Oliveira, Roberto Imbuzeiro},
title = {Mixing of the symmetric exclusion processes in terms of the corresponding single-particle random walk},
journal = {Ann. Probab.},
volume = {41},
number = {2},
year = {2013},
pages = {871--913},
doi = {10.1214/11-AOP714},
eprint = {1007.2669},
archivePrefix = {arXiv},
}
@article{hermon2020,
author = {Hermon, Jonathan and Pymar, Richard},
title = {The exclusion process mixes (almost) faster than independent particles},
journal = {Ann. Probab.},
volume = {48},
number = {6},
year = {2020},
pages = {3077--3123},
doi = {10.1214/20-AOP1455},
eprint = {1808.10846},
archivePrefix = {arXiv},
}
@article{caputo2010,
author = {Caputo, Pietro and Liggett, Thomas M. and Richthammer, Thomas},
title = {Proof of {A}ldous' spectral gap conjecture},
journal = {J. Amer. Math. Soc.},
volume = {23},
number = {3},
year = {2010},
pages = {831--851},
doi = {10.1090/S0894-0347-10-00659-4},
}
@misc{aim-exclusion-independent-mixing,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Does exclusion equilibrate as fast as independent particles?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/270-exclusion-independent-mixing.md},
note = {Entry 270 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00431, "Does exclusion equilibrate as fast as independent particles?", clopen.solutions.