A degree-two spectral gap for Brownian energy exchange
Fix $n \ge 2$, positive numbers $\alpha_1, \dots, \alpha_n$, and symmetric nonnegative weights $w_{xy}$ with $w_{xx} = 0$ whose positive-weight graph on $\{1, \dots, n\}$ is connected. On the simplex $\Delta_{n-1} = \{\eta \in [0,1]^n : \sum_x \eta_x = 1\}$ let $\mu = \operatorname{Dirichlet}(\alpha_1, \dots, \alpha_n)$, and for real polynomials $f$ on $\Delta_{n-1}$ let $$\mathcal{E}(f,f) = \sum_{x < y} w_{xy} \int \eta_x \eta_y \bigl((\partial_x - \partial_y) f\bigr)^2 \, d\mu,$$ the Dirichlet form of the Brownian energy process, the diffusion on $\Delta_{n-1}$ with generator $\sum_{x < y} w_{xy} \{ \eta_x \eta_y (\partial_x - \partial_y)^2 - (\alpha_y \eta_x - \alpha_x \eta_y)(\partial_x - \partial_y) \}$, which is reversible with respect to $\mu$. Let $$\lambda = \inf_{\operatorname{Var}_\mu(f) > 0} \frac{\mathcal{E}(f,f)}{\operatorname{Var}_\mu(f)}, \qquad \lambda_{\le 2} = \min_{\substack{\deg f \le 2 \\ \operatorname{Var}_\mu(f) > 0}} \frac{\mathcal{E}(f,f)}{\operatorname{Var}_\mu(f)},$$ the first infimum over polynomials of every degree, so that $\lambda$ is the spectral gap of the process. The generator preserves polynomial degree and is intertwined with the symmetric inclusion process, whose spectral gap equals $\lambda$; polynomials of degree at most $k$ correspond to $k$ particles. Kim and Sau proved that $\lambda$ is attained at degree one when every $\alpha_x \ge 1$, that this can fail otherwise, and that when $\alpha$ is replaced by $\varepsilon\alpha$ with $\varepsilon \to 0$, the spectral gap of $k$ particles is asymptotic to that of two particles, for every $k \ge 2$. Caputo, Quattropani and Sau proved that the spectral gap is always attained at degree at most two for the Kipnis–Marchioro–Presutti model, in which the energies on a block are resampled from the conditional law of $\mu$ at random times, on arbitrary weighted graphs and hypergraphs.
Problem. Is $\lambda = \lambda_{\le 2}$ for every $n \ge 2$, every choice of positive $\alpha_1, \dots, \alpha_n$ and every choice of weights as above?
Source: Kim and Sau (2026), §1.3, as the two-particle spectral gap identity for the symmetric inclusion process, after a conjecture of Caputo for the Beta-binomial splitting process; asked for the Brownian energy process by Caputo, Quattropani and Sau (2026), §1.4.6. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- SP
- Keywords
- AIM Open Applied Problems, Brownian energy process, interacting particle systems, spectral gaps
References
@article{kim2026,
author = {Kim, Seonwoo and Sau, Federico},
title = {One- and two-particle spectral gap identities for the symmetric inclusion process and related models},
journal = {Comm. Math. Phys.},
year = {2026},
eprint = {2412.01489},
archivePrefix = {arXiv},
note = {To appear},
}
@misc{caputo2026,
author = {Caputo, Pietro and Quattropani, Matteo and Sau, Federico},
title = {Aldous' spectral gap phenomena in stochastic exchange models},
year = {2026},
eprint = {2609.10450},
archivePrefix = {arXiv},
}
@article{kim2024,
author = {Kim, Seonwoo and Sau, Federico},
title = {Spectral gap of the symmetric inclusion process},
journal = {Ann. Appl. Probab.},
volume = {34},
number = {5},
year = {2024},
pages = {4899--4920},
doi = {10.1214/24-AAP2085},
eprint = {2303.16607},
archivePrefix = {arXiv},
}
@misc{aim-brownian-energy-quadratic-gap,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A degree-two spectral gap for Brownian energy exchange}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/608-brownian-energy-quadratic-gap.md},
note = {Entry 608 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-00279, "A degree-two spectral gap for Brownian energy exchange", clopen.solutions.