bibkey: chen2017movingparticle
authors: Joe P. Chen
year: 2017
title: "The moving particle lemma for the exclusion process on a weighted graph"
doi: 10.1214/17-ECP82
url: "https://arxiv.org/abs/1606.01577v4"
claim: "For the symmetric exclusion process on a finite weighted graph and every density α∈[0,1], the cost ½ν_α[(∇_{xy}f)²] of exchanging the occupations at x and y is at most the effective resistance R_eff(x,y) times the exclusion Dirichlet energy; the paper conjectures that this moving particle lemma is optimal on any finite weighted graph."
strata_touched:
  - D5/S3/StatisticalMechanics/ParticleProcesses/ExclusionMovingParticleOptimality
license: citation-only
triage: anchor

Chen, The moving particle lemma for the exclusion process on a weighted graph

J. P. Chen, Electron. Commun. Probab. 22 (2017), no. 47, 1–13; arXiv:1606.01577v4. Quotations are from the v4 TeX source.

On a finite connected weighted graph $(G,\mathbf{c})$ with nonnegative conductances, the symmetric exclusion process is reversible with respect to the product Bernoulli measures ${\nu }_{\alpha }$ on $\{0,1{\}}^V$, and its Dirichlet energy is ${\mathcal{E}}_{{\nu }_{\alpha }}^{\mathrm{EX}}(f)=\frac{1}{2}{\sum }_{xy\in E}{c}_{xy}{\nu }_{\alpha }[({\nabla }_{xy}f{)}^2]$ with $({\nabla }_{xy}f)(\zeta )=f({\zeta }^{xy})-f(\zeta )$. Theorem 1 states $\frac{1}{2}{\nu }_{\alpha }[({\nabla }_{xy}f{)}^2]\le R_{\mathrm{eff}}(x,y){\mathcal{E}}_{{\nu }_{\alpha }}^{\mathrm{EX}}(f)$ for every $\alpha \in [0,1]$, $x,y\in V$ and $f:\{0,1{\}}^V\to \mathbb{R}$, where $[R_{\mathrm{eff}}(x,y){]}^{-1}=\inf \{{\sum }_{zw\in E}{c}_{zw}[h(z)-h(w){]}^2:h(x)=1,h(y)=0\}$. The proof uses the octopus inequality of Caputo, Liggett and Richthammer. After noting that Theorem 1 gives $R_{\mathrm{eff}}(x,y)\ge ({\inf }_f{\mathcal{J}}^{x,y}(f){)}^{-1}$ (1.9), the paper states:

However, it is unclear in general when (1.9) becomes an equality. That being said, we conjecture that Theorem 1 is optimal on any finite weighted graph, and more generally, on resistance spaces [24], in light of the connection between the moving particle lemma and the spectral gap of an interacting particle system [22,33,34].

Verified locator

Source: https://arxiv.org/abs/1606.01577v4

• DOI: https://doi.org/10.1214/17-ECP82 (Electronic Communications in Probability 22, pages 1–13, 2017, title and author from Crossref; article number 47 from the arXiv journal reference).
• URL: https://arxiv.org/abs/1606.01577v4. The e-print member OctopusMoving_v3.tex has 1061 lines and md5 39f69aeb26c41617f292bc17b1771bca. Locations:
  • the exclusion generator, the exchange ${\zeta }^{xy}$ and ${\nu }_{\alpha }$, l. 110–124;
  • the Dirichlet energy, l. 125–128;
  • Theorem 1 and the effective resistance, l. 131–145;
  • the functional ${\mathcal{J}}^{x,y}$, (1.9) and the conjecture, l. 168–179.