Spectral gap of the symmetric inclusion process

TL;DR

The paper provides universal bounds for the spectral gap of the symmetric inclusion process on finite graphs in terms of the RW gap, with sharp equality in the log-concave regime (α_min≥1). The authors combine self-duality of SIP with Dirichlet-form decompositions and a min-max framework to derive both upper and lower bounds, deriving an Aldous-type identity for SIP in the α_min≥1 case and extending the results to the Brownian energy process via intertwining. The approach yields a robust method to relate interacting particle systems with gamma-like reversible measures to simpler RW dynamics, with implications for metastability and scaling limits. Open questions remain for α_min∈(0,1) and for general Gibbs-sampler perturbations, suggesting further extensions to related beta-binomial, harmonic, and BEP-type models. Overall, the work advances quantitative understanding of convergence to equilibrium for gamma-like interacting systems and connects spectral-gap identities to duality and intertwinings.

Abstract

We consider the symmetric inclusion process on a general finite graph. Our main result establishes universal upper and lower bounds for the spectral gap of this interacting particle system in terms of the spectral gap of the random walk on the same graph. In the regime in which the gamma-like reversible measures of the particle systems are log-concave, our bounds match, yielding a version for the symmetric inclusion process of the celebrated Aldous' spectral gap conjecture originally formulated for the interchange process. Finally, by means of duality techniques, we draw analogous conclusions for an interacting diffusion-like unbounded conservative spin system known as Brownian energy process.

Theorems & Definitions (29)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Remark 2.1
• Lemma 2.2
• proof
• Proposition 3.1
• proof
• Lemma 3.2
• Lemma 3.3