Communications in Mathematical Physics· 2026Q1
One- and Two-Particle Spectral Gap Identities for the Symmetric Inclusion Process and Related Models
- 0citations
- Q1SCImago
- 2026year
Short summary
The spectral gap identity between a multi-particle symmetric inclusion process (SIP) and a single particle generally fails outside the log-concave regime, but always holds for the non-conservative SIP.
AI-generated from the title and abstract; the full text is not read.
Key points
- The spectral gap identity between multi-particle SIP and single-particle SIP fails outside the log-concave regime.
- The spectral gap identity always holds for the non-conservative SIP, irrespective of interaction strength.
- Sharp bounds for the spectral gap are derived when the one-particle identity breaks down.
- A two-particle spectral gap identity is identified in the vanishing diffusivity limit.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract The symmetric inclusion process (SIP) models particles diffusing on a graph with mutual attraction. We recently showed [KS24] that, in the log-concave regime (where diffusivity dominates interaction), the spectral gap of the conservative SIP matches that of a single particle. In this paper, our main result demonstrates that this identity generally fails outside this regime, but always holds for the non-conservative SIP, regardless of the interaction strength. When this one-particle spectral gap identity breaks down, we derive sharp bounds for the gap in terms of diffusivity, and reveal a two-particle spectral gap identity in the vanishing diffusivity limit. Our approach leverages the rigid eigenstructure of SIP, refined comparisons of Dirichlet forms for arbitrary diffusivity and particle numbers, and techniques from slow–fast system analysis. These findings extend to the dual interacting diffusion known as Brownian energy process, and shed some light on the spectral gap behavior for related Dirichlet-reversible systems on general, non-mean-field, geometries.
The authors' abstract, as published at the source. Communications in Mathematical Physics, 2026 · DOI ↗
Continue with a free account
Ask the paper: 3 free questions a day about this paper; save it, get its citation, new summaries every day for your field. Takeaways are Premium.
Continue free on the webSign in with Google or Apple; no card needed. You come back to this paper.
On your phone:
Field: Mathematical Physics
Mathematical PhysicsMathematics