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Spectral gap of the KMP and other stochastic exchange models on arbitrary graphs

Seonwoo Kim, Matteo Quattropani, Federico Sau

TL;DR

This work develops a universal framework to bound the spectral gap for reversible stochastic exchange models on arbitrary graphs, including KMP, HP, and IEM. The authors reduce spectral-gap analysis to a finite-dimensional problem via a hidden-parameter intertwinement and contractive dynamics on $[0,1]^V$, yielding bounds that depend only on vertex-weights and a graph-geometry–independent kinetic factor. They prove a general lower bound and, in the reversible Dirichlet setting, an Aldous-type identity that equates the model gap with the random-walk gap under a simple criterion; they also show sharpness in key graphs and regimes. The approach extends beyond reversibility, degeneracy, and open/system-with-reservoir variants, providing a robust tool for spectral analysis in a broad class of conservative stochastic exchange models with practical implications for understanding relaxation to equilibrium on networks.

Abstract

We present a simple strategy to derive universal bounds on the spectral gap of reversible stochastic exchange models on arbitrary graphs. The Kipnis-Marchioro-Presutti (KMP) model, the harmonic process (HP), and the immediate exchange model (IEM) are all examples that fall into this class. Our upper and lower bounds depend only on two features: worst-case linear statistics and a kinetic factor, which is, in essence, graph-independent. For the three aforementioned examples, these bounds are sharp, and even saturate to an identity for HP and IEM in some log-concave regimes. The proof -- which yields bounds for eigenvalues even in the non-reversible context -- crucially exploits the rigidity of the eigenstructure of these models and quantitative contraction rates of the corresponding hidden parameter models recently introduced in [DMFG24, GRT25].

Spectral gap of the KMP and other stochastic exchange models on arbitrary graphs

TL;DR

This work develops a universal framework to bound the spectral gap for reversible stochastic exchange models on arbitrary graphs, including KMP, HP, and IEM. The authors reduce spectral-gap analysis to a finite-dimensional problem via a hidden-parameter intertwinement and contractive dynamics on , yielding bounds that depend only on vertex-weights and a graph-geometry–independent kinetic factor. They prove a general lower bound and, in the reversible Dirichlet setting, an Aldous-type identity that equates the model gap with the random-walk gap under a simple criterion; they also show sharpness in key graphs and regimes. The approach extends beyond reversibility, degeneracy, and open/system-with-reservoir variants, providing a robust tool for spectral analysis in a broad class of conservative stochastic exchange models with practical implications for understanding relaxation to equilibrium on networks.

Abstract

We present a simple strategy to derive universal bounds on the spectral gap of reversible stochastic exchange models on arbitrary graphs. The Kipnis-Marchioro-Presutti (KMP) model, the harmonic process (HP), and the immediate exchange model (IEM) are all examples that fall into this class. Our upper and lower bounds depend only on two features: worst-case linear statistics and a kinetic factor, which is, in essence, graph-independent. For the three aforementioned examples, these bounds are sharp, and even saturate to an identity for HP and IEM in some log-concave regimes. The proof -- which yields bounds for eigenvalues even in the non-reversible context -- crucially exploits the rigidity of the eigenstructure of these models and quantitative contraction rates of the corresponding hidden parameter models recently introduced in [DMFG24, GRT25].
Paper Structure (33 sections, 16 theorems, 136 equations)

This paper contains 33 sections, 16 theorems, 136 equations.

Table of Contents

  1. Introduction. From KMP to more general exchange models
  2. The KMP model
  3. Convergence to equilibrium and spectral gap
  4. ${\rm KMP}$'s main result
  5. Proof strategy and hidden parameter model
  6. Chen-Wang's approach for ${\rm KMP}$
  7. ${\rm KMP}$'s hidden parameter model
  8. Generalizations: HP & IEM, and beyond reversibility
  9. Harmonic process & immediate exchange model
  10. Beyond non-degeneracy and reversibility
  11. Open systems
  12. Organization of the paper
  13. General setting and main results
  14. Stochastic exchange models
  15. Precise assumptions and first properties
  16. ...and 18 more sections

Key Result

Theorem 1.1

For all graphs $G=(V,(c_{xy})_{x,y\in V})$ and weights $\alpha=(\alpha_x)_{x\in V}$, where with

Theorems & Definitions (41)

  • Theorem 1.1: KMP
  • Theorem 1.2: HP
  • Theorem 1.3: IEM
  • Remark 2.1: Hidden parameter model
  • Remark 2.2
  • Proposition 2.4: Well-definition
  • proof
  • Remark 2.6
  • Proposition 2.7: Invariant measure
  • Theorem 2.8
  • ...and 31 more