Spectral independence bound improves mixing times for graph models
Degree-Free Spectral Independence for Log-Concave Holant Measures
Data Structures and Algorithms
Summary
This paper finds a way to measure how quickly certain random processes on graphs settle down without depending on how connected each node is. Specifically, the authors study problems where nodes are linked through complex constraints called log-concave Holant problems. Their findings lead to better predictions of how fast particular sampling methods, like Glauber dynamics, reach equilibrium in models involving matchings, which are ways of pairing edges or nodes. Interestingly, they used an advanced AI tool, GPT-5.6 Sol, to assist in discovering the main proof ideas.
What this means in practice
- •For network engineers: Predict how quickly random updates mix in network matching problems to improve load balancing algorithms.
- •For statistical physicists: Use improved bounds on Glauber dynamics mixing times for simulating molecular matching and dimer coverings more efficiently.
A theory result. No direct application yet.
Authors
Xiaoyu Chen, Zejia Chen, Xinyuan Zhang
Abstract
We establish a degree-independent bound on spectral independence for log-concave Holant problems on simple graphs. As a corollary, we obtain relaxation-time bounds for Glauber dynamics of $O_λ(m)$ for the monomer-dimer model at activity $λ$ and $O_{b,λ}(m)$ for $b$-matchings at fugacity $λ>0$, where $m$ is the number of edges. For uniform $b$-matchings, the relaxation-time bound improves to $O(bm)$. The main proof ideas were found using GPT-5.6 Sol.