Rapid Mixing via Coupling Independence for Spin Systems with Unbounded Degree

We develop a new framework to prove the mixing or relaxation time for the Glauber dynamics on spin systems with unbounded degree. It works for general spin systems including both $2$-spin and multi-spin systems. As applications for this approach: $\bullet$ We prove the optimal $O(n)$ relaxation time for the Glauber dynamics of random $q$-list-coloring on an $n$-vertices triangle-tree graph with maximum degree $Δ$ such that $q/Δ> α^\star$, where $α^\star \approx 1.763$ is the unique positive solution of the equation $α= \exp(1/α)$. This improves the $n^{1+o(1)}$ relaxation time for Glauber dynamics obtained by the previous work of Jain, Pham, and Vuong (2022). Besides, our framework can also give a near-linear time sampling algorithm under the same condition. $\bullet$ We prove the optimal $O(n)$ relaxation time and near-optimal $\widetilde{O}(n)$ mixing time for the Glauber dynamics on hardcore models with parameter $λ$ in $\textit{balanced}$ bipartite graphs such that $λ< λ_c(Δ_L)$ for the max degree $Δ_L$ in left part and the max degree $Δ_R$ of right part satisfies $Δ_R = O(Δ_L)$. This improves the previous result by Chen, Liu, and Yin (2023). At the heart of our proof is the notion of $\textit{coupling independence}$ which allows us to consider multiple vertices as a huge single vertex with exponentially large domain and do a "coarse-grained" local-to-global argument on spin systems. The technique works for general (multi) spin systems and helps us obtain some new comparison results for Glauber dynamics.