Rapid Mixing via Coupling Independence for Spin Systems with Unbounded Degree
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929410727215104 |
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| author | Chen, Xiaoyu Feng, Weiming |
| author_facet | Chen, Xiaoyu Feng, Weiming |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04672 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rapid Mixing via Coupling Independence for Spin Systems with Unbounded Degree Chen, Xiaoyu Feng, Weiming Data Structures and Algorithms Probability 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. |
| title | Rapid Mixing via Coupling Independence for Spin Systems with Unbounded Degree |
| topic | Data Structures and Algorithms Probability |
| url | https://arxiv.org/abs/2407.04672 |