一般图上非平衡对称排斥过程的混合时间与谱
Mixing times and spectra of non-equilibrium symmetric exclusion processes on general graphs
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中文总结 AI 辅助
研究一般图上非平衡对称排斥过程，通过单个粒子到热浴的最坏情况击中时间给出混合时间界，据此导出计算\(k\)个顶点稳态联合占据分布的算法，还证明相关马尔可夫链谱与热浴温度无关。
中文摘要 AI 辅助
对称排斥过程（SEP）是图上相互作用粒子的经典模型，固定粒子数的版本已通过可逆马尔可夫链研究得很清楚。本文研究非平衡SEP，图的一些顶点与不同温度的外部热浴相互作用以介导图中的传输。所得稳态分布仍是自然马尔可夫链的平稳分布，但该链不可逆，稳态分布很难描述，相关文献几乎只关注晶格\(Z^d\)的有限子集图。我们得出任意带任意热浴的图的非平衡SEP的各种定量结果。主要结果是根据单个粒子到任何热浴的最坏情况击中时间给出混合时间的界，此界在图大小上至多相差两个对数因子。然后利用此结果导出一个简单算法，能计算任意\(k\)个顶点集的稳态联合占据分布，这为研究小\(k\)值时的\(k\)元相关性提供了有用工具。最后证明与SEP非可逆动力学相关的马尔可夫链的谱与热浴温度无关，即谱总是实的。
英文摘要
The symmetric exclusion process (SEP) is a classical model of interacting particles on a graph, in which multiple particles execute random walks subject to the constraint that no two particles may occupy the same vertex. The version of the process in which the number of particles is fixed, is by now very well understood, through the study of reversible Markov chains. In this paper, we study the non-equilibrium version of the SEP, in which some vertices of the graph interact with external heat baths, held at different temperatures, that mediate transport through the graph. The resulting steady-state distribution is still the stationary distribution of a natural Markov chain, but the chain is no longer reversible. This means that even the steady-state distribution is very hard to describe, and indeed is the subject of a rich literature, which has focused almost exclusively on graphs which are finite subsets of the lattice $Z^d$. Very little is known about other important quantities, notably the mixing time. We derive various quantitative results about the non-equilibrium SEP for arbitrary graphs with arbitrary heat baths. Our main result is a bound on the mixing time in terms of the worst-case hitting time of a single particle to any of the heat baths; this bound is tight up to two logarithmic factors in the size of the graph. We then use this result to derive a simple algorithm, running in time roughly $n^{O(k)}$, that computes the steady-state joint occupation distribution for any set of $k$ vertices; since the steady-state distribution is very elusive, this provides a potentially useful tool to study its $k$-wise correlations for small values of $k$. Finally, we prove that the spectrum of the Markov chain associated with the non-reversible dynamics for the SEP is independent of the heat bath temperatures; thus in particular the spectrum is always real.
