Uniqueness of Gibbs Measures and Rapidly Mixing Dynamics

[98003511]本文研究了整数晶格上的硬核晶格气体模型，重点研究了方晶格。该项目包括以下问题：(i)在一般图上硬核模型的（无限体积）吉布斯测度的唯一性与某些离散动力学的接近平衡率之间建立严格的定量联系；（ii）分析某些动力学的混合率，目的是改进硬核模型在方格上的临界活度的现有界限。调查渗透和其他技术在这方面的作用。在图论术语中，目标之一是生成大型n × n网格的独立集，其概率倾向于较大的独立集，特别是具有活动1的硬核分布。该提案还包括一个关于估计组合结构上马尔可夫链混合率的学生项目，其计数函数是第n个加泰罗尼亚数。晶格气体模型对统计物理、计算理论、概率组合学和通信网络等领域的研究人员具有独立的兴趣和重要意义。最容易描述的动机是具有泊松到达率和呼叫独立等待时间的电话（损失）网络。硬约束要求，只有当所有相邻单元（根据任意拓扑的邻域）都处于非活动状态时，才能接受/激活调用。PI希望有助于理解空间相关性（即在一个位置的呼叫对在相当远的位置的呼叫的影响）和动态收敛（即网络接近的速率，如果它最终做到的话，一个独特的稳定的主动呼叫配置）之间的联系。

9800351TetaliThis research concerns the hard-core lattice gas model on the integer lattice, with special emphasis on the square lattice. The project includes the following problems:(i) To establish a rigorous, quantitative connection between the rate of approach to equilibrium of certain discrete dynamics and the uniqueness of (infinite--volume) Gibbs measure for the hard-core model on general graphs(ii) To analyze the rate of mixing of certain dynamics with the intention of improving the existing bounds on the critical activity of the hard-core model on the square lattice.(iii) To investigate the role of percolation and other techniques in this context. In graph-theoretic terms, one of the objectives is to generate independent sets of a large n by n grid with probabilities favoring the bigger independent sets, in particular, with hard-core distribution with activity 1. The proposal also includes a student project on estimating the mixing rates of Markov chains on combinatorial structures, whose counting function is the nth Catalan number.The lattice gas model is of independent interest and significance to researchers in statistical physics, theory of computing, probabilistic combinatorics, and communication networks. The easiest-to-describe motivation is that of a telephone (loss) network with Poisson arrival rates and independent holding times for calls. The hard constraint demands that a call be accepted/active only if all the neighboring cells (neighborhood according to an arbitrary topology) are inactive. The PI hopes to contribute to the understanding of the connections between spatial correlations (i.e. the effect of the presence of a call in one location on calls at reasonably far locations) and the convergence of the dynamics (i.e. the rate at which the network approaches, if it eventually does, a unique stable configuration of active calls).