Uniqueness of Gibbs Measures and Rapidly Mixing Dynamics

9800351 Tetali本研究关注的是整数格点上的硬核格气模型，特别强调了正方形格点。 该项目包括以下问题：（i）建立一个严格的，定量的某些离散动力学的接近平衡的速度和（无限体积）吉布斯测度的唯一性之间的联系的硬核模型在一般图（ii）分析某些动力学的混合率，旨在改善现有的边界上的硬核模型的临界活动的正方形格子。(iii)研究渗透和其他技术在这方面的作用。 在图论术语中，目标之一是生成具有有利于较大独立集的概率的大型n乘n网格的独立集，特别是具有活动1的硬核分布。 该提案还包括一个学生项目，估计混合率的马尔可夫链的组合结构，其计数函数是第n个Catalan数。格子气模型是独立的兴趣和意义的研究人员在统计物理，计算理论，概率组合学和通信网络。 最容易描述的动机是，电话（损失）网络的泊松到达率和独立的保持时间的电话。 硬约束要求只有当所有相邻小区（根据任意拓扑的邻域）都不活动时，呼叫才被接受/活动。 PI希望有助于理解空间相关性（即在一个位置的呼叫对合理远位置的呼叫的影响）和动态收敛（即网络接近的速率，如果最终确实如此，则是活动呼叫的唯一稳定配置）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).