arXiv 1708.042632017-08-14math.PR

连续Hardcore模型的Gibbs测度的唯一性

Uniqueness of Gibbs Measures for Continuous Hardcore Models

David Gamarnik, Kavita Ramanan

备注：: 34 pages, 1 figure

摘要：

我们制定了一个连续的版本，众所周知的离散硬核（或独立集）模型的局部有限图，参数化的所谓的活动参数$λ> 0$。在这个版本中，状态或“自旋值”$x_u$的任何节点$u$的图位于区间$[0,1]$，硬核约束$x_u + x_v \leq 1$满足的每一个边缘$(u,v)$的图，和空间的可行的配置是由凸多面体。当图是一个正规的树，我们表明，有一个唯一的吉布斯措施与每个活动参数$λ>0$。我们的结果表明，与标准的离散硬核模型相比，连续硬核模型在无限正则树上不表现出相变。我们还考虑了一个家庭的连续模型，插值之间的离散和连续的硬核模型在一个定期的树时，$λ= 1$和家庭的每个成员有一个独特的吉布斯措施，即使离散模型不。在每种情况下，证明需要分析相关的哈密顿动力系统，描述了一定的限制，边缘分布在一个节点。此外，给定任意序列的正则图与固定的程度和周长发散到无穷大，我们应用我们的结果来计算相应的序列的凸多面体的可行的配置适当规范化的体积的渐近极限。特别是，这产生了一个近似的连续硬核模型的分拆函数的情况下，$λ= 1$的一个规则的图与大围长。

英文摘要：

We formulate a continuous version of the well known discrete hardcore (or independent set) model on a locally finite graph, parameterized by the so-called activity parameter $λ> 0$. In this version, the state or "spin value" $x_u$ of any node $u$ of the graph lies in the interval $[0,1]$, the hardcore constraint $x_u + x_v \leq 1$ is satisfied for every edge $(u,v)$ of the graph, and the space of feasible configurations is given by a convex polytope. When the graph is a regular tree, we show that there is a unique Gibbs measure associated to each activity parameter $λ>0$. Our result shows that, in contrast to the standard discrete hardcore model, the continuous hardcore model does not exhibit a phase transition on the infinite regular tree. We also consider a family of continuous models that interpolate between the discrete and continuous hardcore models on a regular tree when $λ= 1$ and show that each member of the family has a unique Gibbs measure, even when the discrete model does not. In each case, the proof entails the analysis of an associated Hamiltonian dynamical system that describes a certain limit of the marginal distribution at a node. Furthermore, given any sequence of regular graphs with fixed degree and girth diverging to infinity, we apply our results to compute the asymptotic limit of suitably normalized volumes of the corresponding sequence of convex polytopes of feasible configurations. In particular, this yields an approximation for the partition function of the continuous hard core model on a regular graph with large girth in the case $λ= 1$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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