Phase Coexistence and Slow Mixing for the Hard-Core Model on ℤ2
Phase Coexistence and Slow Mixing for the Hard-Core Model on ℤ2
复制标题
ℤ2 上硬核模型的相位共存和慢速混合
DOI：
10.1007/978-3-642-40328-6_27
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
P. Tetali
中科院分区：
文献类型：
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作者：
Antonio Blanca;David J. Galvin;Dana Randall;P. Tetali
在有限图上的硬核模型中，我们给定参数λ>0，并且独立集I以与λ ^成比例的概率出现。|我|.在无限图上，吉布斯分布被定义为具有正确条件概率的适当极限。在无限设置中，我们感兴趣的是确定何时该极限是唯一的以及何时存在相位共存，即，存在多个吉布斯状态。在有限图上，我们感兴趣的是确定局部马尔可夫链的混合时间。
在Z^2上，证明了这些问题是相关的，并且都在λ_c约为3.79的某个临界点处经历相变。对于相共存，迄今为止的大部分工作都集中在唯一性的制度上，最好的结果是雷斯特雷波等人最近的工作，表明所有λ 5.3646都存在唯一的吉布斯状态。我们的证明增加了两个重要的创新，以标准的佩尔斯的论点。首先，建立在由兰德尔介绍的断层线的想法，我们构建了一个事件，区分两个边界条件，总是有长的轮廓与它相关联，避免了需要准确地枚举短轮廓。第二，我们通过将轮廓线与Z^2的有向版本上的一类新的自避免行走联系起来，获得了轮廓线数量的极大改进的界限。
我们扩展我们的故障线的特征，以表明本地马尔可夫链将混合缓慢时，λ> 5.3646的晶格区域与周期性（环形）边界条件，当λ> 7.1031与非周期性（自由）边界条件。这里的论点依赖于一个仔细的分析，将轮廓与出租车步行联系起来，并代表了对以前最知名的\lambda值的七倍改进。
In the hard-core model on a finite graph we are given a parameter lambda>0, and an independent set I arises with probability proportional to lambda^|I|. On infinite graphs a Gibbs distribution is defined as a suitable limit with the correct conditional probabilities. In the infinite setting we are interested in determining when this limit is unique and when there is phase coexistence, i.e., existence of multiple Gibbs states. On finite graphs we are interested in determining the mixing time of local Markov chains.
On Z^2 it is conjectured that these problems are related and that both undergo a phase transition at some critical point lambda_c approx 3.79. For phase coexistence, much of the work to date has focused on the regime of uniqueness, with the best result being recent work of Restrepo et al. showing that there is a unique Gibbs state for all lambda 5.3646. Our proof adds two significant innovations to the standard Peierls argument. First, building on the idea of fault lines introduced by Randall, we construct an event that distinguishes two boundary conditions and always has long contours associated with it, obviating the need to accurately enumerate short contours. Second, we obtain vastly improved bounds on the number of contours by relating them to a new class of self-avoiding walks on an oriented version of Z^2.
We extend our characterization of fault lines to show that local Markov chains will mix slowly when lambda > 5.3646 on lattice regions with periodic (toroidal) boundary conditions and when lambda > 7.1031 with non-periodic (free) boundary conditions. The arguments here rely on a careful analysis that relates contours to taxi walks and represent a sevenfold improvement to the previously best known values of \lambda.