Sequential Cavity Method for Computing Free Energy and Surface Pressure
Sequential Cavity Method for Computing Free Energy and Surface Pressure
复制标题
计算自由能和表面压力的顺序腔法
DOI：
10.1007/s10955-009-9849-3
复制
发表时间：
2008
影响因子：
1.6
通讯作者：
Dmitriy A. Katz
中科院分区：
文献类型：
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作者：
D. Gamarnik;Dmitriy A. Katz
我们提出了一种新方法来计算晶格 ℤd 上各种统计力学模型的自由能和表面压力问题。我们的方法基于在适当修改的ℤd 子晶格中以某些边际概率表示自由能和表面压力。然后，使用最近计算边际概率的确定性算法来获得感兴趣的数量的数值估计。该方法在强空间混合（SSP）的假设下工作，这是相关衰减的一种形式。我们在硬核模型和单体二聚体模型上说明了我们的方法，在此基础上我们改进了一些早期的估计。例如，我们表明ℤ3的单体-二聚体覆盖指数属于区间[0.78595,0.78599]，改进了先前已知的最佳估计[0.7850,0.7862]（Friedland and Peled in Adv. Appl. Math. 34:486–522, 2005；Friedland et al. in J. Stat. 2005）。物理学，2009）。此外，我们还表明，给定目标加性误差 ε > 0，我们的方法对于这两个模型的自由能和表面压力值的计算量均为 (1/ε)O(1)。相比之下，现有方法（例如转移矩阵方法）需要 exp ((1/ε)O(1)) 计算工作。
We propose a new method for the problems of computing free energy and surface pressure for various statistical mechanics models on a lattice ℤd. Our method is based on representing the free energy and surface pressure in terms of certain marginal probabilities in a suitably modified sublattice of ℤd. Then recent deterministic algorithms for computing marginal probabilities are used to obtain numerical estimates of the quantities of interest. The method works under the assumption of Strong Spatial Mixing (SSP), which is a form of a correlation decay.We illustrate our method on the hard-core and monomer-dimer models, on which we improve several earlier estimates. For example we show that the exponential of the monomer-dimer coverings of ℤ3 belongs to the interval [0.78595,0.78599], improving best previously known estimate of [0.7850,0.7862] obtained in (Friedland and Peled in Adv. Appl. Math. 34:486–522, 2005; Friedland et al. in J. Stat. Phys., 2009). Moreover, we show that given a target additive error ε>0, the computational effort of our method for these two models is (1/ε)O(1)both for the free energy and surface pressure values. In contrast, prior methods, such as the transfer matrix method, require exp ((1/ε)O(1)) computation effort.