Sequential Cavity Method for Computing Free Energy and Surface Pressure

We propose a new method for the problems of computing free energy and surface pressure for various statistical mechanics models on a lattice a"currency sign (d) . Our method is based on representing the free energy and surface pressure in terms of certain marginal probabilities in a suitably modified sublattice of a"currency sign (d) . 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 a"currency sign(3) 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 epsilon > 0, the computational effort of our method for these two models is (1/epsilon) (O(1)) both for the free energy and surface pressure values. In contrast, prior methods, such as the transfer matrix method, require exp ((1/epsilon) (O(1))) computation effort.