Replica cluster variational method: the replica symmetric solution for the 2D random bond Ising model

Abstract

We present and solve the Replica Symmetric equations in the context of the Replica Cluster Variational Method for the 2D random bond Ising model (including the 2D Edwards-Anderson spin glass model). First we solve a linearized version of these equations to obtain the phase diagrams of the model on the square and triangular lattices. In both cases the spin-glass transition temperatures and the tricritical point estimations improve largely over the Bethe predictions. Moreover, we show that this phase diagram is consistent with the behavior of inference algorithms on single instances of the problem. Finally, we present a method to consistently find approximate solutions to the equations in the glassy phase. The method is applied to the triangular lattice down to T = 0, also in the presence of an external field. arXiv:1204.0439v3 [cond-mat.dis-nn] 3 Sep 2012

Topologically invariant tensor renormalization group method for the Edwards-Anderson spin glasses model

“…We compared the partition function calculated by our topological invariant TRG with those obtained by the original TRG [12] and mean-field approach, belief propagation and generalized belief propagation (GBP) [24,27,28], on the pure spin glass model without external fields, i.e., p = 0.5 and h i = 0. The exact partition function is calculated by the algorithm [23].…”

“…In the previous study, the mean-field approximation [24][25][26][27][28] and Monte Carlo sampling [29], transfer matrix method [30], and numerical exact algorithm for 2D without the external field [23,31] are used to calculate local properties for individual finite size instances. These methods are combined with finite size scaling to investigate the thermodynamic limit properties.…”

“…We show that the contribution of the negative part is comparable to the error fluctuation for a large enough system and, therefore, it can be neglected. In the high-temperature region, TRG outperforms the mean-field method, belief propagation, and generalized belief propagation [24][25][26][27][28], while the mean-field methods are failed in the lower temperature because of the convergence problems. The Nishimori multicritical point [32,36,37] is calculated by our TRG scheme.…”

Statistical analysis of loopy belief propagation in random fields

“…In this paper, we propose an approximate method based on the idea of the RCVM [13,14]. Figure 2(b) shows the procedure of our method.…”

“…In this paper, we propose a new analytical method for evaluating the quenched average of LBP over random fields based on the idea of the replica cluster variation method (RCVM) [13,14]. The presented method allows the quenched average of the Bethe free energies over random fields in general pair-wise MRFs to be evaluated, unlike the previous method.…”