How to Compute Loop Corrections to Bethe Approximation

Any spanning tree in a loopy interaction graph can be used for communicating the effect of the loopy interactions by introducing messages that are passed along the edges in the spanning tree. This defines an exact mapping of the problem on the loopy interaction graph onto an extended problem on a tree interaction graph, where the thermodynamic quantities can be computed by a message-passing algorithm based on the Bethe equations. We propose an approximation loop correction al...

We discuss analytical approximation schemes for the dynamics of diluted spin models. The original dynamics of the complete set of degrees of freedom is replaced by a hierarchy of equations including an increasing number of global observables, which can be closed approximately at different levels of the hierarchy. We illustrate this method on the simple example of the Ising ferromagnet on a Bethe lattice, investigating the first three possible closures, which are all exact in ...

We apply for the first time a new one-loop topological expansion around the Bethe solution to the spin-glass model with field in the high connectivity limit, following the methodological scheme proposed in a recent work. The results are completely equivalent to the well known ones, found by standard field theoretical expansion around the fully connected model (Bray and Roberts 1980, and following works). However this method has the advantage that the starting point is the ori...

We study the d-dimensional random Ising model using a Bethe-Peierls approximation in the framework of the replica method. We take into account the correct interaction only inside replicated clusters of spins. Our ansatz is that the interaction of the borders of the clusters with the external world can be described via an effective interaction among replicas. The Bethe-Peierls model is mapped into a single Ising model with a random gaussian field, whose strength (related to th...

The inverse Ising problem consists in inferring the coupling constants of an Ising model given the correlation matrix. The fastest methods for solving this problem are based on mean-field approximations, but which one performs better in the general case is still not completely clear. In the first part of this work, I summarize the formulas for several mean- field approximations and I derive new analytical expressions for the Bethe approximation, which allow to solve the inver...

We study the dynamical low temperature behaviour of the Ising spin glass on the Bethe lattice. Starting from Glauber dynamics we propose a cavity like Ansatz that allows for the treatment of the slow (low temperature) part of dynamics. Assuming a continuous phase transitions and ultrametricity with respect to long time scales we approach the problem perturbatively near the critical temperature. The theory is formulated in terms of correlation-response-functions of arbitrary o...

The spin-glass transition in a field in finite dimension is analyzed directly at zero temperature using a perturbative loop expansion around the Bethe lattice solution. The loop expansion is generated by the $M$-layer construction whose first diagrams are evaluated numerically and analytically. The generalized Ginzburg criterion reveals that the upper critical dimension below which mean-field theory fails is $D_U \le 8$, at variance with the classical result $D_U = 6$ yielded...

We study spin glasses on random lattices with finite connectivity. In the infinite connectivity limit they reduce to the Sherrington Kirkpatrick model. In this paper we investigate the expansion around the high connectivity limit. Within the replica symmetry breaking scheme at two steps, we compute the free energy at the first order in the expansion in inverse powers of the average connectivity (z), both for the fixed connectivity and for the fluctuating connectivity random l...

We study spin systems on Bethe lattices constructed from d-dimensional hypercubes. Although these lattices are not tree-like, and therefore closer to real cubic lattices than Bethe lattices or regular random graphs, one can still use the Bethe-Peierls method to derive exact equations for the magnetization and other thermodynamic quantities. We compute phase diagrams for ferromagnetic Ising models on hypercubic Bethe lattices with dimension d=2, 3, and 4. Our results are in go...

Recently, it has been shown that, when the dimension of a graph turns out to be infinite dimensional in a broad sense, the upper critical surface and the corresponding critical behavior of an arbitrary Ising spin glass model defined over such a graph, can be exactly mapped on the critical surface and behavior of a non random Ising model. A graph can be infinite dimensional in a strict sense, like the fully connected graph, or in a broad sense, as happens on a Bethe lattice an...