The Bethe lattice spin glass revisited

By Marc Mézard, Giorgio Parisi. 2000

636 citations recorded by the source corpus.

Abstract:So far the problem of a spin glass on a Bethe lattice has been solved only at the replica symmetric level, which is wrong in the spin glass phase. Because of some technical difficulties, attempts at deriving a replica symmetry breaking solution have been confined to some perturbative regimes, high connectivity lattices or temperature close to the critical temperature. Using the cavity method, we propose a general non perturbative solution of the Bethe lattice spin glass problem at a level of approximation which is equivalent to a one step replica symmetry breaking solution. The results compare well with numerical simulations. The method can be used for many finite connectivity problems appearing in combinatorial optimization.

Subfields: Condensed Matter Physics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e575db1bcdfd35fc0afcf8dda26f6464d0105e27

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Condensed Matter Physics

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

Statistical physics of inference: thresholds and algorithms (2015, methodology): “With the work of Mézard, Parisi and Zecchina on survey propagation [204], the statistical physics community learned that the replica and cavity calculations are as powerful when used as algorithms as they are when used as tools for analyzing the phase diagram.”
A hierarchical version of the de Finetti and Aldous-Hoover representations (2013, methodology): “One application of this representation can be found in [14], where it is explained how (12) is related to the predictions about the structure ofthe Gibbs measure in diluted spin glass models that originate in the work of Mézard and Parisi[12].”
Control Principles of Complex Networks (2015, methodology): “To answer these questions we can turn to the cavity method, a versatile tool of statistical physics (Mézard and Parisi, 2001; Zdeborová and Mézard, 2006; Zhou and Ou-Yang, 2003).”
Predicting protein functions with message passing algorithms (2004, methodology): “The Gibbs potential extremizing equations used in this work are commonly known under the name of Belief Propagation (BP) equations and can be easily found via a procedure called Cavity Method 20 .”
ARACNE: An Algorithm for the Reconstruction of Gene Regulatory Networks in a Mammalian Cellular Context (2004, methodology): “networks [33,34], particularly if the gi are binary (such discretization of expression levels is a common technique to”
Reconstruction on Trees and Spin Glass Transition (2005, background): “We shall not try to explain here its physical origin, which is related to spin glass theory, (19) but just discuss the relation with the reconstruction problem.”
Edwards Statistical Mechanics for Jammed Granular Matter (2016, methodology): “A second mean-field theory of choice consists in approximating the contact network by a sparse random graph (Mézard and Parisi, 2001), which allows one to preserve an essential property of real finite dimensional packings: the finite coordination number z.”
Critical phenomena in complex networks (2007, methodology): “The advantage of a regular tree is that we can calculate the parameters gji(Si) at a given vertex i by using the following recursion relation: gji(Si) = ∑ Sj=±1 exp[βJijSiSj + βHjSj ] ∏ m∈N(j)\i gmj(Sj).”