Replica Symmetry Breaking and the Spin-Glass on a Bethe Lattice

Mottishaw, Peter

doi:10.1209/0295-5075/4/3/013

Cited by 58 publications

(56 citation statements)

References 16 publications

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“…We shall not present here the details of the replica approach to this problem, for which we refer the reader to [19,20,32,23,27]. Let us recall the main results of [23].…”

Section: Equivalence With the Replica Formalismmentioning

confidence: 99%

“…Unfortunately, getting the replica symmetry broken solution in the low temperature phase is difficult. In general the problem involves an infinity of order parameters which are multi-spin overlaps [24,9,19,20,23]. As we saw, the replica symmetric solution already involves an order parameter which is a whole function (the distribution of local fields); going to a 'one step RSB' solution [3], the replica order parameter becomes now a functional, the probability distribution over the space of local field distributions [27] (the reason will be discussed in details in the next section).…”

Section: The Rsb Instabilitymentioning

confidence: 99%

“…While one can write formally some integral equations satisfied by this order parameter, solving them is in general a formidable task. The solution is known only in the neighborhood of the critical temperature [20], or in the limit of large connectivities [23], where the overlaps involving three spins or more are small and can be treated perturbatively, allowing for some expansion around the SK solution. In the general case, the only tractable method so far has been an approximation which parametrizes the functional by a small enough number of parameters and optimizes the one step RSB free energy inside this subspace [32,33].…”

Section: The Rsb Instabilitymentioning

confidence: 99%

“…However one knows that there exists a replica symmetry breaking (RSB) instability similar to the one found in the SK model [17,[19][20][21][22]. Unfortunately, in the replica formalism, the RSB solution could be found only in some rather limited regimes: expansion around the high connectivity (SK-like) limit [23], or close to the critical temperature [20]. The main problem encountered in all these attempts is a very general one, common to all disordered problems with a finite connectivity.…”

Section: Introductionmentioning

confidence: 99%

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The Bethe lattice spin glass revisited

2001

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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.

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“…We shall not present here the details of the replica approach to this problem, for which we refer the reader to [19,20,32,23,27]. Let us recall the main results of [23].…”

Section: Equivalence With the Replica Formalismmentioning

confidence: 99%

Section: The Rsb Instabilitymentioning

confidence: 99%

Section: The Rsb Instabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Bethe lattice spin glass revisited

2001

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“…To calculate the partition function Z(β) of the Ising model at inverse temperature β on the graph of figure 1, we consider the leaves of the uncovered local tree i.e. vertices at distance D from the root (in figure 1 the maximum drawn distance is D = 2) [11,8,9,12,10,13]. At sufficiently large β, a spontaneous, say, positive magnetization m is expected to be present in the bulk.…”

Section: Loops With Multiple Crossingsmentioning

confidence: 99%

Circuits in random graphs: from local trees to global loops

Marinari¹,

Monasson

2004

J. Stat. Mech.: Theor. Exp.

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Abstract. We compute the number of circuits and of loops with multiple crossings in random regular graphs. We discuss the importance of this issue for the validity of the cavity approach. On the one side we obtain analytic results for the infinite volume limit in agreement with existing exact results. On the other side we implement a counting algorithm, enumerate circuits at finite N and draw some general conclusions about the finite N behavior of the circuits. IntroductionThe study of random graphs, initiated more than four decades ago, has been since an issue of major interest in probability theory and in statistical mechanics. Examples of random graphs abound [1], from the original Erdös-Renyi model [2], where edges are chosen independently of each other between pairs of a set of N vertices (with a fixed probability of O(1/N)), to the scale-free graphs with power law degree distribution [3], only introduced in recent times. An interesting and useful distribution is the one that generates random c-regular graphs [4]. These are uniformly drawn from the set of all graphs over N vertices, each restricted to have degree c. Random regular graphs can be easily generated when N is large (and c finite) [5], and an instance of 3-regular graph is symbolized in figure 1. Around a randomly picked up vertex called root the graph looks like a regular tree. The probability that there exists a circuit (a self-avoiding closed path) of length L going through the root vanishes when N is sent to infinity and L is kept finite ‡ [4].The purpose of this article is to reach some quantitative understanding of the presence of long circuits in random graphs. Our motivation is at least two-fold.First, improving our knowledge on circuits in random graphs would certainly have positive fall-out on the understanding of equilibrium properties of models of interacting ‡ More precisely, this probability asymptotically departs from zero when log N = O(L). Finite-length loops may be present in the graph, but not in extensive number.

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Spin Glass on a Bethe Lattice: Order Parameter and Critical Susceptibility

Timonin

1992

Physica Status Solidi (b)

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Since de Almeida and Thouless (AT) /1/ have shown the existence of the phase transition in the symmetric Sherrington-Kirkpatrick (SK) model 121 at finite external field H manifesting itself by the divergence at H = HAT(T) of the spin glass susceptibility where <. . . > T denotes a Gibbs average and <. . . > J is an average over random exchange interactions J.. , it has become clear that the Edwards-Anderson (EA) parameter 13 I 11 could not be the order parameter of this transition because q f 0 at H f 0 for any temperature T. The search for the adequate order parameter of the AT transition has lead to the replica symmetry breaking scheme by Parisi / 4 / in which the spin glass phase at H < HAT is described by the order parameter function q ( x ) , 0 < x < 1. The physical interpretation of q ( x ) in terms of the probability distribution function for the overlaps of the magnetizations in various "pure states" / 5 / leaves still open the question: why just x i G (1) diverges at the AT line? The clear physical arguments on this point are quite necessary as the mathematical rigorousness of the results obtained in 11 to 5 / using the so-called "replica trick" (the continuation from the integer replicas number n to the real n + 0) is rather doubtful. Also the attempts to consider the S K model without resorting to the "replica trick" 16 to 8 / have not elucidated the problem of the order parameter of the AT transition and the cause of the x '~~ divergence. The same is true f o r the results of the Bethe spin glass treatments / 9 to 151 in which the existence of the AT transition in finite field has been established using only two replicas / 9 , 13/. Thus the order parameter definitions considered in /14, 15/ rely essentially on the specific features of the Bethe lattice geometry which does ) Prospekt Stachki 194, 344104 Rostov-on-Don, Russia.

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Replica Symmetry Breaking and the Spin-Glass on a Bethe Lattice

Cited by 58 publications

References 16 publications

The Bethe lattice spin glass revisited

The Bethe lattice spin glass revisited

Circuits in random graphs: from local trees to global loops

Spin Glass on a Bethe Lattice: Order Parameter and Critical Susceptibility

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