Quenched random graphs

Bachas, K.; Calan, C de; Petropoulos, P. Marios

doi:10.1088/0305-4470/27/18/020

Cited by 27 publications

(61 citation statements)

References 20 publications

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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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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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“…It would obviously be better to have a more trustworthy formalism for investigating quenched problems where such troubles did not arise and one could confidently calculate with an arbitrary number of replicas. Such a formalism is actually inherent in the matrix model approach in a limit that is opposite in a sense to that usually considered, namely N → 1 [8]. In this limit the fat graphs that build the surfaces or their duals in the matrix model integrals degenerate to "thin" graphs -in other words zero dimensional Feynman diagrams.…”

Section: Introductionmentioning

confidence: 99%

“…In this limit the fat graphs that build the surfaces or their duals in the matrix model integrals degenerate to "thin" graphs -in other words zero dimensional Feynman diagrams. Such graphs have in fact been considered before in the context of spin glass theories [9], without utilizing the methods of field theory and large order expansions of [8]. The generic problems in the matrix model approach for c > 1 can be traced to the difficulty in evaluating angular integrals that arise from diagonalizing the matrices, and these disappear for N = 1.…”

Section: Introductionmentioning

confidence: 99%

“…In section 4 we discuss the anti-ferromagnetic Ising model on thin graphs and describe our simulations of both the Ising and three and four state Potts models with purely negative couplings on quenched ensembles of Feynman diagrams. The calculations of [8] suggest that the antiferromagnetic model may have a spin glass phase at sufficiently negative coupling, and we investigate this numerically by histogramming the overlaps between replicas. In sections 5 and 6 we discuss briefly some exploratory simulations of the Ising antiferromagnet on quenched fat graphs and of Potts antiferromagnets on thin graphs respectively, largely to compare the results with those for the Ising model.…”

Section: Introductionmentioning

confidence: 99%

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Ising spins on thin graphs

Baillie¹,

Johnston²,

Kownacki³

1994

Nuclear Physics B

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The Ising model on "thin" graphs (standard Feynman diagrams) displays several interesting properties. For ferromagnetic couplings there is a mean field phase transition at the corresponding Bethe lattice transition point. For antiferromagnetic couplings the replica trick gives some evidence for a spin glass phase. In this paper we investigate both the ferromagnetic and antiferromagnetic models with the aid of simulations. We confirm the Bethe lattice values of the critical points for the ferromagnetic model on φ 3 and φ 4 graphs and examine the putative spin glass phase in the antiferromagnetic model by looking at the overlap between replicas in a quenched ensemble of graphs. We also compare the Ising results with those for higher state Potts models and Ising models on "fat" graphs, such as those used in 2D gravity simulations.

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“…In particular, one finds mean-field like behaviour for the spin models due to the locally-tree-like structure of the graphs. This approach is based on the simple observation [3] that the thin graphs appear as the Feynman diagrams in the perturbative expansion of matrix models when the N → 1 (scalar) limit is taken. An alternative N → ∞ limit is perhaps more familiar for matrix models in the context of two-dimensional gravity, where the resulting "fat" graphs are of interest because of their relation to surfaces and string worldsheets.…”

Section: Introductionmentioning

confidence: 99%

Vertex models on Feynman diagrams

Johnston¹,

Plecháč²

1998

Physics Letters A

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The statistical mechanics of spin models, such as the Ising or Potts models, on generic random graphs can be formulated economically by considering the N → 1 limit of N × N Hermitian matrix models. In this paper we consider the N → 1 limit in complex matrix models, which describes vertex models of different sorts living on random graphs. From the graph theoretic perspective one is using matrix model and field theory inspired methods to count various classes of directed graphs.We also make some remarks on vertex models on planar random graphs (the N → ∞ limit) where the resulting matrix models are not generally soluble using currently known methods. Nonetheless, some particular cases may be mapped onto known models and hence solved.

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Quenched random graphs

Abstract: Spin models on quenched random graphs are related to many important optimization problems. We give a new derivation of their mean-field equations that elucidates the role of the natural order parameter in these models.

Cited by 27 publications

References 20 publications

Circuits in random graphs: from local trees to global loops

Circuits in random graphs: from local trees to global loops

Ising spins on thin graphs

Vertex models on Feynman diagrams

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