Phase diagram and metastability of the Ising model on two coupled networks

Bolfe, Maíra; Nicolao, Lucas; Metz, Fernando L.

doi:10.1088/1742-5468/aad6c7

Cited by 4 publications

(3 citation statements)

References 39 publications

(110 reference statements)

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“…Secondly, in the same manner that the percolation transition is inherited in the magnetic properties of the Ising model on random graphs, it is pertinent to ask what is the impact on the magnetic properties of the topological first order transition we have observed here. In a similar context, it would be also interesting to study whether large deviations in the degree sequence can trigger the decay of the metastable states found in coupled ER random graphs [33]. Finally, in light of recent advances in the study of large deviations on diluted random matrices [34][35][36], we should consider the spectral properties corresponding to the constrained graph ensemble studied here.…”

Section: Discussionmentioning

confidence: 98%

Condensation of degrees emerging through a first-order phase transition in classical random graphs

Metz

Castillo

2019

Phys. Rev. E

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Due to their conceptual and mathematical simplicity, Erdös-Rényi or classical random graphs remain as a fundamental paradigm to model complex interacting systems in several areas. Although condensation phenomena have been widely considered in complex network theory, the condensation of degrees has hitherto eluded a careful study. Here we show that the degree statistics of the classical random graph model undergoes a firstorder phase transition between a Poisson-like distribution and a condensed phase, the latter characterized by a large fraction of nodes having degrees in a limited sector of their configuration space. The mechanism underlying the first-order transition is discussed in light of standard concepts in statistical physics. We uncover the phase diagram characterizing the ensemble space of the model and we evaluate the rate function governing the probability to observe a condensed state, which shows that condensation of degrees is a rare statistical event akin to similar condensation phenomena recently observed in several other systems. Monte Carlo simulations confirm the exactness of our theoretical results.

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Section: Discussionmentioning

confidence: 98%

Condensation of degrees emerging through a first-order phase transition in classical random graphs

Metz

Castillo

2019

Phys. Rev. E

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“…For example, to give new insights over the finite size effects [58][59][60], critical phenomena [61,62], and randomlink matching problems on random regular graphs [63]. On the other hand we can use the DZFM on multiplex networks to investigate critical phenomena and collective behavior [64], and finally use our formalism to enlarge the set of statistical field theory toolbox that is been currently used for simplicial complex [65,66]. These issues are under investigation by the authors.…”

Section: Discussionmentioning

confidence: 99%

The Sherrington-Kirkpatrick model for spin glasses: A new approach for the solution

Rodríguez-Camargo,

Mojica-Nava,

Svaiter

2021

Preprint

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We discuss the Sherrington-Kirkpatrick mean-field version of a spin glass within the distributional zeta-function method (DZFM). In the DZFM, since the dominant contribution to the average free energy is written as a series of moments of the partition function of the model, the spin-glass multivalley structure is obtained. Also, an exact expression for the saddle points corresponding to each valley and a global critical temperature showing the existence of many stables or at least metastables equilibrium states is presented. Near the critical point we obtain analytical expressions of the order parameters that are in agreement with phenomenological results. We evaluate the linear and nonlinear susceptibility and we find the expected singular behavior at the spin-glass critical temperature. Furthermore, we obtain a positive definite expression for the entropy and we show that ground-state entropy tends to zero as the temperature goes to zero. We show that our solution is stable for each term in the expansion. Finally, we analyze the behavior of the overlap distribution, where we find a general expression for each moment of the partition function.

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“…The authors in [62] define and solve an Ising model on a network composed by two Erdös-Renyi random graphs.…”

Section: Acknowledgmentsmentioning

confidence: 99%

Sparse random block matrices

Cicuta¹,

Pernici²

2022

J. Phys. A: Math. Theor.

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The spectral moments of ensembles of sparse random block matrices are analytically evaluated in the limit of large order. The structure of the sparse matrix corresponds to the Erdös–Renyi random graph. The blocks are i.i.d. random matrices of the classical ensembles GOE or GUE. The moments are evaluated for finite or infinite dimension of the blocks. The correspondences between sets of closed walks on trees and classes of irreducible partitions studied in free probability together with functional relations are powerful tools for analytic evaluation of the limiting moments. They are helpful to identify probability laws for the blocks and limits of the parameters which allow the evaluation of all the spectral moments and of the spectral density.

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Phase diagram and metastability of the Ising model on two coupled networks

Cited by 4 publications

References 39 publications

Condensation of degrees emerging through a first-order phase transition in classical random graphs

Condensation of degrees emerging through a first-order phase transition in classical random graphs

The Sherrington-Kirkpatrick model for spin glasses: A new approach for the solution

Sparse random block matrices

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