The random field XY model on sparse random graphs shows replica symmetry breaking and marginally stable ferromagnetism

Lupo, Cosimo; Parisi, G.; Ricci-Tersenghi, Federico

doi:10.1088/1751-8121/ab2287

Cited by 18 publications

(19 citation statements)

References 49 publications

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“…The assumption that the distribution of rescaled degrees attains a well-defined limit ν(g) for c → ∞ underlies the validity of equation (33). The next step is to calculate η[t] from equation (34). Since the coupling strengths become very weak for c 1, we expand equation (34) in powers of J r and in turn compute the limit c → ∞, finding…”

Section: The Heterogeneous Mean-field Theorymentioning

confidence: 99%

“…The next step is to calculate η[t] from equation (34). Since the coupling strengths become very weak for c 1, we expand equation (34) in powers of J r and in turn compute the limit c → ∞, finding…”

Section: The Heterogeneous Mean-field Theorymentioning

confidence: 99%

“…Models of scalar spins on networks have a vast number of applications in a variety of research fields, such as opinion dynamics [9,10], models of socio-economic phenomena [11], artificial neural networks [12][13][14], agent-based models of the market behavior [15][16][17], dynamics of biological neural networks [18][19][20], information theory and computer science [8], sparse random-matrix theory [21,22], and the stability of large dynamical systems [23,24]. Models of vector spins on networks are relevant for the study of synchronization phenomena [25][26][27][28], random lasers [29][30][31], vector spin-glasses (SGs) [32][33][34], and the collective dynamics of swarms [35][36][37][38].…”

Section: Introductionmentioning

confidence: 99%

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Mean-field theory of vector spin models on networks with arbitrary degree distributions

Metz

Peron

2022

J. Phys. Complex.

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Understanding the relationship between the heterogeneous structure of complex networks and cooperative phenomena occurring on them remains a key problem in network science. Mean-ﬁeld theories of spin models on networks constitute a fundamental tool to tackle this problem and a cornerstone of statistical physics, with an impressive number of applications in condensed matter, biology, and computer science. In this work we derive the mean-ﬁeld equations for the equilibrium behavior of vector spin models on high-connectivity random networks with an arbitrary degree distribution and with randomly weighted links. We demonstrate that the high-connectivity limit of spin models on networks is not universal in that it depends on the full degree distribution. Such nonuniversal behavior is akin to a remarkable mechanism that leads to the breakdown of the central limit theorem when applied to the distribution of effective local ﬁelds. Traditional mean-ﬁeld theories on fully-connected models, such as the Curie-Weiss, the Kuramoto, and the Sherrington-Kirkpatrick model, are only valid if the network degree distribution is highly concentrated around its mean degree. We obtain a series of results that highlight the importance of degree ﬂuctuations to the phase diagram of mean-ﬁeld spin models by focusing on the Kuramoto model of synchronization and on the Sherrington-Kirkpatrick model of spin-glasses. Numerical simulations corroborate our theoretical ﬁndings and provide compelling evidence that the present mean-ﬁeld theory describes an intermediate regime of connectivity, in which the average degree $c$ scales as a power $c \propto N^{b}$ ($b < 1$) of the total number $N \gg 1$ of spins. Our ﬁndings put forward a novel class of spin models that incorporate the effects of degree ﬂuctuations and, at the same time, are amenable to exact analytic solutions.

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Section: The Heterogeneous Mean-field Theorymentioning

confidence: 99%

Section: The Heterogeneous Mean-field Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Mean-field theory of vector spin models on networks with arbitrary degree distributions

Metz

Peron

2022

J. Phys. Complex.

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“…For example, in the Ising and Potts models the variables take discrete values and so the space of configuration is not continuous, while in the O(n) models (e.g. with XY or Heisenberg spins) each variable is continuous, but needs to satisfy a local constraint of unit norm in an n-dimensional space, and this in turn makes the analytic solution much more complicated; see for instance [3][4][5].…”

Section: Introductionmentioning

confidence: 99%

Solving the spherical p -spin model with the cavity method: equivalence with the replica results

Gradenigo¹,

Angelini²,

Leuzzi³

et al. 2020

J. Stat. Mech.

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The spherical p-spin is a fundamental model for glassy physics, thanks to its analytical solution achievable via the replica method. Unfortunately, the replica method has some drawbacks: it is very hard to apply to diluted models and the assumptions beyond it are not immediately clear. Both drawbacks can be overcome by the use of the cavity method; however, this needs to be applied with care to spherical models. Here, we show how to write the cavity equations for spherical p-spin models, both in the replica symmetric (RS) ansatz (corresponding to belief propagation) and in the one-step replica-symmetry-breaking (1RSB) ansatz (corresponding to survey propagation). The cavity equations can be solved by a Gaussian RS and multivariate Gaussian 1RSB ansatz for the distribution of the cavity fields. We compute the free energy in both ansatzes and check that the results are identical to the replica computation, predicting a phase transition to a 1RSB phase at low temperatures. The advantages of solving the

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“…The influence of the graph structure on dynamical processes and on the cooperative behavior of models defined on random graphs is a key topic in network theory, which has been attracting a huge interest in recent decades [12,[33][34][35][36][37][38][39][40][41][42][43][44][45][46]. The degree statistics plays a pivotal role on the long-time behavior of random walks on graphs [37], on the critical threshold for epidemic spreading [33,34], on the linear stability of large interacting systems [44,45], and on the critical properties of cooperative systems defined on random graphs, such as the Ising model [35,36], the Kuramoto model [38,[41][42][43]46], and the classical Heisenberg model [42,46]. Since condensation of degrees emerges through a discontinuous transition in the degree distribution, it is therefore compelling to ask how this structural transition impacts the macroscopic behavior of systems interacting through the edges of random graphs.…”

Section: Introductionmentioning

confidence: 99%

Phase transitions in atypical systems induced by a condensation transition on graphs

2020

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Random graphs undergo structural phase transitions that are crucial for dynamical processes and cooperative behavior of models defined on graphs. In this work we investigate the impact of a first-order structural transition on the thermodynamics of the Ising model defined on Erdős-Rényi random graphs, as well as on the eigenvalue distribution of the adjacency matrix of the same graphical model. The structural transition in question yields graph samples exhibiting condensation, characterized by a large number of nodes having degrees in a narrow interval. We show that this condensation transition induces distinct thermodynamic first-order transitions between the paramagnetic and the ferromagnetic phases of the Ising model. The condensation transition also leads to an abrupt change in the global eigenvalue statistics of the adjacency matrix, which renders the second moment of the eigenvalue distribution discontinuous. As a side result, we derive the critical line determining the percolation transition in Erdős-Rényi graph samples that feature condensation of degrees. arXiv:1909.10564v1 [cond-mat.dis-nn]

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The random field XY model on sparse random graphs shows replica symmetry breaking and marginally stable ferromagnetism

Cited by 18 publications

References 49 publications

Mean-field theory of vector spin models on networks with arbitrary degree distributions

Mean-field theory of vector spin models on networks with arbitrary degree distributions

Solving the spherical p -spin model with the cavity method: equivalence with the replica results

Phase transitions in atypical systems induced by a condensation transition on graphs

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