Probability Distributions with Singularities

Corberi, Federico; Sarracino, Alessandro

doi:10.3390/e21030312

Cited by 13 publications

(20 citation statements)

References 74 publications

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“…Such rate function exhibits two non-analytic points akin to the presence of the first-order transition. We point out that similar properties have been identified in the probability distributions describing the condensation of fluctuations in other disordered systems [5,9]. Our theoretical results for the rate function are fully confirmed by Monte Carlo simulations.…”

Section: Discussionsupporting

confidence: 82%

“…We also compute the rate function characterizing the large deviation probability of F N [a, b], whose striking property is the non-analytic behaviour. The calculation of the rate function shows that condensation of degrees is a rare statistical event in line with the condensation of fluctuations exhibited by other random systems [9]. The theoretical results for the rate function exhibit an excellent agreement with Monte Carlo simulations.…”

Section: Introductionsupporting

confidence: 73%

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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: Discussionsupporting

confidence: 82%

Section: Introductionsupporting

confidence: 73%

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

Metz

Castillo

2019

Phys. Rev. E

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“…An important example of this singularityphase coexistence correspondence in equilibrium is the 2D Ising model below its critical temperature [1][2][3]8]. In dynamical models, singularities (kinks) of large-deviation functions develop in certain limits and can signal the emergence of a dynamical phase transition and the coexistence of distinct dynamical phases [9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Varied phenomenology of models displaying dynamical large-deviation singularities

Whitelam

Jacobson

2021

Phys. Rev. E

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Singularities of dynamical large-deviation functions are often interpreted as the signal of a dynamical phase transition and the coexistence of distinct dynamical phases, by analogy with the correspondence between singularities of free energies and equilibrium phase behavior. Here we study models of driven random walkers on a lattice. These models display large-deviation singularities in the limit of large lattice size, but the extent to which each model's phenomenology resembles a phase transition depends on the details of the driving. We also compare the behavior of ergodic and nonergodic models that present large-deviation singularities. We argue that dynamical large-deviation singularities indicate the divergence of a model timescale, but not necessarily one associated with cooperative behavior or the existence of distinct phases.

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“…The formation of such condensed configurations has been coined condensation of degrees. These are large de-viation events triggered by atypical fluctuations in the graph structure, similar to other random systems that exhibit condensation transitions driven by rare fluctuations [31,32].…”

Section: Introductionmentioning

confidence: 62%

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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Probability Distributions with Singularities

Cited by 13 publications

References 74 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

Varied phenomenology of models displaying dynamical large-deviation singularities

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

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