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

Edgar Guzmán-González, Isaac Pérez Castillo, and Fernando L. Metz

Phys. Rev. E 101, 012133 – Published 28 January 2020

Abstract

Received 23 September 2019
Corrected 12 October 2020

DOI:https://doi.org/10.1103/PhysRevE.101.012133

Physics Subject Headings (PhySH)

First order phase transitions Network phase transitions

Random graphs

Ising model Replica methods

Statistical Physics & ThermodynamicsNetworks

Corrections

12 October 2020

Correction: The affiliation listing for author I.P.C. required reformatting and has been fixed.

Authors & Affiliations

Edgar Guzmán-González^*

Departamento de Física Cuántica y Fotónica, Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México C.P. 04510, México and London Mathematical Laboratory, 18 Margravine Gardens, London W6 8RH, United Kingdom

Isaac Pérez Castillo^†

Departamento de Física Cuántica y Fotónica, Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México C.P. 04510, México

Fernando L. Metz ^‡

Physics Institute, Federal University of Rio Grande do Sul, 91501-970 Porto Alegre, Brazil and London Mathematical Laboratory, 18 Margravine Gardens, London W6 8RH, United Kingdom

^*edgar.guzman@fisica.unam.mx
^†isaacpc@fisica.unam.mx
^‡fmetzfmetz@gmail.com

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Vol. 101, Iss. 1 — January 2020

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Figure 1
Phase diagram illustrating the second-order percolation transition and the first-order condensation transition for the constrained ER graph ensemble generated from Eq. (5), where the degrees are conditioned to lie in the interval $[k_{*} - 1, k_{*} + 1]$ . The percolation transitions are indicated on the figure. The condensation transitions are those which terminate at critical points (circles). For $k_{*} = 2$ , the second-order percolation transition terminates at a given value $y < 0$ , below which it becomes discontinuous, coinciding with the condensation transition. There is no condensation transition for $k_{*} \geq 5$ in the region of small $c$ (see Fig. 3).
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Figure 2
Degree distribution for $c = 2, a = 1, b = 3$ , and two different values of $y$ : $y = - 1$ (green circles) and $y = - 2.5$ (brown squares). One value of $y$ lies in the condensed phase ( $| y | > | y_{c} |$ ), while the other lies in the Poisson-like phase ( $| y | < | y_{c} |$ ). As a comparison, we also show the Poisson degree distribution for $c = 2$ .
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Figure 3
Average degree ${〈 k 〉}_{y}$ of the constrained ER ensemble as a function of $c$ for fixed $y = - 5$ and different intervals $[k_{*} - 1, k_{*} + 1]$ . The first-order condensation transition disappears at $k_{*} = 5$ .
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Figure 4
Critical temperature $T_{c}$ [see Eq. (23)] identifying the continuous phase transition between the paramagnetic and the ferromagnetic phases. The results for $T_{c}$ are shown as a function of $y$ for different values of the average degree $c$ . The vertical part of the curves mark the critical values $y_{c}$ at which the constrained ER ensemble undergoes a first-order structural transition to a phase exhibiting condensation of degrees (see Fig. 1).
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Figure 5
Magnetization $m$ , internal energy $u$ , and magnetic susceptibility $χ$ of the Ising model as a function of $y$ for average degree $c = 13$ , zero external magnetic field ( $h = 0$ ), and different temperatures $T$ . The theoretical results (solid lines) are derived from the numerical solution of Eq. (13) using the population dynamics algorithm, while the different symbols are results obtained from Monte Carlo simulations of the model by considering 500 samples of the system with a total number of $N = 1000$ spins. The inset shows the behavior of $χ$ around the second-order phase transition between the paramagnetic and the ferromagnetic phases for $T = 2$ .
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Figure 6
Theoretical results (different line styles) for the spectral density of constrained ER random graphs for different values of $y$ [see Eq. (5)], interval [1, 3], and average degrees (a) $c = 13$ and (b) $c = 2$ . Panels (a) and (b) show the behavior of the eigenvalue distribution as we cross the condensation transition for high and low $c$ , respectively. The theoretical results are obtained from the numerical solution of Eq. (29) using the population dynamics algorithm. The square symbols are obtained from the direct diagonalization of 1000 independent realizations of the $1000 \times 1000$ adjacency matrix characterizing atypical graph configurations generated through a reweighted Monte Carlo method [30]. To present the comparison between theory and numerical diagonalization in a clear way, we have chosen to display numerical diagonalization results only for two values of $y$ in each graph. The eigenvalues have been rescaled as $λ_{i} \to λ_{i} / \sqrt{c}$ in panel (a). The inset shows the second moment of the spectral density $ρ_{y} (λ)$ for $c = 13$ .
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Physical Review E

covering statistical, nonlinear, biological, and soft matter physics

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

Edgar Guzmán-González, Isaac Pérez Castillo, and Fernando L. Metz

Phys. Rev. E 101, 012133 – Published 28 January 2020

Abstract

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Physical Review E

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

Edgar Guzmán-González, Isaac Pérez Castillo, and Fernando L. Metz

Phys. Rev. E 101, 012133 – Published 28 January 2020

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