arXiv subjects

Fernando L. Metz

Publications and source records attributed to Fernando L. Metz.

27 records · Page 2Linked to original sources

arXiv2018

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

We explore the cooperative behaviour and phase transitions of interacting networks by studying a simplified model consisting of Ising spins placed on the nodes of two coupled Erdös-Rényi random graphs. We derive analytical expressions for the free-energy of the system and the magnetization of each graph, from which the phase diagrams, the stability of the different states, and the nature of the transitions among them, are clearly characterized. We show that a metastable state appears discontinuously by varying the model parameters, yielding a region in the phase diagram where two solutions coexist. By performing Monte-Carlo simulations, we confirm the exactness of our main theoretical results and show that the typical time the system needs to escape from a metastable state grows exponentially fast as a function of the temperature, characterizing ergodicity breaking in the thermodynamic limit.

cond-mat.stat-mech↗

arXiv2018

Theory for the conditioned spectral density of non-invariant random matrices

We develop a theoretical approach to compute the conditioned spectral density of $N \times N$ non-invariant random matrices in the limit $N \rightarrow \infty$. This large deviation observable, defined as the eigenvalue distribution conditioned to have a fixed fraction $k$ of eigenvalues smaller than $x \in \mathbb{R}$, provides the spectrum of random matrix samples that deviate atypically from the average behavior. We apply our theory to sparse random matrices and unveil strikingly new and generic properties, namely: (i) their conditioned spectral density has compact support; (ii) it does not experience any abrupt transition for $k$ around its typical value; (iii) its eigenvalues do not accumulate at $x$. Moreover, our work points towards other types of transitions in the conditioned spectral density for values of $k$ away from its typical value. These properties follow from the weak or absent eigenvalue repulsion in sparse ensembles and they are in sharp contrast to those displayed by classic or rotationally invariant random matrices. The exactness of our theoretical findings are confirmed through numerical diagonalization of finite random matrices.

cond-mat.dis-nn↗

arXiv2018

Large deviation theory for diluted Wishart random matrices

Wishart random matrices with a sparse or diluted structure are ubiquitous in the processing of large datasets, with applications in physics, biology and economy. In this work we develop a theory for the eigenvalue fluctuations of diluted Wishart random matrices, based on the replica approach of disordered systems. We derive an analytical expression for the cumulant generating function of the number of eigenvalues $\mathcal{I}_N(x)$ smaller than $x\in\mathbb{R}^{+}$, from which all cumulants of $\mathcal{I}_N(x)$ and the rate function $Ψ_{x}(k)$ controlling its large deviation probability $\text{Prob}[\mathcal{I}_N(x)=kN] \asymp e^{-NΨ_{x}(k)}$ follow. Explicit results for the mean value and the variance of $\mathcal{I}_N(x)$, its rate function, and its third cumulant are discussed and thoroughly compared to numerical diagonalization, showing a very good agreement. The present work establishes the theoretical framework put forward in a recent letter [Phys. Rev. Lett. 117, 104101] as an exact and compelling approach to deal with eigenvalue fluctuations of sparse random matrices.

cond-mat.dis-nn↗

arXiv2017

Replica-symmetric approach to the typical eigenvalue fluctuations of Gaussian random matrices

We discuss an approach to compute the first and second moments of the number of eigenvalues $I_N$ that lie in an arbitrary interval of the real line for $N \times N$ Gaussian random matrices. The method combines the standard replica-symmetric theory with a perturbative expansion of the saddle-point action up to $O(1/N)$ ($N \gg 1$), leading to the correct logarithmic scaling of the variance $\langle I_{N}^{2} \rangle - \langle I_N \rangle^2 = O(\ln N)$ as well as to an analytical expression for the $O(1/N)$ correction to the average $\langle I_N \rangle/N$. Standard results for the number variance at the local scaling regime are recovered in the limit of a vanishing interval. The limitations of the replica-symmetric method are unveiled by comparing our results with those derived through exact methods. The present work represents an important step to study the fluctuations of $I_N$ in non-invariant random matrix ensembles, where the joint distribution of eigenvalues is not known.

cond-mat.stat-mech↗

arXiv2017

Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase

We calculate the level compressibility $χ(W,L)$ of the energy levels inside $[-L/2,L/2]$ for the Anderson model on infinitely large random regular graphs with on-site potentials distributed uniformly in $[-W/2,W/2]$. We show that $χ(W,L)$ approaches the limit $\lim_{L \rightarrow 0^+} χ(W,L) = 0$ for a broad interval of the disorder strength $W$ within the extended phase, including the region of $W$ close to the critical point for the Anderson transition. These results strongly suggest that the energy levels follow the Wigner-Dyson statistics in the extended phase, consistent with earlier analytical predictions for the Anderson model on an Erdös-Rényi random graph. Our results are obtained from the accurate numerical solution of an exact set of equations valid for infinitely large regular random graphs.

cond-mat.dis-nn↗

arXiv2016

Large deviation function for the number of eigenvalues of sparse random graphs inside an interval

We present a general method to obtain the exact rate function $Ψ_{[a,b]}(k)$ controlling the large deviation probability $\text{Prob}[\mathcal{I}_N[a,b]=kN] \asymp e^{-NΨ_{[a,b]}(k)}$ that a $N \times N$ sparse random matrix has $\mathcal{I}_N[a,b]=kN$ eigenvalues inside the interval $[a,b]$. The method is applied to study the eigenvalue statistics in two distinct examples: (i) the shifted index number of eigenvalues for an ensemble of Erdös-Rényi graphs and (ii) the number of eigenvalues within a bounded region of the spectrum for the Anderson model on regular random graphs. A salient feature of the rate function in both cases is that, unlike rotationally invariant random matrices, it is asymmetric with respect to its minimum. The asymmetric character depends on the disorder in a way that is compatible with the distinct eigenvalue statistics corresponding to localized and delocalized eigenstates. The results also show that the number variance $σ_{N}^{2}$ for the Anderson model on a regular graph scales as $σ_{N}^{2} \propto N$ ($N \gg 1$) for any nonzero disorder, which is consistent with the absence of level-repulsion in the extended phase. Our theoretical findings are thoroughly compared to numerical diagonalization in both cases, showing a reasonable good agreement.

cond-mat.dis-nn↗

arXiv2015

Index statistical properties of sparse random graphs

Using the replica method, we develop an analytical approach to compute the characteristic function for the probability $\mathcal{P}_N(K,λ)$ that a large $N \times N$ adjacency matrix of sparse random graphs has $K$ eigenvalues below a threshold $λ$. The method allows to determine, in principle, all moments of $\mathcal{P}_N(K,λ)$, from which the typical sample to sample fluctuations can be fully characterized. For random graph models with localized eigenvectors, we show that the index variance scales linearly with $N \gg 1$ for $|λ| > 0$, with a model-dependent prefactor that can be exactly calculated. Explicit results are discussed for Erdös-Rényi and regular random graphs, both exhibiting a prefactor with a non-monotonic behavior as a function of $λ$. These results contrast with rotationally invariant random matrices, where the index variance scales only as $\ln N$, with an universal prefactor that is independent of $λ$. Numerical diagonalization results confirm the exactness of our approach and, in addition, strongly support the Gaussian nature of the index fluctuations.

cond-mat.stat-mech↗

arXiv2014

Finite size correction to the spectrum of regular random graphs: an analytical solution

We develop a thorough analytical study of the $O(1/N)$ correction to the spectrum of regular random graphs with $N \rightarrow \infty$ nodes. The finite size fluctuations of the resolvent are given in terms of a weighted series over the contributions coming from loops of all possible lengths, from which we obtain the isolated eigenvalue as well as an analytical expression for the $O(1/N)$ correction to the continuous part of the spectrum. The comparison between this analytical formula and direct diagonalization results exhibits an excellent agreement, confirming the correctness of our expression.

cond-mat.dis-nn↗

arXiv2014

Statistical mechanics of the spherical hierarchical model with random fields

We study analytically the equilibrium properties of the spherical hierarchical model in the presence of random fields. The expression for the critical line separating a paramagnetic from a ferromagnetic phase is derived. The critical exponents characterising this phase transition are computed analytically and compared with those of the corresponding $D$-dimensional short-range model, leading to conclude that the usual mapping between one dimensional long-range models and $D$-dimensional short-range models holds exactly for this system, in contrast to models with Ising spins. Moreover, the critical exponents of the pure model and those of the random field model satisfy a relationship that mimics the dimensional reduction rule. The absence of a spin-glass phase is strongly supported by the local stability analysis of the replica symmetric saddle-point as well as by an independent computation of the free-energy using a renormalization-like approach. This latter result enlarges the class of random field models for which the spin-glass phase has been recently ruled out.

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