arXiv subjects

Isaac Pérez Castillo

Publications and source records attributed to Isaac Pérez Castillo.

At least 55 records · Page 3Linked to original sources

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↗

arXiv2016

When does inequality freeze an economy?

Inequality and its consequences are the subject of intense recent debate. Using a simplified model of the economy, we address the relation between inequality and liquidity, the latter understood as the frequency of economic exchanges. Assuming a Pareto distribution of wealth for the agents, that is consistent with empirical findings, we find an inverse relation between wealth inequality and overall liquidity. We show that an increase in the inequality of wealth results in an even sharper concentration of the liquid financial resources. This leads to a congestion of the flow of goods and the arrest of the economy when the Pareto exponent reaches one.

econ.GN↗

arXiv2015

On the passage properties of the gradual capture of a diffusive particle in the presence of drift

We investigate a stochastic process consisting of a two-dimensional particle with anisotropic diffusion, mortality rate and a drift velocity, in the presence of an absorbing boundary. After the particle has encountered the boundary, the process is restarted with updated values of its diffusion constants and drift velocity. We then derive the joint probability that, after $M$ encounters, the particle is absorbed at a point of the boundary at a given time and show that, under certain conditions, the eventual hitting probability corresponds to a Bessel distribution. In the context of the man-mosquitoes problem, the mosquito is able to gradually capture the man, after which the mosquito follows a diffusion process with no drift. Our results are compared thoroughly with Monte Carlo simulations showing excellent agreement.

cond-mat.stat-mech↗

arXiv2015

Minimal cooling speed for glass transition in a simple solvable energy landscape model

The minimal cooling speed required to form a glass is obtained for a simple solvable energy landscape model. The model, made from a two-level system modified to include the topology of the energy landscape, is able to capture either a glass transition or a crystallization depending on cooling rate. In this setup, the minimal cooling speed to achieve glass formation is then found to be related with the relaxation time and with the thermal history. In particular, we obtain that the thermal history encodes small fluctuations around the equilibrium population which are exponentially amplified near the glass transition, which mathematically corresponds to the boundary layer of the master equation. Finally, to verify our analytical results, a kinetic Monte-Carlo simulation was implemented.

cond-mat.soft↗

arXiv2015

A unified approach for large deviations of bulk and extreme eigenvalues of the Wishart ensemble

Within the framework of the Coulomb fluid picture, we present a unified approach to derive the large deviations of bulk and extreme eigenvalues of large Wishart matrices. By analysing the statistics of the shifted index number we are able to derive a rate function $Ψ(c, x)$ depending on two variables: the fraction $c$ of eigenvalues to the left of an infinite energetic barrier at position $x$. For a fixed value of $c$, the rate function gives the large deviations of the bulk eigenvalues. In particular, in the limits $c\to 0$ or $c\to 1$ it is possible to extract the left and right deviations of the smallest and largest eigenvalues, respectively. Alternatively, for a fixed value $x$ of the barrier, the rate function provides the large deviations of the shifted index number. All our analytical findings are compared with Metropolis Monte Carlo simulations, obtaining excellent agreement.

cond-mat.stat-mech↗

arXiv2015

Large deviations of the shifted index number in the Gaussian ensemble

We show that, using the Coulomb fluid approach, we are able to derive a rate function $Ψ(c,x)$ of two variables that captures: (i) the large deviations of bulk eigenvalues; (ii) the large deviations of extreme eigenvalues (both left and right large deviations); (iii) the statistics of the fraction $c$ of eigenvalues to the left of a position $x$. Thus, $Ψ(c,x)$ explains the full order statistics of the eigenvalues of large random Gaussian matrices as well as the statistics of the shifted index number. All our analytical findings are thoroughly compared with Monte Carlo simulations, obtaining excellent agreement. A summary of preliminary results was already presented in [22] in the context of one-dimensional trapped spinless fermions in a harmonic potential.

cond-mat.stat-mech↗

arXiv2015

A generalised Airy distribution function for the accumulated area swept by $N$ vicious Brownian paths

In this work exact expressions for the distribution function of the accumulated area swept by reunions and meanders of $N$ vicious Brownian particles up to time $T$ are derived. The results are expressed in terms of a generalised Airy distribution function, containing the Vandermonde determinant of the Airy roots. By mapping the problem to an Random Matrix Theory ensemble we are able to perform Monte Carlo simulations finding perfect agreement with the theoretical results.

cond-mat.stat-mech↗

arXiv2014

Cavity approach for modeling and fitting polymer stretching

The mechanical properties of molecules are today captured by single molecule manipulation experiments, so that polymer features are tested at a nanometric scale. Yet devising mathematical models to get further insight beyond the commonly studied force--elongation relation is typically hard. Here we draw from techniques developed in the context of disordered systems to solve models for single and double--stranded DNA stretching in the limit of a long polymeric chain. Since we directly derive the marginals for the molecule local orientation, our approach allows us to readily calculate the experimental elongation as well as other observables at wish. As an example, we evaluate the correlation length as a function of the stretching force. Furthermore, we are able to fit successfully our solution to real experimental data. Although the model is admittedly phenomenological, our findings are very sound. For single--stranded DNA our solution yields the correct (monomer) scale and, yet more importantly, the right persistence length of the molecule. In the double--stranded case, our model reproduces the well-known overstretching transition and correctly captures the ratio between native DNA and overstretched DNA. Also in this case the model yields a persistence length in good agreement with consensus, and it gives interesting insights into the bending stiffness of the native and overstretched molecule, respectively.

physics.bio-ph↗

arXiv2014

Spectral order statistics of Gaussian random matrices: large deviations for trapped fermions and associated phase transitions

We compute the full order statistics of a one-dimensional gas of fermions in a harmonic trap at zero temperature, including its large deviation tails. The problem amounts to computing the probability distribution of the $k$th smallest eigenvalue $λ_{(k)}$ of a large dimensional Gaussian random matrix. We find that this probability behaves for large $N$ as $\mathcal{P}[λ_{(k)}=x]\approx \exp\left(-βN^2 ψ(k/N,x)\right)$, where $β$ is the Dyson index of the ensemble. The rate function $ψ(c,x)$, computed explicitly as a function of $x$ in terms of the intensive label $c=k/N$, has a quadratic behavior modulated by a weak logarithmic singularity at its minimum. This is shown to be related to phase transitions in the associated Coulomb gas problem. The connection with statistics of extreme eigenvalues of random matrices is also elucidated.

cond-mat.stat-mech↗

arXiv2014

Spectral density of products of Wishart dilute random matrices. Part I: the dense case

In this work we study the spectral density of products of Wishart diluted random matrices of the form $X(1)\cdots X(M)(X(1)\cdots X(M))^T$ using the Edwards-Jones trick to map this problem into a system of interacting particles with random couplings on a multipartite graph. We apply the cavity method to obtain recursive relations in typical instances from which to obtain the spectral density. As this problem is fairly rich, we start by reporting in part I a lengthy analysis for the case of dense matrices. Here we derive that the spectral density is a solution of a polynomial equation of degree $M+1$ and obtain exact expressions of it for $M=1$, $2$ and $3$. For general $M$, we are able to find the exact expression of the spectral density only when all the matrices $X(t)$ for $t=1,\ldots, M$ are square. We also make some observations for general $M$, based admittedly on some weak numerical evidence, which we expect to be correct.

cond-mat.dis-nn↗

arXiv2014

Phase transitions in the condition number distribution of Gaussian random matrices

We study the statistics of the condition number $κ=λ_{\mathrm{max}}/λ_{\mathrm{min}}$ (the ratio between largest and smallest squared singular values) of $N\times M$ Gaussian random matrices. Using a Coulomb fluid technique, we derive analytically and for large $N$ the cumulative $\mathcal{P}[κ x]$ distributions of $κ$. We find that these distributions decay as $\mathcal{P}[κ x]\approx\exp\left(-βN Φ_{+}(x)\right)$, where $β$ is the Dyson index of the ensemble. The left and right rate functions $Φ_{\pm}(x)$ are independent of $β$ and calculated exactly for any choice of the rectangularity parameter $α=M/N-1>0$. Interestingly, they show a weak non-analytic behavior at their minimum $\langleκ\rangle$ (corresponding to the average condition number), a direct consequence of a phase transition in the associated Coulomb fluid problem. Matching the behavior of the rate functions around $\langleκ\rangle$, we determine exactly the scale of typical fluctuations $\sim\mathcal{O}(N^{-2/3})$ and the tails of the limiting distribution of $κ$. The analytical results are in excellent agreement with numerical simulations.

cond-mat.stat-mech↗

arXiv2013

Reunion probabilities of $N$ one-dimensional random walkers with mixed boundary conditions

In this work we extend the results of the reunion probability of $N$ one-dimensional random walkers to include mixed boundary conditions between their trajectories. The level of the mixture is controlled by a parameter $c$, which can be varied from $c=0$ (independent walkers) to $c\to\infty$ (vicious walkers). The expressions are derived by using Quantum Mechanics formalism (QMf) which allows us to map this problem into a Lieb-Liniger gas (LLg) of $N$ one-dimensional particles. We use Bethe ansatz and Gaudin's conjecture to obtain the normalized wave-functions and use this information to construct the propagator. As it is well-known, depending on the boundary conditions imposed at the endpoints of a line segment, the statistics of the maximum heights of the reunited trajectories have some connections with different ensembles in Random Matrix Theory (RMT). Here we seek to extend those results and consider four models: absorbing, periodic, reflecting, and mixed. In all four cases, the probability that the maximum height is less or equal than $L$ takes the form $F_N(L)=A_N\sum_{k\inΩ_{B}}\int Dz e^{-\sum_{j=1}^Nk_j^2+G_N(k)-\sum_{j,\ell=1}^N z_jV_{j\ell}(k)\overline{z}_\ell}$, where $A_N$ is a normalization constant, $G_N(k)$ and $V_{j\ell}(k)$ depend on the type of boundary condition, and $Ω_{B}$ is the solution set of quasi-momenta $k$ obeying the Bethe equations for that particular boundary condition.

cond-mat.stat-mech↗

arXiv2013

A Novel Methodology to Estimate Metabolic Flux Distributions in Constraint-Based Models

Quite generally, constraint-based metabolic flux analysis describes the space of viable flux configurations for a metabolic network as a high-dimensional polytope defined by the linear constraints that enforce the balancing of production and consumption fluxes for each chemical species in the system. In some cases, the complexity of the solution space can be reduced by performing an additional optimization, while in other cases, knowing the range of variability of fluxes over the polytope provides a sufficient characterization of the allowed configurations. There are cases, however, in which the thorough information encoded in the individual distributions of viable fluxes over the polytope is required. Obtaining such distributions is known to be a highly challenging computational task when the dimensionality of the polytope is sufficiently large, and the problem of developing cost-effective {\it ad hoc} algorithms has recently seen a major surge of interest. Here, we propose a method that allows us to perform the required computation heuristically in a time scaling {\it linearly} with the number of reactions in the network, overcoming some limitations of similar techniques employed in recent years. As a case study, we apply it to the analysis of the human red blood cell metabolic network, whose solution space can be sampled by different exact techniques, like Hit-and-Run Monte Carlo (scaling roughly like the third power of the system size). Remarkably accurate estimates for the true distributions of viable reaction fluxes are obtained, suggesting that, although further improvements are desirable, our method enhances our ability to analyze the space of allowed configurations for large biochemical reaction networks.

q-bio.MN↗

arXiv2012

Spectral Properties of the Jacobi Ensembles via the Coulomb Gas approach

Using the Coulomb gas method and standard methods of statistical physics, we compute analytically the joint cumulative probability distribution of the extreme eigenvalues of the Jacobi-MANOVA ensemble of random matrices, in the limit of large matrices. This allows us to derive the rate functions for the large fluctuations to the left and the right of the expected values of the smallest and largest eigenvalues analytically. Our findings are compared with some available known exact results as well as with numerical simulations finding good agreement.

cond-mat.stat-mech↗

arXiv2012

A weighted belief-propagation algorithm to estimate volume-related properties of random polytopes

In this work we introduce a novel weighted message-passing algorithm based on the cavity method to estimate volume-related properties of random polytopes, properties which are relevant in various research fields ranging from metabolic networks, to neural networks, to compressed sensing. Unlike the usual approach consisting in approximating the real-valued cavity marginal distributions by a few parameters, we propose an algorithm to faithfully represent the entire marginal distribution. We explain various alternatives to implement the algorithm and benchmark the theoretical findings by showing concrete applications to random polytopes. The results obtained with our approach are found to be in very good agreement with the estimates produced by the Hit-and-Run algorithm, known to produce uniform sampling.

cond-mat.dis-nn↗

arXiv2011

A weighted message-passing algorithm to estimate volume-related properties of random polytopes

In this letter, we introduce a novel message-passing algorithm for a class of problems which can be mathematically understood as estimating volume-related properties of random polytopes. Unlike the usual approach consisting in approximating the real-valued cavity marginal distributions by a few parameters, we propose a weighted message-passing algorithm to deal with the entire function. Various alternatives of how to implement our approach are discussed and numerical results for random polytopes are compared with results using the Hit-and-Run algorithm.

cs.DS↗

arXiv2011

The Storm and Nelson's model for polymer stretching revisited

The quite recent technological rise in molecular biology allowed single molecule manipulation experiments, where molecule stretching plays a primary role. In order to understand the experimental data, it is felt the urge of some physical and mathematical models to quantitatively express the mechanical properties of the observed molecules. In this paper we reconsider a simple phenomenological model which reproduces the behaviour of single and double stranded DNA under tension. The problem is easily solved via the cavity method both in the small forces range and in presence of overstretching transition, so that some properties such as bending stiffness and elasticity of DNA emerge in a very clear manner. Our theoretical findings are successfully fitted to real measurements and compared to Monte Carlo simulations, confirming the quality of the approach.

physics.bio-ph↗

arXiv2010

Large Deviations of the Smallest Eigenvalue of the Wishart-Laguerre Ensemble

We consider the large deviations of the smallest eigenvalue of the Wishart-Laguerre Ensemble. Using the Coulomb gas picture we obtain rate functions for the large fluctuations to the left and the right of the hard edge. Our findings are compared with known exact results for $β=1$ finding good agreement. We also consider the case of almost square matrices finding new universal rate functions describing large fluctuations.

cond-mat.dis-nn↗