arXiv subjects

Jamal Najim

Publications and source records attributed to Jamal Najim.

At least 19 recordsLinked to original sources

arXiv2026

Extreme eigenvalues and eigenvectors for finite rank additive deformations of non-hermitian sparse random matrices

Consider a $n\times n$ sparse non-Hermitian random matrix $X_n$ defined as the Hadamard product between a random matrix with centered independent and identically distributed entries and a sparse Bernoulli matrix with success probability $K_n/n$ where $K_n\le n$ (and possibly $K_n\ll n$) and $K_n\to \infty$ as $n\to \infty$. Let $E_n$ be a deterministic $n\times n$ finite-rank matrix. We prove that the outlier eigenvalues of $Y_n= X_n +E_n$ asymptotically match those of $E_n$. In the special case of a rank-one deformation, assuming further that the sparsity parameter satisfies $K_n \gg \log^9(n)$ and that the entries of the random matrix are sub-Gaussian, we describe the limiting behavior of the projection of the right eigenvector associated with the leading eigenvalue onto the right eigenvector of the rank-one deformation. In particular, we prove that the projection behaves as in the Hermitian case. To that end, we rely on the recent universality results of Brailovskaya and van Handel (2024) relating the singular value spectra of deformations of $X_n$ to Gaussian analogues of these matrices. Our analysis builds upon a recent framework introduced by Bordenave et.al. (2022), and amounts to showing the asymptotic equivalence between the reverse characteristic polynomial of the random matrix and a random analytic function on the unit disc with explicit dependence on the finite-rank deformation.

math.PR↗

arXiv2025

Approximate Message Passing for general non-Symmetric random matrices

Approximate Message Passing (AMP) algorithms are a family of iterative algorithms based on large random matrices with the special property of tracking the statistical properties of their iterates. They are used in various fields such as Statistical Physics, Machine learning, Communication systems, Theoretical ecology, etc. In this article we consider AMP algorithms based on non-Symmetric random matrices with a general variance profile, possibly sparse, a general covariance profile, and non-Gaussian entries. We hence substantially extend the results on Elliptic random matrices that we developed in [Gueddari et al., 2024]. From a technical point of view, we enhance the combinatorial techniques developed in [Bayati et al., 2015] and in [Hachem, 2024]. Our main motivation is the understanding of equilibria of large food-webs described by Lotka-Volterra systems of ODE, in the continuation of the works of [Hachem, 2024], [Akjouj et al., 2024] and [Gueddari et al., 2024], but the versatility of the model studied might be of interest beyond these particular applications.

math.PR↗

arXiv2024

Impact of a block structure on the Lotka-Volterra model

The Lotka-Volterra (LV) model is a simple, robust, and versatile model used to describe large interacting systems such as food webs or microbiomes. The model consists of $n$ coupled differential equations linking the abundances of $n$ different species. We consider a large random interaction matrix with independent entries and a block variance profile. The $i$th diagonal block represents the intra-community interaction in community $i$, while the off-diagonal blocks represent the inter-community interactions. The variance remains constant within each block, but may vary across blocks. We investigate the important case of two communities of interacting species, study how interactions affect their respective equilibrium. We also describe equilibrium with feasibility (i.e., whether there exists an equilibrium with all species at non-zero abundances) and the existence of an attrition phenomenon (some species may vanish) within each community. Information about the general case of $b$ communities ($b> 2$) is provided in the appendix.

q-bio.PE↗

arXiv2024

Elliptic Approximate Message Passing and an application to theoretical ecology

Approximate Message Passing (AMP) algorithmshave recently gathered significant attention across disciplines such as statistical physics, machine learning, and communication systems. This study aims to extend AMP algorithms to non-symmetric (elliptic) matrices, motivated by analyzing equilibrium properties in ecological systems featuring elliptic interaction matrices.In this article, we provide the general form of an AMP algorithm associated to a random elliptic matrix, the main change lying in a modification of the corrective (Onsager) term. In order to establish the statistical properties of this algorithm, we use and prove a generalized form of Bolthausen conditioning argument, pivotal to proceed by a Gaussian-based induction.We finally address the initial motivating question from theoretical ecology. Large foodwebs are often described by Lotka-Volterra systems of coupled differential equations, where the interaction matrix is elliptic random. In this context, we design an AMP algorithm to analyze the statistical properties of the equilibrium point in a high-dimensional regime. We rigorously recover the results established by [Bunin, 2017] and [Galla,2018] who used techniques from theoretical physics, and extend them with the help of propagation of chaos type arguments.

math.PR↗

arXiv2023

Equilibria of large random Lotka-Volterra systems with vanishing species: a mathematical approach

Ecosystems with a large number of species are often modelled as Lotka-Volterra dynamical systems built around a large random interaction matrix. Under some known conditions, a global equilibrium exists and is unique. In this article, we rigorously study its statistical properties in the large dimensional regime. Such an equilibrium vector is known to be the solution of a so-called Linear Complementarity Problem (LCP). We describe its statistical properties by designing an Approximate Message Passing (AMP) algorithm, a technique that has recently aroused an intense research effort in the fields of statistical physics, Machine Learning, or communication theory. Interaction matrices taken from the Gaussian Orthogonal Ensemble, or following a Wishart distribution are considered. Beyond these models, the AMP approach developed in this article has the potential to describe the statistical properties of equilibria associated to more involved interaction matrix models.

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arXiv2022

Complex systems in Ecology: a guided tour with large Lotka-Volterra models and random matrices

Ecosystems represent archetypal complex dynamical systems, often modelled by coupled differential equations of the form $$ \frac{d x_i}{d t} = x_i φ_i(x_1,\cdots, x_N)\ , $$ where $N$ represents the number of species and $x_i$, the abundance of species $i$. Among these families of coupled diffential equations, Lotka-Volterra (LV) equations $$ \frac{d x_i}{d t} = x_i ( r_i - x_i +(Γ\mathbf{x})_i)\ , $$ play a privileged role, as the LV model represents an acceptable trade-off between complexity and tractability. Here, $r_i$ represents the intrinsic growth of species $i$ and $Γ$ stands for the interaction matrix: $Γ_{ij}$ represents the effect of species $j$ over species $i$. For large $N$, estimating matrix $Γ$ is often an overwhelming task and an alternative is to draw $Γ$ at random, parametrizing its statistical distribution by a limited number of model features. Dealing with large random matrices, we naturally rely on Random Matrix Theory (RMT). The aim of this review article is to present an overview of the work at the junction of theoretical ecology and large random matrix theory. It is intended to an interdisciplinary audience spanning theoretical ecology, complex systems, statistical physics and mathematical biology.

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arXiv2022

Equilibrium in a large Lotka-Volterra system with pairwise correlated interactions

We study the equilibria of a large Lokta-Volterra system of coupled differential equations in the case where the interaction coefficients form a large random matrix. In the case where this random matrix follows an elliptic model , we study the existence of a (componentwise) positive equilibrium and describe a phase transition for the matrix normalization.If there is no positive equilibrium, we provide conditions on the model parameters for the existence of a stable equilibrium (with vanishing components) and state heuristics to compute the number of positive components of the equilibrium. Lotka-Volterra systems are important in mathematical biology/ theoretical ecology.

math.PR↗

arXiv2022

Equilibrium and surviving species in a large Lotka-Volterra system of differential equations

Lotka-Volterra (LV) equations play a key role in the mathematical modeling of various ecological, biological and chemical systems. When the number of species (or, depending on the viewpoint, chemical components) becomes large, basic but fundamental questions such as computing the number of surviving species still lack theoretical answers. In this paper, we consider a large system of LV equations where the interactions between the various species are a realization of a random matrix. We provide conditions to have a unique equilibrium and present a heuristics to compute the number of surviving species. This heuristics combines arguments from Random Matrix Theory, mathematical optimization (LCP), and standard extreme value theory. Numerical simulations, together with an empirical study where the strength of interactions evolves with time, illustrate the accuracy and scope of the results.

math.PR↗

arXiv2021

Feasibility of sparse large Lotka-Volterra ecosystems

Consider a large ecosystem (foodweb) with n species, where the abundances follow a Lotka-Volterra system of coupled differential equations. We assume that each species interacts with d other species and that their interaction coefficients are independent random variables. This parameter d reflects the connectance of the foodweb and the sparsity of its interactions especially if d is much smaller that n. We address the question of feasibility of the foodweb, that is the existence of an equilibrium solution of the Lotka-Volterra system with no vanishing species. We establish that for a given range of d with an extra condition on the sparsity structure, there exists an explicit threshold depending on n and d and reflecting the strength of the interactions, which guarantees the existence of a positive equilibrium as the number of species n gets large. From a mathematical point of view, the study of feasibility is equivalent to the existence of a positive solution (component-wise) to the equilibrium linear equation. The analysis of such positive solutions essentially relies on large random matrix theory for sparse matrices and Gaussian concentration of measure. The stability of the equilibrium is established. The results in this article extend to a sparse setting the results obtained by Bizeul and Najim in Proc. AMS 2021.

math.PR↗

arXiv2021

Unbounded Largest Eigenvalue of Large Sample Covariance Matrices: Asymptotics, Fluctuations and Applications

Given a large sample covariance matrix $S_N=\frac 1nΓ_N^{1/2}Z_N Z_N^*Γ_N^{1/2}\, ,$ where $Z_N$ is a $N\times n$ matrix with i.i.d. centered entries, and $Γ_N$ is a $N\times N$ deterministic Hermitian positive semidefinite matrix, we study the location and fluctuations of $λ_{\max}(S_N)$, the largest eigenvalue of $S_N$ as $N,n\to\infty$ and $Nn^{-1} \to r\in(0,\infty)$ in the case where the empirical distribution $μ^{Γ_N}$ of eigenvalues of $Γ_N$ is tight (in $N$) and $λ_{\max}(Γ_N)$ goes to $+\infty$. These conditions are in particular met when $μ^{Γ_N}$ weakly converges to a probability measure with unbounded support on $\mathbb{R}^+$. We prove that asymptotically $λ_{\max}(S_N)\sim λ_{\max}(Γ_N)$. Moreover when the $Γ_N$'s are block-diagonal, and the following {\em spectral gap condition} is assumed:$$\limsup_{N\to\infty} \frac{λ_2(Γ_N)}{λ_{\max}(Γ_N)}<1,$$where $λ_2(Γ_N)$ is the second largest eigenvalue of $Γ_N$, we prove Gaussian fluctuations for $λ_{\max}(S_N)/λ_{\max}(Γ_N)$ at the scale $\sqrt{n}$.In the particular case where $Z_N$ has i.i.d. Gaussian entries and $Γ_N$ is the $N\times N$ autocovariance matrix of a long memory Gaussian stationary process $({\mathcal X}_t)_{t\in\mathbb{Z}}$, the columns of $Γ_N^{1/2} Z_N$ can be considered as $n$ i.i.d. samples of the random vector $({\mathcal X}_1,\dots,{\mathcal X}_N)^T$. We then prove that $Γ_N$ is similar to a diagonal matrix which satisfies all the required assumptions of our theorems, hence our results apply to this case.

math.PR↗

arXiv2020

Non-Hermitian random matrices with a variance profile (I): Deterministic equivalents and limiting ESDs

For each $n$, let $A_n=(σ_{ij})$ be an $n\times n$ deterministic matrix and let $X_n=(X_{ij})$ be an $n\times n$ random matrix with i.i.d. centered entries of unit variance. We study the asymptotic behavior of the empirical spectral distribution $μ_n^Y$ of the rescaled entry-wise product \[ Y_n = \left(\frac1{\sqrt{n}} σ_{ij}X_{ij}\right). \] For our main result we provide a deterministic sequence of probability measures $μ_n$, each described by a family of Master Equations, such that the difference $μ^Y_n - μ_n$ converges weakly in probability to the zero measure. A key feature of our results is to allow some of the entries $σ_{ij}$ to vanish, provided that the standard deviation profiles $A_n$ satisfy a certain quantitative irreducibility property. An important step is to obtain quantitative bounds on the solutions to an associate system of Schwinger--Dyson equations, which we accomplish in the general sparse setting using a novel graphical bootstrap argument.

math.PR↗

arXiv2020

Non-Hermitian random matrices with a variance profile (II): properties and examples

For each $n$, let $A_n=(σ_{ij})$ be an $n\times n$ deterministic matrix and let $X_n=(X_{ij})$ be an $n\times n$ random matrix with i.i.d. centered entries of unit variance. In the companion article Cook et al., we considered the empirical spectral distribution $μ_n^Y$ of the rescaled entry-wise product \[ Y_n = \frac 1{\sqrt{n}} A_n\odot X_n = \left(\frac1{\sqrt{n}} σ_{ij}X_{ij}\right) \] and provided a deterministic sequence of probability measures $μ_n$ such that the difference $μ^Y_n - μ_n$ converges weakly in probability to the zero measure. A key feature in Cook et al. was to allow some of the entries $σ_{ij}$ to vanish, provided that the standard deviation profiles $A_n$ satisfy a certain quantitative irreducibility property. In the present article, we provide more information on the sequence $(μ_n)$, described by a family of Master Equations. We consider these equations in important special cases such as separable variance profiles $σ^2_{ij}=d_i \widetilde d_j$ and sampled variance profiles $σ^2_{ij} = σ^2\left(\frac in, \frac jn \right)$ where $(x,y)\mapsto σ^2(x,y)$ is a given function on $[0,1]^2$. Associate examples are provided where $μ_n^Y$ converges to a genuine limit. We study $μ_n$'s behavior at zero and provide examples where $μ_n$'s density is bounded, blows up, or vanishes while an atom appears. As a consequence, we identify the profiles that yield the circular law. Finally, building upon recent results from Alt et al., we prove that except maybe in zero, $μ_n$ admits a positive density on the centered disc of radius $\sqrt{ρ(V_n)}$, where $V_n=(\frac 1n σ_{ij}^2)$ and $ρ(V_n)$ is its spectral radius.

math.PR↗

arXiv2019

Positive solutions for large random linear systems

Consider a large linear system where $A_n$ is a $n\times n$ matrix with independent real standard Gaussian entries, $\boldsymbol{1}_n$ is a $n\times 1$ vector of ones and with unknown the $n\times 1$ vector $\boldsymbol{x}_n$ satisfying$$\boldsymbol{x}_n = \boldsymbol{1}_n +\frac 1{α_n\sqrt{n}} A_n \boldsymbol{x}_n\, .$$We investigate the (componentwise) positivity of the solution $\boldsymbol{x}_n$ depending on the scaling factor $α_n$ as the dimension $n$ goes to $\infty$. We prove that there is a sharp phase transition at the threshold $α^*_n =\sqrt{2\log n}$: below the threshold ($α_n\ll \sqrt{2\log n}$), $\boldsymbol{x}_n$ has negative components with probability tending to 1 while above ($α_n\gg \sqrt{2\log n}$), all the vector's components are eventually positive with probability tending to 1. At the critical scaling $α^*_n$, we provide a heuristics to evaluate the probability that $\boldsymbol{x}_n$ is positive.Such linear systems arise as solutions at equilibrium of large Lotka-Volterra systems of differential equations, widely used to describe large biological communities with interactions such as foodwebs for instance. In the domaine of positivity of the solution $\boldsymbol{x}_n$, that is when $α_n\gg \sqrt{2\log n}$, we establish that the Lotka-Volterra system of differential equations whose solution at equilibrium is precisely $\boldsymbol{x}_n$ is stable in the sense that its jacobian $${\mathcal J}(\boldsymbol{x}_n) = \mathrm{diag}(\boldsymbol{x}_n)\left(-I_n + \frac {A_n}{α_n\sqrt{n}}\right)$$ has all its eigenvalues with negative real part with probability tending to one. Our results shed a new light and complement the understanding of feasibility and stability issues for large biological communities with interaction.

math.PR↗

arXiv2016

Gaussian fluctuations for linear spectral statistics of large random covariance matrices

Consider a $N\times n$ matrix $Σ_n=\frac{1}{\sqrt{n}}R_n^{1/2}X_n$, where $R_n$ is a nonnegative definite Hermitian matrix and $X_n$ is a random matrix with i.i.d. real or complex standardized entries. The fluctuations of the linear statistics of the eigenvalues \[\operatorname {Trace}f \bigl(Σ_nΣ_n^*\bigr)=\sum_{i=1}^Nf(λ_i),\qquad (λ_i)\ eigenvalues\ of\ Σ_nΣ_n^*,\] are shown to be Gaussian, in the regime where both dimensions of matrix $Σ_n$ go to infinity at the same pace and in the case where $f$ is of class $C^3$, that is, has three continuous derivatives. The main improvements with respect to Bai and Silverstein's CLT [Ann. Probab. 32 (2004) 553-605] are twofold: First, we consider general entries with finite fourth moment, but whose fourth cumulant is nonnull, that is, whose fourth moment may differ from the moment of a (real or complex) Gaussian random variable. As a consequence, extra terms proportional to $ \vert \mathcal{V}\vert ^2=\bigl|\mathbb{E}\bigl(X_{11}^n\bigr) ^2\bigr|^2$ and $κ=\mathbb{E}\bigl \vert X_{11}^n\bigr \vert ^4-\vert {\mathcal{V}}\vert ^2-2$ appear in the limiting variance and in the limiting bias, which not only depend on the spectrum of matrix $R_n$ but also on its eigenvectors. Second, we relax the analyticity assumption over $f$ by representing the linear statistics with the help of Helffer-Sjöstrand's formula. The CLT is expressed in terms of vanishing Lévy-Prohorov distance between the linear statistics' distribution and a Gaussian probability distribution, the mean and the variance of which depend upon $N$ and $n$ and may not converge.

math.PR↗

arXiv2016

Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges

We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show that for every generic positive edge of that support, there exists an extremal eigenvalue which converges almost surely toward that edge and fluctuates according to the Tracy-Widom law at the scale $N^{2/3}$. Moreover, given several generic positive edges, we establish that the associated extremal eigenvalue fluctuations are asymptotically independent. Finally, when the leftmost edge is the origin (hard edge), the fluctuations of the smallest eigenvalue are described by mean of the Bessel kernel at the scale $N^2$.

math.PR↗

arXiv2015

Large Complex Correlated Wishart Matrices: The Pearcey Kernel and Expansion at the Hard Edge

We study the eigenvalue behaviour of large complex correlated Wishart matrices near an interior point of the limiting spectrum where the density vanishes (cusp point), and refine the existing results at the hard edge as well. More precisely, under mild assumptions for the population covariance matrix, we show that the limiting density vanishes at generic cusp points like a cube root, and that the local eigenvalue behaviour is described by means of the Pearcey kernel if an extra decay assumption is satisfied. As for the hard edge, we show that the density blows up like an inverse square root at the origin. Moreover, we provide an explicit formula for the $1/N$ correction term for the fluctuation of the smallest random eigenvalue.

math.PR↗

arXiv2015

A Survey on the Eigenvalues Local Behavior of Large Complex Correlated Wishart Matrices

The aim of this note is to provide a pedagogical survey of the recent works by the authors ( arXiv:1409.7548 and arXiv:1507.06013) concerning the local behavior of the eigenvalues of large complex correlated Wishart matrices at the edges and cusp points of the spectrum: Under quite general conditions, the eigenvalues fluctuations at a soft edge of the limiting spectrum, at the hard edge when it is present, or at a cusp point, are respectively described by mean of the Airy kernel, the Bessel kernel, or the Pearcey kernel. Moreover, the eigenvalues fluctuations at several soft edges are asymptotically independent. In particular, the asymptotic fluctuations of the matrix condition number can be described. Finally, the next order term of the hard edge asymptotics is provided.

math.PR↗

arXiv2012

A Subspace Estimator for Fixed Rank Perturbations of Large Random Matrices

This paper deals with the problem of parameter estimation based on certain eigenspaces of the empirical covariance matrix of an observed multidimensional time series, in the case where the time series dimension and the observation window grow to infinity at the same pace. In the area of large random matrix theory, recent contributions studied the behavior of the extreme eigenvalues of a random matrix and their associated eigenspaces when this matrix is subject to a fixed-rank perturbation. The present work is concerned with the situation where the parameters to be estimated determine the eigenspace structure of a certain fixed-rank perturbation of the empirical covariance matrix. An estimation algorithm in the spirit of the well-known MUSIC algorithm for parameter estimation is developed. It relies on an approach recently developed by Benaych-Georges and Nadakuditi, relating the eigenspaces of extreme eigenvalues of the empirical covariance matrix with eigenspaces of the perturbation matrix. First and second order analyses of the new algorithm are performed.

math.PR↗