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Michail Louvaris

Publications and source records attributed to Michail Louvaris.

8 recordsLinked to original sources
arXiv2026

On the growth spectrum of hyperbolic groups

We study the growth spectrum of groups acting on hyperbolic spaces, i.e.\ the set of exponential growth rates achieved by subgroups. For a finitely generated free group or a surface group acting convex-cocompactly on a proper geodesic hyperbolic metric space, we prove that the growth spectrum is the full interval $[0, \omega_G]$. For any hyperbolic group, we prove that the growth spectrum contains a large interval $[0, \omega_{\mathcal{F}}]$ where $\omega_{\mathcal{F}} \geq \omega_G / 2$, with strict inequality when the action is divergent. In the case of the Cayley graph of a free group, we also present an approach via the non-backtracking matrix of the configuration model, connecting the density of growth rates to a spectral concentration result for random graphs.

math.GR↗
arXiv2026

A note on outlier eigenvectors for sparse non-Hermitian perturbations

We consider a sparse i.i.d.\ non-Hermitian random matrix model $X_n$ (with sparsity parameter $K_n$) and a deterministic finite-rank perturbation $E_n$. Assuming biorthogonality for $E_n$ and a growth condition on $K_n$, we outline a finite-rank resolvent reduction leading to asymptotics for the overlap between an outlier eigenvector of $Y_n:=X_n+E_n$ and the corresponding spike eigenspace. In particular, for an outlier spike $\mu$ with $|\mu|>1$, the squared projection of the associated (right) eigenvector onto the spike eigenspace converges in probability to $1-|\mu|^{-2}$. Our result generalizes Theorem 1.6 of [HLN26] to general finite rank case solving Open Problem 5.

math.PR↗
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

On the spectral radius and the characteristic polynomial of a random matrix with independent elements and a variance profile

In this paper, it is shown that with large probability, the spectral radius of a large non-Hermitian random matrix with a general variance profile does not exceed the square root of the spectral radius of the variance profile matrix. A minimal moment assumption is considered and sparse variance profiles are covered. Following an approach developed recently by Bordenave, Chafa{\"i} and Garc{\'i}a-Zelada, the key theorem states the asymptotic equivalence between the reverse characteristic polynomial of the random matrix at hand and a random analytic function which depends on the variance profile matrix. The result is applied to the case of a non-Hermitian random matrix with a variance profile given by a piecewise constant or a continuous non-negative function, the inhomogeneous (centered) directed Erd\H{o}s-R{\'e}nyi model, and more.

math.PR↗
arXiv2024

The limit of the operator norm for random matrices with a variance profile

In this work we study symmetric random matrices with variance profile satisfying certain conditions. We establish the convergence of the operator norm of these matrices to the largest element of the support of the limiting empirical spectral distribution. We prove that it is sufficient for the entries of the matrix to have finite only the $4$-th moment or the $4+ε$ moment in order for the convergence to hold in probability or almost surely respectively. Our approach determines the behaviour of the operator norm for random symmetric or non-symmetric matrices whose variance profile is given by a step or a continuous function, random band matrices whose bandwidth is proportional to their dimension, random Gram matrices, triangular matrices and more.

math.PR↗
arXiv2024

On the spectral radius of the non-backtracking matrix of the configuration model

We prove a concentration result for the leading eigenvalue of the non--backtracking matrix of the configuration model under the assumption of uniformly bounded degrees. Let $P$ denote the limiting degree distribution. Assuming polynomial approximation, we show that as the number of vertices tends to infinity, the leading eigenvalue of the non--backtracking matrix concentrates around \[ \frac{\mathbb{E}[P(P-1)]}{\mathbb{E}[P]}. \] This quantity corresponds to the mean offspring number of the excess--degree branching process associated with the local limit of the configuration model. As a byproduct of our work we explain how this result can be applied to prove the density of the growth rates of the subgroups of the free group.

math.GR↗
arXiv2023

Universality of the least singular value and singular vector delocalisation for Lévy non-symmetric random matrices

In this paper we consider $N \times N $ matrices $D_{N}$ with i.i.d. entries all following an $a-$stable law divided by $N^{1/a}$. We prove that the least singular value of $D_{N}$, multiplied by $N$, tends to the same law as in the Gaussian case, for almost all $a \in (0,2)$. This is proven by considering the symmetrization of the matrix $D_{N}$ and using a version of the three step strategy, a well known strategy in the random matrix theory literature. In order to apply the three step strategy, we also prove an isotropic local law for the symmetrization of matrices after slightly perturbing them by a Gaussian matrix with a similar structure. The isotropic local law is proven for a general class of matrices that satisfy some regularity assumption. We also prove the complete delocalization for the left and right singular vectors of $D_{N}$ at small energy, i.e., for energies at a small interval around $0$.

math.PR↗