arXiv subjects

Charles Bordenave

Publications and source records attributed to Charles Bordenave.

At least 19 recordsLinked to original sources

arXiv2026

Logarithmic regularity of spectral measures on infinite graphs

We study the regularity of spectral measures of self-adjoint operators on infinite weighted graphs in the unimodular setting. This framework encompasses operators in the group algebra of a finitely generated group, random operators whose distribution is quasi-invariant under a group action, and Benjamini--Schramm limits of operators on finite graphs. Under a natural geometric condition on the underlying graph, we prove that the expected spectral measure satisfies a logarithmic Hölder regularity estimate. The proof relies on a strengthened version of the monotone labelling method previously introduced with Sen and Virág to control the pure point part of the spectal measure. Applications include operators in group algebras of indicable groups, Anderson-type models with arbitrary compactly supported potentials on Cayley graphs, anisotropic percolation operators, and operators on quasi-transitive graphs. In particular, our results extend the classical Craig--Simon theorem beyond the Euclidean lattice.

math-ph↗

arXiv2026

Freeness for tensors

We pursue the current developments in random tensor theory by laying the foundations of a free probability theory for tensors and establish its relevance in the study of random tensors of high dimension. We give a definition of freeness associated to a collection of tensors of possibly different orders. Our definition reduces to the usual freeness when only tensors of order 2 are concerned. We define the free cumulants which are associated to this notion of tensor freeness. We prove that the basic models of random tensors are asymptotically free as the dimension goes to infinity. On the way, we establish Schwinger-Dyson loop equations associated to random tensors.

math.PR↗

arXiv2026

Quadratic differentials and random walks on the dual graph of a pants decomposition

Let X be an infinite Riemann surface with an upper-bounded geodesic pants decomposition. The vertices of the corresponding dual graph G are pairs of pants and edges are cuffs with conductances equal to their lengths. We prove that the geodesic flow on X is ergodic if and only if the random walk on G is recurrent. This yields explicit criteria for deciding, in terms of cuff-length growth, whether the geodesic flow is ergodic. We provide concrete and new families of Riemann surfaces with an explicit understanding of the phase transitions from recurrent to non-recurrent geodesic flows. In addition, we show that rough isometry of surfaces does not preserve the ergodicity of the geodesic flow while rough isometry of their dual graphs does. The above equivalence result uses a characterization of the measured geodesic laminations on X that arise as straightened horizontal foliations of finite-area holomorphic quadratic differentials. The conditions on the measured laminations are translated into the conditions on the existence of a square summable flow function on G.

math.DS↗

arXiv2026

Cutoff for geodesic paths on hyperbolic manifolds

We establish new instances of the cutoff phenomenon for geodesic paths and for the Brownian motion on compact hyperbolic manifolds. We prove that for any fixed compact hyperbolic manifold, the geodesic path started on a spatially localized initial condition exhibits cutoff. Our work also extends results obtained by Golubev and Kamber on hyperbolic surfaces of large volume to any dimension. Our proof builds upon a spectral strategy introduced by Lubetzky and Peres for Ramanujan graphs and on a detailed spectral analysis of the spherical mean operator.

math.PR↗

arXiv2026

Quantum mixing on large Schreier graphs

We prove quantum ergodicity and quantum mixing for sequences of finite Schreier graphs converging to an infinite Cayley graph whose adjacency operator has absolutely continuous spectrum. Under Benjamini-Schramm convergence (or strong convergence in distribution), we show that correlations between eigenvectors at distinct energies vanish asymptotically when tested against a broad class of local observables. Our results apply to all orthonormal eigenbases and do not require tree-like structure or periodicity of the limiting graph, unlike previous approaches based on non-backtracking operators or Floquet theory. The proof introduces a new framework for quantum ergodicity, based on trace identities, resolvent approximations and representation-theoretic techniques and extends to certain families of non-regular graphs. We illustrate the assumptions and consequences of our theorems on Schreier graphs arising from free products of groups, right-angled Coxeter groups and lifts of a fixed base graph.

math.SP↗

arXiv2025

Sparse graphs and their Benjamini-Schramm limits: a spectral tour

Sparse graphs with bounded average degree form a rich class of discrete structures where local geometry strongly influences global behavior. The Benjamini-Schramm (BS) convergence offers a natural framework to describe their asymptotic local structure. In this note, we survey spectral aspects of BS convergence and their applications, with a focus on random Schreier graphs and covering graphs. We review some recent progress on the spectral decomposition of the local operators on graphs. We discuss the behavior of extreme eigenvalues and the growing role of strong convergence in distribution, which rules out spectral outliers. We also give a new application of strong convergence to the typical graph distance between vertices in Schreier graphs

math.PR↗

arXiv2024

Outliers of perturbations of banded Toeplitz matrices

Toeplitz matrices form a rich class of possibly non-normal matrices whose asymptotic spectral analysis in high dimension is well-understood. The spectra of these matrices are notoriously highly sensitive to small perturbations. In this work, we analyze the spectrum of a banded Toeplitz matrix perturbed by a random matrix with iid entries of variance $σ_n^2 / n$ in the asymptotic of high dimension and $σ_n$ converging to $σ\geq 0$. Our results complement and provide new proofs on recent progresses in the case $σ= 0$. For any $σ\geq 0$, we show that the point process of outlier eigenvalues is governed by a low-dimensional random analytic matrix field, typically Gaussian, alongside an explicit deterministic matrix that captures the algebraic structure of the resonances responsible for the outlier eigenvalues. On our way, we prove a new functional central limit theorem for trace of polynomials in deterministic and random matrices and present new variations around Szego's strong limit theorem.

math.PR↗

arXiv2024

Large deviations for macroscopic observables of heavy-tailed matrices

We consider a finite collection of independent Hermitian heavy-tailed random matrices of growing dimension. Our model includes the Lévy matrices proposed by Bouchaud and Cizeau, as well as sparse random matrices with O(1) non-zero entries per row. By representing these matrices as weighted graphs, we derive a large deviations principle for key macroscopic observables. Specifically, we focus on the empirical distribution of eigenvalues, the joint neighborhood distribution, and the joint traffic distribution. As an application, we define a notion of microstates entropy for traffic distributions which is additive for free traffic convolution.

math.PR↗

arXiv2024

Norm of matrix-valued polynomials in random unitaries and permutations

We consider a non-commutative polynomial in several independent $N$-dimensional random unitary matrices, uniformly distributed over the unitary, orthogonal or symmetric groups, and assume that the coefficients are $n$-dimensional matrices. The main purpose of this paper is to study the operator norm of this random non-commutative polynomial. We compare it with its counterpart where the the random unitary matrices are replaced by the unitary generators of the free group von Neumann algebra. Our first result is that these two norms are overwhelmingly close to each other in the large $N$ limit, and this estimate is uniform over all matrix coefficients as long as $n \le\exp (N^α)$ for some explicit $α>0$. Such results had been obtained by very different techniques for various regimes, all falling in the category $n\ll N$. Our result provides a new proof of the Peterson-Thom conjecture. Our second result is a universal quantitative lower bound for the operator norm of polynomials in independent $N$-dimensional random unitary and permutation matrices with coefficients in an arbitrary $C^*$-algebra. A variant of this result for permutation matrices generalizes the Alon-Boppana lower bound in two directions. Firstly, it applies for arbitrary polynomials and not only linear polynomials, and secondly, it applies for coefficients of an arbitrary $C^*$-algebra with non-negative joint moments and not only for non-negative real numbers.

math.PR↗

arXiv2023

Eigenvalues of random lifts and polynomials of random permutation matrices

Consider a finite sequence of independent random permutations, chosen uniformly either among all permutations or among all matchings on n points. We show that, in probability, as n goes to infinity, these permutations viewed as operators on the (n-1) dimensional vector space orthogonal to the vector with all coordinates equal to 1, are asymptotically strongly free. Our proof relies on the development of a matrix version of the non-backtracking operator theory and a refined trace method. As a byproduct, we show that the non-trivial eigenvalues of random n-lifts of a fixed based graphs approximately achieve the Alon-Boppana bound with high probability in the large n limit. This result generalizes Friedman's Theorem stating that with high probability, the Schreier graph generated by a finite number of independent random permutations is close to Ramanujan. Finally, we extend our results to tensor products of random permutation matrices. This extension is especially relevant in the context of quantum expanders.

math.PR↗

arXiv2023

Mobility Edge for Lévy Matrices

Lévy matrices are symmetric random matrices whose entry distributions lie in the domain of attraction of an $α$-stable law. For $α< 1$, predictions from the physics literature suggest that high-dimensional Lévy matrices should display the following phase transition at a point $E_{\mathrm{mob}}$. Eigenvectors corresponding to eigenvalues in $(-E_{\mathrm{mob}},E_{\mathrm{mob}})$ should be delocalized, while eigenvectors corresponding to eigenvalues outside of this interval should be localized. Further, $E_{\mathrm{mob}}$ is given by the (presumably unique) positive solution to $λ(E,α) =1$, where $λ$ is an explicit function of $E$ and $α$. We prove the following results about high-dimensional Lévy matrices. (1) If $λ(E,α) > 1$ then eigenvectors with eigenvalues near $E$ are delocalized. (2) If $E$ is in the connected components of the set $\big\{ x : λ(x,α) < 1 \big\}$ containing $\pm \infty$, then eigenvectors with eigenvalues near $E$ are localized. (3) For $α$ sufficiently near $0$ or $1$, there is a unique positive solution $E = E_{\mathrm{mob}}$ to $λ(E,α) = 1$, demonstrating the existence of a (unique) phase transition. (a) If $α$ is close to $0$, then $E_{\mathrm{mob}}$ scales approximately as $|\log α|^{-2/α}$. (b) If $α$ is close to $1$, then $E_{\mathrm{mob}}$ scales as $(1-α)^{-1}$. Our proofs proceed through an analysis of the local weak limit of a Lévy matrix, given by a certain infinite-dimensional, heavy-tailed operator on the Poisson weighted infinite tree.

math.PR↗

arXiv2022

Strong asymptotic freeness for independent uniform variables on compact groups associated to non-trivial representations

Asymptotic freeness of independent Haar distributed unitary matrices was discovered by Voiculescu. Many refinements have been obtained, including strong asymptotic freeness of random unitaries and strong asymptotic freeness of random permutations acting on the orthogonal of the Perron-Frobenius eigenvector. In this paper, we consider a new matrix unitary model appearing naturally from representation theory of compact groups. We fix a non-trivial signature $ρ$, i.e. two finite sequences of non-increasing natural numbers, and for $n$ large enough, consider the irreducible representation $V_{n,ρ}$ of $\mathbb{U}_n$ associated to the signature $ρ$. We consider the quotient $\mathbb{U}_{n,ρ}$ of $\mathbb{U}_n$ viewed as a matrix subgroup of $\mathbb{U}(V_{n,ρ})$, and show that strong asymptotic freeness holds in this generalized context when drawing independent copies of the Haar measure. We also obtain the orthogonal variant of this result. Thanks to classical results in representation theory, this result is closely related to strong asymptotic freeness for tensors, which we establish as a preliminary. In order to achieve this result, we need to develop four new tools, each of independent theoretical interest: (i) a centered Weingarten calculus and uniform estimates thereof, (ii) a systematic and uniform comparison of Gaussian moments and unitary moments of matrices, (iii) a generalized and simplified operator valued non-backtracking theory in a general $C^*$-algebra, and finally, (iv) combinatorics of tensor moment matrices.

math.PR↗

arXiv2022

Noise sensitivity for the top eigenvector of a sparse random matrix

We investigate the noise sensitivity of the top eigenvector of a sparse random symmetric matrix. Let $v$ be the top eigenvector of an $N\times N$ sparse random symmetric matrix with an average of $d$ non-zero centered entries per row. We resample $k$ randomly chosen entries of the matrix and obtain another realization of the random matrix with top eigenvector $v^{[k]}$. Building on recent results on sparse random matrices and a noise sensitivity analysis previously developed for Wigner matrices, we prove that, if $d\geq N^{2/9}$, with high probability, when $k \ll N^{5/3}$, the vectors $v$ and $v^{[k]}$ are almost collinear and, on the contrary, when $k\gg N^{5/3}$, the vectors $v$ and $v^{[k]}$ are almost orthogonal. A similar result holds for the eigenvector associated to the second largest eigenvalue of the adjacency matrix of an Erdős-Rényi random graph with average degree $d \geq N^{2/9}$.

math.PR↗

arXiv2022

Markovian linearization of random walks on groups

In operator algebra, the linearization trick is a technique that reduces the study of a non-commutative polynomial evaluated at elements of an algebra A to the study of a polynomial of degree one, evaluated on the enlarged algebra A x M r (C), for some integer r. We introduce a new instance of the linearization trick which is tailored to study a finitely supported random walk on a group G by studying instead a nearest-neighbor colored random walk on G x {1,. .. , r}, which is much simpler to analyze. As an application we extend well-known results for nearest-neighbor walks on free groups and free products of finite groups to colored random walks, thus showing how one can obtain explicit formulas for the drift and entropy of a finitely supported random walk.

math.PR↗

arXiv2021

Typicality and entropy of processes on infinite trees

Consider a uniformly sampled random $d$-regular graph on $n$ vertices. If $d$ is fixed and $n$ goes to $\infty$ then we can relate typical (large probability) properties of such random graph to a family of invariant random processes (called "typical" processes) on the infinite $d$-regular tree $T_d$. This correspondence between ergodic theory on $T_d$ and random regular graphs is already proven to be fruitful in both directions. This paper continues the investigation of typical processes with a special emphasis on entropy. We study a natural notion of micro-state entropy for invariant processes on $T_d$. It serves as a quantitative refinement of the notion of typicality and is tightly connected to the asymptotic free energy in statistical physics. Using entropy inequalities, we provide new sufficient conditions for typicality for edge Markov processes. We also extend these notions and results to processes on unimodular Galton-Watson random trees.

math.PR↗

arXiv2021

Convergence of the spectral radius of a random matrix through its characteristic polynomial

Consider a square random matrix with independent and identically distributed entries of mean zero and unit variance. We show that as the dimension tends to infinity, the spectral radius is equivalent to the square root of the dimension in probability. This result can also be seen as the convergence of the support in the circular law theorem under optimal moment conditions. In the proof we establish the convergence in law of the reciprocal characteristic polynomial to a random analytic function outside the unit disc, related to a hyperbolic Gaussian analytic function. The proof is short and differs from the usual approaches for the spectral radius. It relies on a tightness argument and a joint central limit phenomenon for traces of fixed powers.

math.PR↗

arXiv2021

Existence of absolutely continuous spectrum for Galton-Watson random trees

We establish a quantitative criterion for an operator defined on a Galton-Watson random tree for having an absolutely continuous spectrum. For the adjacency operator, this criterion requires that the offspring distribution has a relative variance below a threshold. As a byproduct, we prove that the adjacency operator of a supercritical Poisson Galton-Watson tree has a non-trivial absolutely continuous part if the average degree is large enough. We also prove that its Karp and Sipser core has purely absolutely spectrum on an interval if the average degree is large enough. We finally illustrate our criterion on the Anderson model on a d-regular infinite tree with d $\ge$ 3 and give a quantitative version of Klein's Theorem on the existence of absolutely continuous spectrum at disorder smaller that C $\sqrt$ d for some absolute constant C.

math.PR↗

arXiv2021

Spectral radii of sparse random matrices

We establish bounds on the spectral radii for a large class of sparse random matrices, which includes the adjacency matrices of inhomogeneous Erdős-Rényi graphs. Our error bounds are sharp for a large class of sparse random matrices. In particular, for the Erdős-Rényi graph $G(n,d/n)$, our results imply that the smallest and second-largest eigenvalues of the adjacency matrix converge to the edges of the support of the asymptotic eigenvalue distribution provided that $d \gg \log n$. Together with the companion paper [3], where we analyse the extreme eigenvalues in the complementary regime $d \ll \log n$, this establishes a crossover in the behaviour of the extreme eigenvalues around $d \sim \log n$. Our results also apply to non-Hermitian sparse random matrices, corresponding to adjacency matrices of directed graphs. The proof combines (i) a new inequality between the spectral radius of a matrix and the spectral radius of its nonbacktracking version together with (ii) a new application of the method of moments for nonbacktracking matrices.

math.PR↗