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Pierre Bousseyroux

Publications and source records attributed to Pierre Bousseyroux.

13 recordsLinked to original sources
arXiv2026

Phase transitions in non-Hermitian spherical integrals

We study the large-$N$ asymptotics of a constrained spherical integral for non-Hermitian random matrices, in which the norms and mutual scalar product of two vectors are fixed. In the delocalized regime, the asymptotics are governed by the non-Hermitian transforms $\mathcal R_1$ and $\mathcal R_2$. At saddle-point level, the constrained integral exhibits a transition to a localized regime controlled by the largest singular value of a shifted matrix and by the overlap of its associated left and right singular vectors. Motivated by the Hermitian spherical-integral mechanism and by the Coulomb-gas picture, we formulate conjectures for one-eigenvalue large deviations and boundary fluctuations. Throughout, our analysis is carried out in the spirit of mathematical physics.

math-ph↗
arXiv2026

Finite-rank multiplicative perturbations of rotationally invariant non-Hermitian random matrices

We study finite-rank multiplicative deformations of rotationally invariant non-Hermitian random matrices. More precisely, we consider models of the form $\mathbf{A}(\mathbf{I}+\mathbf{T})$, where $\mathbf{A}$ is a large rotationally invariant non-Hermitian random matrix, $\mathbf{T}$ is a finite-rank normal perturbation, and $\mathbf{I}$ denotes the identity matrix. We characterize the emergence of outlier eigenvalues, their fluctuations, and the associated eigenvector overlaps. Our results provide a multiplicative non-Hermitian counterpart to the classical Baik--Ben Arous--Péché framework.

cond-mat.dis-nn↗
arXiv2026

Spectral boundaries of deterministic matrices deformed by rotationally invariant random non-Hermitian ensembles

One of the great miracles of random matrix theory is that, in the $N \to \infty$ limit, many otherwise intractable matrix problems with horrendously complicated finite-$N$ expressions admit remarkably simple and elegant asymptotic solutions. In this paper, we illustrate this phenomenon in the context of spectral boundaries (or spectral edges) for deformed random matrices. Specifically, we consider matrices of the form $\mathbf{A} + \mathbf{B}$, where $\mathbf{A}$ is a deterministic $N\times N$ matrix (not necessarily Hermitian) and $\mathbf{B}$ is a rotationally invariant random matrix. In the large-$N$ limit, we show that the complex eigenvalue distribution of $\mathbf{A} + \mathbf{B}$ satisfies remarkably simple boundary equations that depend on the $\mathcal{R}_1$ and $\mathcal{R}_2$ transforms of $\mathbf{B}$. We illustrate our results on several explicit random matrix ensembles and support them with numerical simulations.

cond-mat.dis-nn↗
arXiv2026

Spectral properties of deterministic matrices multiplied by rotationally invariant random non-Hermitian ensembles

In this paper, we study spectral properties of multiplicative deformations of non-Hermitian random matrices. We consider matrices of the form $\mathbf{A}\mathbf{B}$, where $\mathbf{A}$ is a deterministic $N\times N$ matrix (not necessarily Hermitian) and $\mathbf{B}$ is a rotationally invariant random matrix. We show that, as $N\to\infty$, the boundary of the complex eigenvalue distribution of $\mathbf{A}\mathbf{B}$ is governed by simple equations involving the $\mathcal{R}_1$ and $\mathcal{R}_2$ transforms of $\mathbf{B}$.

cond-mat.dis-nn↗
arXiv2026

R-transforms for non-Hermitian matrices: a spherical integral approach

In this paper, we establish a connection between the formalism of $\mathcal{R}$-transforms for non-Hermitian random matrices and the framework of spherical integrals, using the replica method. This connection was previously proved in the Hermitian setting and in the case of bi-invariant random matrices. We show that the $\mathcal{R}$-transforms used in the non-Hermitian context in fact originate from a single scalar function of two variables. This provides a new and transparent way to compute $\mathcal{R}$-transforms, which until now had been known only in restricted cases such as bi-invariant, Hermitian, or elliptic ensembles.

cond-mat.dis-nn↗
arXiv2026

The eigenvalues and eigenvectors of finite-rank normal perturbations of large rotationally invariant non-Hermitian matrices

We study finite-rank normal deformations of rotationally invariant non-Hermitian random matrices. Extending the classical Baik-Ben Arous-Péché (BBP) framework, we characterize the emergence and fluctuations of outlier eigenvalues in models of the form $\mathbf{A} + \mathbf{T}$, where $\mathbf{A}$ is a large rotationally invariant non-Hermitian random matrix and $\mathbf{T}$ is a finite-rank normal perturbation. We also describe the corresponding eigenvector behavior. Our results provide a unified framework encompassing both Hermitian and non-Hermitian settings, thereby generalizing several known cases.

cond-mat.dis-nn↗
arXiv2026

Spectra of high-dimensional Spearman correlation matrices under scale-mixture dependence

We study the asymptotic spectral properties of high-dimensional Spearman correlation matrices for scale-mixture data. We consider observations of the form $x_t=σ_t ξ_t \in \mathbb{R}^N,$ where the coordinates of $ξ_t$ are i.i.d.\ and the scalar mixture variable $σ_t$ is shared by all coordinates. Under natural symmetry assumptions, the coordinates of $x_t$ are pairwise uncorrelated in both the Pearson and Spearman sense. Nevertheless, they are not independent when the mixture variable is non-degenerate. We show that this higher-order dependence survives the rank transformation and leaves a nontrivial spectral signature. In the proportional regime $N/T\to q\in(0,\infty),$ the empirical spectral distribution of the Spearman correlation matrix converges almost surely to a generalized Marčenko--Pastur law governed by the limiting distribution of an effective rank variance. We also formulate a broader latent-variable extension, which covers, in particular, some scale-mixture models with correlated directional components. We discuss solvable examples and numerical approximations, motivated in part by heavy-tailed data in robust multivariate statistics, econometrics, and finance.

math.ST↗
arXiv2026

Another Marcenko-Pastur law for Kendall's tau

Bandeira et al. (2017) show that the eigenvalues of the Kendall correlation matrix of $n$ i.i.d. random vectors in $\mathbb{R}^p$ are asymptotically distributed like $1/3 + (2/3)Y_q$, where $Y_q$ has a Marčenko-Pastur law with parameter $q=\lim(p/n)$ if $p, n\to\infty$ proportionately to one another. Here we show that another Marčenko-Pastur law emerges in the "ultra-high dimensional" scaling limit where $p\sim q'\, n^2/2$ for some $q'>0$: in this quadratic scaling regime, Kendall correlation eigenvalues converge weakly almost surely to $(1/3)Y_{q'}$.

math.PR↗
arXiv2025

Free Convolution and Generalized Dyson Brownian Motion

The eigenvalue spectrum of the sum of large random matrices that are mutually "free", i.e., randomly rotated, can be obtained using the formalism of R-transforms, with many applications in different fields. We provide a direct interpretation of the otherwise abstract additivity property of R-transforms for the sum in terms of a dynamical evolution of "particles" (the eigenvalues), interacting through two-body and higher-body forces and subject to a Gaussian noise, generalizing the usual Dyson Brownian motion with Coulomb interaction. Interestingly, the appearance of an outlier outside of the bulk of the spectrum is signalled by a divergence of the "velocity" of the generalized Dyson motion. We extend our result to products of free matrices.

cond-mat.dis-nn↗
arXiv2025

Group-Level Imitation May Stabilize Cooperation

Stabilizing cooperation among self-interested individuals presents a fundamental challenge in evolutionary theory and social science. While classical models predict the dominance of defection in social dilemmas, empirical and theoretical studies have identified various mechanisms that promote cooperation, including kin selection, reciprocity, and spatial structure. In this work, we investigate the role of localized imitation in the evolutionary dynamics of cooperation within an optional Public Goods Game (PGG). We introduce a model where individuals belong to distinct groups and adapt their strategies based solely on comparisons within their own group. We identify different dynamical regimes, including stable fixed points, limit cycles, and Rock-Scissors-Paper-type oscillations. Our analysis, grounded in a replicator-type framework, reveals that such group-level imitation can stabilize cooperative behavior, provided that groups are not initially polarized around a single strategy. In other words, restricting imitation to group-level interactions mitigates the destabilizing effects of global competition, providing a potential explanation for the resilience of cooperation in structured populations.

physics.soc-ph↗
arXiv2024

Distribution of the Diagonal Entries of the Resolvent of a Complex Ginibre Matrix

The study of eigenvalue distributions in random matrix theory is often conducted by analyzing the resolvent matrix $ \mathbf{G}_{\mathbf{M}}^N(z) = (z \mathbf{1} - \mathbf{M})^{-1} $. The normalized trace of the resolvent, known as the Stieltjes transform $ \mathfrak{g}_{\mathbf{M}}^N(z) $, converges to a limit $ \mathfrak{g}_{\mathbf{M}}(z) $ as the matrix dimension $ N $ grows, which provides the eigenvalue density $ ρ_{\mathbf{M}} $ in the large-$ N $ limit. In the Hermitian case, the distribution of $ \mathfrak{g}_{\mathbf{M}}^N(z) $, now regarded as a random variable, is explicitly known when $ z $ lies within the limiting spectrum, and it coincides with the distribution of any diagonal entry of $ \mathbf{G}_{\mathbf{M}}^N(z) $. In this paper, we investigate what becomes of these results when $ \mathbf{M} $ is non-Hermitian. Our main result is the exact computation of the diagonal elements of $ \mathbf{G}_{\mathbf{M}}^N(z) $ when $ \mathbf{M} $ is a Ginibre matrix of size $ N $, as well as the high-dimensional limit for different regimes of $ z $, revealing a tail behavior connected to the statistics of the left and right eigenvectors. Interestingly, the limit distribution is stable under inversion, a property previously observed in the symmetric case. We then propose two general conjectures regarding the distribution of the diagonal elements of the resolvent and its normalized trace in the non-Hermitian case, both of which reveal a symmetry under inversion.

math-ph↗
arXiv2024

Spectral Initialization for High-Dimensional Phase Retrieval with Biased Spatial Directions

We explore a spectral initialization method that plays a central role in contemporary research on signal estimation in nonconvex scenarios. In a noiseless phase retrieval framework, we precisely analyze the method's performance in the high-dimensional limit when sensing vectors follow a multivariate Gaussian distribution for two rotationally invariant models of the covariance matrix C. In the first model C is a projector on a lower dimensional space while in the second it is a Wishart matrix. Our analytical results extend the well-established case when C is the identity matrix. Our examination shows that the introduction of biased spatial directions leads to a substantial improvement in the spectral method's effectiveness, particularly when the number of measurements is less than the signal's dimension. This extension also consistently reveals a phase transition phenomenon dependent on the ratio between sample size and signal dimension. Surprisingly, both of these models share the same threshold value.

cond-mat.dis-nn↗