arXiv subjects

Guillaume Dubach

Publications and source records attributed to Guillaume Dubach.

18 recordsLinked to original sources

arXiv2026

On stability of outliers from the circular law

This work investigates the stability of outliers from the circular law, via the convergence of their associated diagonal overlaps between eigenvectors - also known as the squared eigenvalue condition numbers. We consider and compare two paradigmatic cases, namely: 1) the Complex Ginibre Ensemble conditioned on the existence of an outlier, and 2) the outlier induced by a rank-one Hermitian perturbation of a Complex Ginibre matrix. In both cases, we prove almost sure convergence towards a specific constant that only depends on the radius of the outlier and its status - either conditioned or induced. These results can be generalized to other complex integrable ensembles with the same techniques, and complement our understanding of eigenvalue stability in non-Hermitian ensembles.

math.PR↗

arXiv2026

On a random matrix proof of a bipartite Harer-Zagier formula

This work establishes a bipartite generalization of the Harer-Zagier formula using non-Hermitian Random Matrix Theory. More specifically, we use a decomposition of powers of Ginibre eigenvalues as a superposition of independent point processes to identify all coefficients of the generating function of the genus of a surface obtained by a random bipartite pairing of the sides of one polygon with $kM$ sides and $k$ polygons with $M$ sides.

math.PR↗

arXiv2026

The delocalization of eigenvectors of real elliptic matrices

We investigate delocalization phenomena for eigenvectors of real random matrices that are invariant by orthogonal transformations. A specific phenomenon with these ensembles is that an eigenvector is typically more localized when its eigenvalue is closer to the real axis while for unitarily invariant ensembles, all eigenvectors are delocalized at the same level. More precisely, we measure the delocalization level of a vector $x\in \mathbb{C}^N$ using the Inverse Participation Ratio $\mathrm{IPR}(x) = N|x|_4^4 / |x|_2^4 \geqslant 1$. A higher IPR means a more localized vector. Using the exact distribution of the Schur decomposition of some paradigmatic rotation-invariant matrix models, we prove that conditionally on having an eigenvalue $λ$ with $|\mathfrak{Im}(λ)| = y / \sqrt{N}$, the IPR of the associated eigenvector converges in distribution towards a random variable $\ell_y$ with an explicit density depending only on $y$. We then prove that $\ell_y \to 3$ when $y \to 0$ and $\ell_y \to 2$ when $y\to +\infty$, coherently with the observed phenomenon. This result is explicitly proved for higher-order IPRs and for the real Elliptic Ginibre ensemble at every non-symmetry parameter $τ\in [0,1[$, including the classical real Ginibre ensemble ($τ=0$).

math.PR↗

arXiv2026

Fisher-Hartwig asymptotics for non-Hermitian random matrices

We prove the two-dimensional analogue of the asymptotics for Toeplitz determinants with Fisher-Hartwig singularities, for general real symbols. This formula has applications to random normal matrices with complex spectra: (i) the characteristic polynomial converges to a Gaussian multiplicative chaos random measure on the limiting droplet, in the subcritical phase; (ii) the electric potential converges pointwise to a logarithmically correlated field; (iii) the measure of its level sets (i.e. thick points) is identified; (iv) the associated free energy undergoes a freezing transition. This establishes emergence of the Liouville quantum gravity measure from free fermions in 2d, and universality with respect to the external potential.

math.PR↗

arXiv2026

Une brève histoire des perturbations non-hermitiennes de rang un

Les perturbations de faible rang de matrices aléatoires ont été au cœur de nombreux travaux ces vingt dernières années. En particulier, les cas non-hermitiens, moins représentés dans la littérature en règle générale, font ici l'objet d'une attention spéciale en raison de leurs applications à la physique et à l'étude des réseaux de neurones. Petit tour d'horizon. -- A brief history of non-Hermitian perturbations of rank one: Low-rank perturbations of random matrices have been the focus of active research over the past twenty years. We give an overview of different non-Hermitian models, which are generally less represented in the literature, as well as some of their applications in physics and the study of neural networks.

math.PR↗

arXiv2023

On the derivation of the homogeneous kinetic wave equation for a nonlinear random matrix model

We consider a nonlinear system of ODEs, where the underlying linear dynamics are determined by a Hermitian random matrix ensemble. We prove that the leading order dynamics in the weakly nonlinear, infinite volume limit are determined by a solution to the corresponding kinetic wave equation on a non-trivial timescale. Our proof relies on estimates for Haar-distributed unitary matrices obtained from Weingarten calculus, which may be of independent interest.

math.AP↗

arXiv2023

Dynamics of a rank-one multiplicative perturbation of a unitary matrix

We provide a dynamical study of a model of multiplicative perturbation of a unitary matrix introduced by Fyodorov. In particular, we identify a flow of deterministic domains that bound the spectrum with high probability, separating the outlier from the typical eigenvalues at all sub-critical timescales. These results are obtained under generic assumptions on $U$ that hold for a variety of unitary random matrix models.

math.PR↗

arXiv2023

Dynamics of a rank-one perturbation of a Hermitian matrix

We study the eigenvalue trajectories of a time dependent matrix $ G_t = H+i t vv^*$ for $t \geq 0$, where $H$ is an $N \times N$ Hermitian random matrix and $v$ is a unit vector. In particular, we establish that with high probability, an outlier can be distinguished at all times $t>1+N^{-1/3+ε}$, for any $ε>0$. The study of this natural process combines elements of Hermitian and non-Hermitian analysis, and illustrates some aspects of the intrinsic instability of (even weakly) non-Hermitian matrices.

math.PR↗

arXiv2023

Explicit formulas concerning eigenvectors of weakly non-unitary matrices

We investigate eigenvector statistics of the Truncated Unitary ensemble $\mathrm{TUE}(N,M)$ in the weakly non-unitary case $M=1$, that is when only one row and column are removed. We provide an explicit description of generalized overlaps as deterministic functions of the eigenvalues, as well as a method to derive an exact formula for the expectation of diagonal overlaps (or squared eigenvalue condition numbers), conditionally on one eigenvalue. This complements recent results obtained in the opposite regime when $M \geq N$, suggesting possible extensions to $\mathrm{TUE}(N,M)$ for all values of $M$.

math.PR↗

arXiv2022

Maxima of a Random Model of the Riemann Zeta Function over Intervals of Varying Length

We consider a model of the Riemann zeta function on the critical axis and study its maximum over intervals of length $(\log T)^θ$, where $θ$ is either fixed or tends to zero at a suitable rate. It is shown that the deterministic level of the maximum interpolates smoothly between the ones of log-correlated variables and of i.i.d. random variables, exhibiting a smooth transition 'from $\frac34$ to $\frac14$' in the second order. This provides a natural context where extreme value statistics of log-correlated variables with time-dependent variance and rate occur. A key ingredient of the proof is a precise upper tail tightness estimate for the maximum of the model on intervals of size one, that includes a Gaussian correction. This correction is expected to be present for the Riemann zeta function and pertains to the question of the correct order of the maximum of the zeta function in large intervals.

math.PR↗

arXiv2021

Characteristic polynomials of random truncations: moments, duality and asymptotics

We study moments of characteristic polynomials of truncated Haar distributed matrices from the three classical compact groups O(N), U(N) and Sp(2N). For finite matrix size we calculate the moments in terms of hypergeometric functions of matrix argument and give explicit integral representations highlighting the duality between the moment and the matrix size as well as the duality between the orthogonal and symplectic cases. Asymptotic expansions in strong and weak non-unitarity regimes are obtained. Using the connection to matrix hypergeometric functions, we establish limit theorems for the log-modulus of the characteristic polynomial evaluated on the unit circle.

math-ph↗

arXiv2021

On the number of cycles in commutators of random permutations

We present general links between statistics of non-Hermitian random matrices and the distribution of the number of cycles of some specific random permutations. In particular, we derive explicit formulas for the generating functions of the number of cycles in the commutator $[σ,τ] = στσ^{-1} τ^{-1}$ where $σ$ is uniformly distributed, and $τ$ is either one cycle, the product of two cycles of same size, or the product of many transpositions.

math.PR↗

arXiv2021

Formal verification of Zagier's one-sentence proof

We comment on two formal proofs of Fermat's sum of two squares theorem, written using the Mathematical Components libraries of the Coq proof assistant. The first one follows Zagier's celebrated one-sentence proof; the second follows David Christopher's more recent proof relying on partition-theoretic arguments. Both formal proofs rely on a general property of involutions of finite sets, of independent interest. The proof technique consists for the most part of automating recurrent tasks (such as case distinctions and computations on natural numbers) via ad hoc tactics. With the same method, we also provide a formal proof of another classical result on primes of the form $a^2 + 2 b^2$.

cs.LO↗

arXiv2021

On Words of non-Hermitian Random Matrices

We consider words $G_{i_1} \cdots G_{i_m}$ involving i.i.d. complex Ginibre matrices, and study tracial expressions of their eigenvalues and singular values. We show that the limit distribution of the squared singular values of every word of length $m$ is a Fuss-Catalan distribution with parameter $m+1$. This generalizes previous results concerning powers of a complex Ginibre matrix, and products of independent Ginibre matrices. In addition, we find other combinatorial parameters of the word that determine the second-order limits of the spectral statistics. For instance, the so-called coperiod of a word characterizes the fluctuations of the eigenvalues. We extend these results to words of general non-Hermitian matrices with i.i.d. entries under moment-matching assumptions, band matrices, and sparse matrices. These results rely on the moments method and genus expansion, relating Gaussian matrix integrals to the counting of compact orientable surfaces of a given genus. This allows us to derive a central limit theorem for the trace of any word of complex Ginibre matrices and their conjugate transposes, where all parameters are defined topologically.

math.PR↗

arXiv2019

The distribution of overlaps between eigenvectors of Ginibre matrices

We study the overlaps between eigenvectors of nonnormal matrices. They quantify the stability of the spectrum, and characterize the joint eigenvalues increments under Dyson-type dynamics. Well known work by Chalker and Mehlig calculated the expectation of these overlaps for complex Ginibre matrices. For the same model, we extend their results by deriving the distribution of diagonal overlaps (the condition numbers), and their correlations. We prove: (i) convergence of condition numbers for bulk eigenvalues to an inverse Gamma distribution; more generally, we decompose the quenched overlap (i.e. conditioned on eigenvalues) as a product of independent random variables; (ii) asymptotic expectation of off-diagonal overlaps, both for microscopic or mesoscopic separation of the corresponding eigenvalues; (iii) decorrelation of condition numbers associated to eigenvalues at mesoscopic distance, at polynomial speed in the dimension; (iv) second moment asymptotics to identify the fluctuations order for off-diagonal overlaps, when the related eigenvalues are separated by any mesoscopic scale; (v) a new formula for the correlation between overlaps for eigenvalues at microscopic distance, both diagonal and off-diagonal. These results imply estimates on the extreme condition numbers, the volume of the pseudospectrum and the diffusive evolution of eigenvalues under Dyson-type dynamics, at equilibrium.

math.PR↗

arXiv2019

On eigenvector statistics in the spherical and truncated unitary ensembles

We study the overlaps between right and left eigenvectors for random matrices of the spherical and truncated unitary ensembles. Conditionally on all eigenvalues, diagonal overlaps are shown to be distributed as a product of independent random variables. This enables us to prove that the scaled diagonal overlaps, conditionally on one eigenvalue, converge in distribution to a heavy-tail limit, namely, the inverse of a $γ_2$ distribution. These results are analogous to what is known for the complex Ginibre ensemble. We also provide formulae for the conditional expectation of diagonal and off-diagonal overlaps, with respect to all eigenvalues.

math.PR↗

arXiv2018

Symmetries of the Quaternionic Ginibre Ensemble

We establish a few properties of eigenvalues and eigenvectors of the quaternionic Ginibre ensemble (QGE), analogous to what is known in the complex Ginibre case. We first recover a version of Kostlan's theorem that was already noticed by Rider: the set of the squared radii of the eigenvalues is distributed as a set of independent gamma variables. Our proof technique uses the De Bruijn identity and properties of Pfaffians; it also allows to prove that the high powers of these eigenvalues are independent. These results extend to any potential beyond the Gaussian case, as long as radial symmetry holds; this includes for instance truncations of quaternionic unitary matrices, products of quaternionic Ginibre matrices, and the quaternionic spherical ensemble. We then study the eigenvectors of quaternionic Ginibre matrices. The angle between eigenvectors and the matrix of overlaps both exhibit some specific features that can be compared to the complex case. In particular, we compute the distribution and the limit of the diagonal overlap associated to an eigenvalue that is conditioned to be zero.

math.PR↗

arXiv2018

Powers of Ginibre Eigenvalues

We study the images of the complex Ginibre eigenvalues under the power maps $π_M: z \mapsto z^M$, for any integer $M$. We establish the following equality in distribution, $$ {\rm{Gin}}(N)^M \stackrel{d}{=} \bigcup_{k=1}^M {\rm{Gin}} (N,M,k), $$ where the so-called Power-Ginibre distributions ${\rm{Gin}}(N,M,k)$ form $M$ independent determinantal point processes. The decomposition can be extended to any radially symmetric normal matrix ensemble, and generalizes Rains' superposition theorem for the CUE and Kostlan's independence of radii to a wider class of point processes. Our proof technique also allows us to recover a result by Edelman and La Croix for the GUE. Concerning the Power-Ginibre blocks, we prove convergence of fluctuations of their smooth linear statistics to independent gaussian variables, coherent with the link between the complex Ginibre Ensemble and the Gaussian Free Field. Finally, some partial results about two-dimensional beta ensembles with radial symmetry and even parameter $β$ are discussed, replacing independence by conditional independence.

math.PR↗