@article{17047,
  abstract     = {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.},
  author       = {Dubach, Guillaume and Reker, Jana},
  issn         = {2010-3271},
  journal      = {Random Matrices: Theory and Applications},
  number       = {2},
  publisher    = {World Scientific Publishing},
  title        = {{Dynamics of a rank-one multiplicative perturbation of a unitary matrix}},
  doi          = {10.1142/s2010326324500072},
  volume       = {13},
  year         = {2024},
}

@article{12683,
  abstract     = {We study the eigenvalue trajectories of a time dependent matrix Gt=H+itvv∗ for t≥0, where H is an N×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.},
  author       = {Dubach, Guillaume and Erdös, László},
  issn         = {1083-589X},
  journal      = {Electronic Communications in Probability},
  pages        = {1--13},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Dynamics of a rank-one perturbation of a Hermitian matrix}},
  doi          = {10.1214/23-ECP516},
  volume       = {28},
  year         = {2023},
}

@article{17079,
  abstract     = {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.},
  author       = {Serebryakov, Alexander and Simm, Nick and Dubach, Guillaume},
  issn         = {2010-3271},
  journal      = {Random Matrices: Theory and Applications},
  number       = {01},
  publisher    = {World Scientific Publishing},
  title        = {{Characteristic polynomials of random truncations: Moments, duality and asymptotics}},
  doi          = {10.1142/s2010326322500496},
  volume       = {12},
  year         = {2023},
}

@article{15259,
  abstract     = {We consider words Gi1⋯Gim 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.},
  author       = {Dubach, Guillaume and Peled, Yuval},
  issn         = {0091-1798},
  journal      = {The Annals of Probability},
  keywords     = {Statistics, Probability and Uncertainty, Statistics and Probability},
  number       = {4},
  pages        = {1886--1916},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{On words of non-Hermitian random matrices}},
  doi          = {10.1214/20-aop1496},
  volume       = {49},
  year         = {2021},
}

@article{10285,
  abstract     = {We study the overlaps between right and left eigenvectors for random matrices of the spherical ensemble, as well as truncated unitary ensembles in the regime where half of the matrix at least is truncated. These two integrable models exhibit a form of duality, and the essential steps of our investigation can therefore be performed in parallel. In every case, conditionally on all eigenvalues, diagonal overlaps are shown to be distributed as a product of independent random variables with explicit distributions. 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 γ2 distribution. We also provide formulae for the conditional expectation of diagonal and off-diagonal overlaps, either with respect to one eigenvalue, or with respect to the whole spectrum. These results, analogous to what is known for the complex Ginibre ensemble, can be obtained in these cases thanks to integration techniques inspired from a previous work by Forrester & Krishnapur.},
  author       = {Dubach, Guillaume},
  issn         = {1083-6489},
  journal      = {Electronic Journal of Probability},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{On eigenvector statistics in the spherical and truncated unitary ensembles}},
  doi          = {10.1214/21-EJP686},
  volume       = {26},
  year         = {2021},
}

@unpublished{9230,
  abstract     = {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
3/4 to 1/4’ 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.},
  author       = {Arguin, Louis-Pierre and Dubach, Guillaume and Hartung, Lisa},
  booktitle    = {arXiv},
  title        = {{Maxima of a random model of the Riemann zeta function over intervals of varying length}},
  doi          = {10.48550/arXiv.2103.04817},
  year         = {2021},
}

@unpublished{9281,
  abstract     = {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 recent new 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.},
  author       = {Dubach, Guillaume and Mühlböck, Fabian},
  booktitle    = {arXiv},
  title        = {{Formal verification of Zagier's one-sentence proof}},
  doi          = {10.48550/arXiv.2103.11389},
  year         = {2021},
}