François Chapon | Searcharxiv

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↗

arXiv2020

Quantum $SL_2$, Infinite curvature and Pitman's 2M-X theorem

The classical theorem by Pitman states that a Brownian motion minus twice its running infimum enjoys the Markov property. On the one hand, Biane understood that Pitman's theorem is intimately related to the representation theory of the quantum group $\mathcal{U}_q\left( \mathfrak{sl}_2 \right)$, in the so-called crystal regime $q \rightarrow 0$. On the other hand, Bougerol and Jeulin showed the appearance of exactly the same Pitman transform in the infinite curvature limit $r \rightarrow \infty$ of a Brownian motion on the hyperbolic space $\mathbb{H}^3 = SL_2(\mathbb{C})/SU_2$. This paper aims at understanding this phenomenon by giving a unifying point of view. In order to do so, we exhibit a presentation $\mathcal{U}_q^\hbar\left( \mathfrak{sl}_2 \right)$ of the Jimbo-Drinfeld quantum group which isolates the role of curvature $r$ and that of the Planck constant $\hbar$. The simple relationship between parameters is $q=e^{-r}$. The semi-classical limits $\hbar \rightarrow 0$ are the Poisson-Lie groups dual to $SL_2(\mathbb{C})$ with varying curvatures $r \in \mathbb{R}_+$. We also construct classical and quantum random walks, drawing a full picture which includes Biane's quantum walks and the construction of Bougerol-Jeulin. Taking the curvature parameter $r$ to infinity leads indeed to the crystal regime at the level of representation theory ($\hbar>0$) and to the Bougerol-Jeulin construction in the classical world ($\hbar=0$). All these results are neatly in accordance with the philosophy of Kirillov's orbit method.

math.PR↗