이지운 교수

We consider N × N random matrices of the form H = W + V where W is a real symmetric Wigner matrix and V a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W and we choose V so that the eigenvalues of W and V are typically of the same order. For a large class of diagonal matrices V, we show that the rescaled distribution of the extremal eigenvalues is given by the Tracy–Widom distribution F 1 in the limit o

We consider $N\times N$ random matrices of the form $H=W+V$ where $W$ is a real symmetric or complex Hermitian Wigner matrix and $V$ is a random or deterministic, real, diagonal matrix whose entries are independent of $W$. We assume subexponential decay for the matrix entries of $W$, and we choose $V$ so that the eigenvalues of $W$ and $V$ are typically of the same order. For a large class of diagonal matrices $V$, we show that the local statistics in the bulk of the spectrum are universal in th

The mean field dynamics of an N-particle weekly interacting Boson system can be described by the nonlinear Hartree equation. In this paper, we present estimates on the 1/N rate of convergence of many-body Schrödinger dynamics to the one-body nonlinear Hartree dynamics with factorized initial data with two-body interaction potential V in \documentclass[12pt]{minimal}\begin{document}$L^3 (\mathbb {R}^3)+ L^{\infty } (\mathbb {R}^3)$\end{document}L3(R3)+L∞(R3).

We consider the semi-relativistic system of N gravitating Bosons with gravitation constant G. The time evolution of the system is described by the relativistic dispersion law, and we assume the mean-field scaling of the interaction where N → ∞ and G → 0 while GN = λ fixed. In the super-critical regime of large λ, we introduce the regularized interaction where the cutoff vanishes as N → ∞. We show that the difference between the many-body semi-relativistic Schrödinger dynamics and the correspondi

We consider large-dimensional Hermitian or symmetric random matrices of the form [Formula: see text], where [Formula: see text] is a Wigner matrix and [Formula: see text] is a real diagonal matrix whose entries are independent of [Formula: see text]. For a large class of diagonal matrices [Formula: see text], we prove that the fluctuations of linear spectral statistics of [Formula: see text] for [Formula: see text] test function can be decomposed into that of [Formula: see text] and of [Formula:

We consider sample covariance matrices of the form $\mathcal{Q}=(\Sigma^{1/2}X)(\Sigma^{1/2}X)^{*}$, where the sample $X$ is an $M\times N$ random matrix whose entries are real independent random variables with variance $1/N$ and where $\Sigma$ is an $M\times M$ positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of $\mathcal{Q}$ when both $M$ and $N$ tend to infinity with $N/M\to d\in(0,\infty)$. For a large class of populations $\Si

We consider the spectral properties of sparse stochastic block models, where $N$ vertices are partitioned into $K$ balanced communities. Under an assumption that the intra-community probability and inter-community probability are of similar order, we prove a local semicircle law up to the spectral edges, with an explicit formula on the deterministic shift of the spectral edge. We also prove that the fluctuation of the extremal eigenvalues is given by the GOE Tracy–Widom law after rescaling and c

Nous considérons l’énergie libre du modèle sphérique bipartite de Sherrington–Kirkpatrick et déterminons l’énergie libre limite à chaque température. Nous prouvons également la convergence de la loi des fluctuations de l’énergie libre à température non critique. La limite est donnée par la distribution Gaussienne pour toutes les températures élevées et par la distribution de Tracy–Widom GOE pour toutes les températures basses. Le résultat est universel et l’analyse est applicable à un cadre plus

We study the statistical decision process of detecting the signal from a `signal+noise' type matrix model with an additive Wigner noise. We propose a hypothesis test based on the linear spectral statistics of the data matrix, which does not depend on the distribution of the signal or the noise. The test is optimal under the Gaussian noise if the signal-to-noise ratio is small, as it minimizes the sum of the Type-I and Type-II errors. Under the non-Gaussian noise, the test can be improved with an