Related papers: Recovering Parameters from Edge Fluctuations: Beta…
The soft and hard edge scaling limits of $\beta$-ensembles can be characterized as the spectra of certain random Sturm-Liouville operators. It has been shown that by tuning the parameter of the hard edge process one can obtain the soft edge…
Estimating the leading principal components of data, assuming they are sparse, is a central task in modern high-dimensional statistics. Many algorithms were developed for this sparse PCA problem, from simple diagonal thresholding to…
We investigate the characteristic polynomials of the Gaussian $\beta$-ensemble for general $\beta>0$ through its transfer matrix recurrence. We show that the rescaled characteristic polynomial converges to a random entire function in a…
Ram\'irez and Rider (2009) established that the hard edge of the spectrum of the $\beta$-Laguerre ensemble converges, in the high-dimensional limit, to the bottom of the spectrum of the stochastic Bessel operator. Using stochastic analysis…
It was shown in [J. A. Ram\'irez, B. Rider and B. Vir\'ag. J. Amer. Math. Soc. 24, 919-944 (2011)] that the edge of the spectrum of $\beta$ ensembles converges in the large $N$ limit to the bottom of the spectrum of the stochastic Airy…
We prove that the spectrum of the stochastic Airy operator is rigid in the sense of Ghosh and Peres (Duke Math. J., 166(10):1789--1858, 2017) for Dirichlet and Robin boundary conditions. This proves the rigidity of the Airy-$\beta$ point…
For the eigenvalues of principal submatrices of stochastically evolving Wigner matrices, we construct and study the edge scaling limit: a random decreasing sequence of continuous functions of two variables, which at every point has the…
The stochastic Airy and sine operators, which are respectively a random Sturm-Liouville operator and a random Dirac operator, characterize the soft edge and bulk scaling limits of $\beta$-ensembles. Dirac and Sturm-Liouville operators are…
We know from Ram{\'i}rez and Rider that the hard edge of the spectrum of the Beta-Laguerre ensemble converges, in the high-dimensional limit, to the bottom of the spectrum of the stochastic Bessel operator. Using stochastic analysis…
The Airy$_\beta$ point process, originally introduced by Ram\'irez, Rider, and Vir\'ag, is defined as the spectrum of the stochastic Airy operator $\mathcal{H}_\beta$ acting on a subspace of $L^2[0,\infty)$ with Dirichlet boundary…
Given a large, high-dimensional sample from a spiked population, the top sample covariance eigenvalue is known to exhibit a phase transition. We show that the largest eigenvalues have asymptotic distributions near the phase transition in…
The Airy$_\beta$ point process, $a_i \equiv N^{2/3} (\lambda_i-2)$, describes the eigenvalues $\lambda_i$ at the edge of the Gaussian $\beta$ ensembles of random matrices for large matrix size $N \to \infty$. We study the probability…
The critical behaviour of $d$-dimensional semi-infinite systems with $n$-component order parameter $\bm{\phi}$ is studied at an $m$-axial bulk Lifshitz point whose wave-vector instability is isotropic in an $m$-dimensional subspace of…
Consider random Schr\"odinger operators $H_n$ defined on $[0,n]\cap\mathbb{Z}$ with zero boundary conditions: $$ (H_n\psi)_\ell=\psi_{\ell-1}+\psi_{\ell+1}+\sigma\frac{\mathfrak{a}(\ell)}{n^{\alpha}}\psi_{\ell},\quad \ell=1,\cdots,n,\quad…
We study the scaling limit of the spectrum of the \beta-Jacobi ensemble at the soft-edge and hard-edge for general values of \beta. We show that the limiting point processes correspond respectively to the stochastic Airy and Bessel point…
We show that in the point process limit of the bulk eigenvalues of $\beta$-ensembles of random matrices, the probability of having no eigenvalue in a fixed interval of size $\lambda$ is given by \[\bigl(\…
Covariances and variances of linear statistics of a point process can be written as integrals over the truncated two-point correlation function. When the point process consists of the eigenvalues of a random matrix ensemble, there are often…
We prove limit theorems for rescaled occupation time fluctuations of a (d,alpha,beta)-branching particle system (particles moving in R^d according to a spherically symmetric alpha-stable Levy process, (1+beta)-branching, 0<beta<1, uniform…
The extremal index $\theta$, a measure of the degree of local dependence in the extremes of a stationary process, plays an important role in extreme value analyses. We estimate $\theta$ semiparametrically, using the relationship between the…
We investigate the marginal distribution of the bottom eigenvalues of the stochastic Airy operator when the inverse temperature $\beta$ tends to $0$. We prove that the minimal eigenvalue, whose fluctuations are governed by the Tracy-Widom…