Related papers: The stochastic Airy operator at large temperature

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…

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…

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…

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…

This paper studies beta ensembles on the real line in a high temperature regime, that is, the regime where $\beta N \to const \in (0, \infty)$, with $N$ the system size and $\beta$ the inverse temperature. In this regime, the convergence to…

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 study the asymptotic edge statistics of the Gaussian $\beta$-ensemble, a collection of $n$ particles, as the inverse temperature $\beta$ tends to zero as $n$ tends to infinity. In a certain decay regime of $\beta$, the associated extreme…

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…

In this article, we consider $\beta$-ensembles, i.e. collections of particles with random positions on the real line having joint distribution $$\frac{1}{Z_N(\beta)}|\Delta(\lambda)|^\beta e^{- \frac{N\beta}{4}\sum_{i=1}^N\lambda_i^2}d…

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…

We study the bottom of the spectrum of the Anderson Hamiltonian $\mathcal{H}_L := -\partial_x^2 + \xi$ on $[0,L]$ driven by a white noise $\xi$ and endowed with either Dirichlet or Neumann boundary conditions. We show that, as…

In previous work, a description of the result of applying the Householder tridiagonalization algorithm to a G$\beta$E random matrix is provided by Edelman and Dumitriu. The resulting tridiagonal ensemble makes sense for all $\beta>0$, and…

For general $\beta \geq 1$, we consider Dyson Brownian motion at equilibrium and prove convergence of the extremal particles to an ensemble of continuous sample paths in the limit $N \to \infty$. For each fixed time, this ensemble is…

Let $\Lambda=\{\Lambda_0,\Lambda_1,\Lambda_2,\ldots\}$ be the point process that describes the edge scaling limit of either (i) "regular" beta-ensembles with inverse temperature $\beta>0$, or (ii) the top eigenvalues of Wishart or Gaussian…

We study the limiting behavior of Gaussian beta ensembles in the regime where $\beta n = const$ as $n \to \infty$. The results are (1) Gaussian fluctuations for linear statistics of the eigenvalues, and (2) Poisson convergence of the bulk…

We consider the Gibbs measure of a general interacting particle system for a certain class of ``weakly interacting" kernels. In particular, we show that the local point process converges to a Poisson point process as long as the inverse…

The hard edge and bulk scaling limits of $\beta$-ensembles are described by the stochastic Bessel and sine operators, which are respectively a random Sturm-Liouville operator and a random Dirac operator. By representing both operators as…

We consider the adjacency matrix $A$ of the Erd\H{o}s-R\'enyi graph on $N$ vertices with edge probability $d/N$. For $(\log \log N)^4 \ll d \lesssim \log N$, we prove that the eigenvalues near the spectral edge form asymptotically a Poisson…

We examine the eigenvalues and eigenvectors of the staggered Dirac operator on thermal ensembles created in QCD with two flavours of staggered quarks. We see that across the phase transition a gap opens in the spectrum. For finite volume…

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…