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We prove the edge universality of the beta ensembles for any $betage 1$, provided that the limiting spectrum is supported on a single interval, and the external potential is $mathscr{C}^4$ and regular. We also prove that the edge universality holds for generalized Wigner matrices for all symmetry classes. Moreover, our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class $mathscr{C}^4$.
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We access the edge of Gaussian beta ensembles with one spike by analyzing high powers of the associated tridiagonal matrix models. In the classical cases beta=1, 2, 4, this corresponds to studying the fluctuations of the largest eigenvalues of additive rank one perturbations of the GOE/GUE/GSE random matrices. In the infinite-dimensional limit, we arrive at a one-parameter family of random Feynman-Kac type semigroups, which features the stochastic Airy semigroup of Gorin and Shkolnikov [13] as an extreme case. Our analysis also provides Feynman-Kac formulas for the spiked stochastic Airy operators, introduced by Bloemendal and Virag [6]. The Feynman-Kac formulas involve functionals of a reflected Brownian motion and its local times, thus, allowing to study the limiting operators by tools of stochastic analysis. We derive a first result in this direction by obtaining a new distributional identity for a reflected Brownian bridge conditioned on its local time at zero.
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 process as a scaling limit. We prove that this limit can be realized on the level of the corresponding random operators. More precisely, the random operators can be coupled in a way so that the scal
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 distributed as the Airy$_beta$ random point field. We prove that the increments of the limiting process are locally Brownian. When $beta >1$ we prove that after subtracting a Brownian motion, the sample paths are almost surely locally $r$-H{o}lder for any $r<1-(1+beta)^{-1}$. Furthermore for all $beta geq 1$ we show that the limiting process solves an SDE in a weak sense. When $beta=2$ this limiting process is the Airy line ensemble.
The spherical orthogonal, unitary, and symplectic ensembles (SOE/SUE/SSE) $S_beta(N,r)$ consist of $N times N$ real symmetric, complex hermitian, and quaternionic self-adjoint matrices of Frobenius norm $r$, made into a probability space with the uniform measure on the sphere. For each of these ensembles, we determine the joint eigenvalue distribution for each $N$, and we prove the empirical spectral measures rapidly converge to the semicircular distribution as $N to infty$. In the unitary case ($beta=2$), we also find an explicit formula for the empirical spectral density for each $N$.
We consider various asymptotic scaling limits $Ntoinfty$ for the $2N$ complex eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble. These are known to be integrable, forming Pfaffian point processes, and we obtain limiting expressions for the corresponding kernel for different potentials. The first part is devoted to the symplectic Ginibre ensemble with a Gaussian potential. We obtain the asymptotic at the edge of the spectrum in the vicinity of the real line. The unifying form of the kernel allows us to make contact with the bulk scaling along the real line and with the edge scaling away from the real line, where we recover the known determinantal process of the complex Ginibre ensemble. Part two covers ensembles of Mittag-Leffler type with a singularity at the origin. For potentials $Q(zeta)=|zeta|^{2lambda}-(2c/N)log|zeta|$, with $lambda>0$ and $c>-1$, the limiting kernel obeys a linear differential equation of fractional order $1/lambda$ at the origin. For integer $m=1/lambda$ it can be solved in terms of Mittag-Leffler functions. In the last part, we derive the Wards equation for a general class of potentials as a tool to investigate universality. This allows us to determine the functional form of kernels that are translation invariant up to its integration domain.
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