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Register a new userEdge of spiked beta ensembles, stochastic Airy semigroups and reflected Brownian motions
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Added by
Mykhaylo Shkolnikov
Publication date
2017
fields
Physics
and research's language is
English
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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.
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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 $mathscr{C}^4$.
The perturbed GUE corners ensemble is the joint distribution of eigenvalues of all principal submatrices of a matrix $G+mathrm{diag}(mathbf{a})$, where $G$ is the random matrix from the Gaussian Unitary Ensemble (GUE), and $mathrm{diag}(mathbf{a})$ is a fixed diagonal matrix. We introduce Markov transitions based on exponential jumps of eigenvalues, and show that their successive application is equivalent in distribution to a deterministic shift of the matrix. This result also leads to a new distributional symmetry for a family of reflected Brownian motions with drifts coming from an arithmetic progression. The construction we present may be viewed as a random matrix analogue of the recent results of the first author and Axel Saenz (arXiv:1907.09155 [math.PR]).
We determine the operator limit for large powers of random tridiagonal matrices as the size of the matrix grows. The result provides a novel expression in terms of functionals of Brownian motions for the Laplace transform of the Airy$_beta$ process, which describes the largest eigenvalues in the $beta$ ensembles of random matrix theory. Another consequence is a Feynman-Kac formula for the stochastic Airy operator of Ram{i}rez, Rider, and Vir{a}g. As a side result, we find that the difference between the area underneath a standard Brownian excursion and one half of the integral of its squared local times is a Gaussian random variable.
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.
We consider the edge statistics of Dyson Brownian motion with deterministic initial data. Our main result states that if the initial data has a spectral edge with rough square root behavior down to a scale $eta_* geq N^{-2/3}$ and no outliers, then after times $t gg sqrt{ eta_*}$, the statistics at the spectral edge agree with the GOE/GUE. In particular we obtain the optimal time to equilibrium at the edge $t = N^{varepsilon} / N^{1/3}$ for sufficiently regular initial data. Our methods rely on eigenvalue rigidity results similar to those appearing in [Lee-Schnelli], the coupling idea of [Bourgade-ErdH{o}s-Yau-Yin] and the energy estimate of [Bourgade-ErdH{o}s-Yau].
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