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Register a new userEdge scaling limit of Dyson Brownian motion at equilibrium for general $beta geq 1$
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Authors
Benjamin Landon
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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.
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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].
We study the averaged products of characteristic polynomials for the Gaussian and Laguerre $beta$-ensembles with external source, and prove Pearcey-type phase transitions for particular full rank perturbations of source. The phases are characterised by determining the explicit functional forms of the scaled limits of the averaged products of characteristic polynomials, which are given as certain multidimensional integrals, with dimension equal to the number of products.
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