Abstract: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 neighborhood of the edge of the limiting spectrum. This random entire function, called the stochastic Airy function, is the unique (up to scaling) $L^2$ solution to the stochastic Airy equation, a family of second order stochastic differential equations. Moreover, we obtain a coupling between the characteristic polynomial and a solution of the stochastic Airy equation which allows us to show that for any $\epsilon>0$, these two function are uniformly close by $N^{-1/6 + \epsilon}$ with overwhelming probability. These results build on the results of the authors in which the hyperbolic portion of the transfer matrix recurrence for the characteristic polynomial is analyzed.
From: Elliot Paquette [view email]
[v1]
Thu, 10 Sep 2020 17:11:56 UTC (203 KB)
[v2]
Fri, 30 Jul 2021 19:12:09 UTC (223 KB)
[v3]
Fri, 21 Aug 2026 17:07:56 UTC (122 KB)
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