Strong approximation of Gaussian $\beta$-ensemble characteristic polynomials: the edge regime and the stochastic Airy function

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.