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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 randomentire function in a neighborhood of the edge of the limiting spectrum. Thisrandom entire function, called the stochastic Airy function, is the unique (upto scaling) $L^2$ solution to the stochastic Airy equation, a family of secondorder stochastic differential equations. Moreover, we obtain a coupling betweenthe characteristic polynomial and a solution of the stochastic Airy equationwhich allows us to show that for any $\epsilon>0$, these two function areuniformly close by $N^{-1/6 + \epsilon}$ with overwhelming probability. Theseresults build on the results of the authors in which the hyperbolic portion ofthe transfer matrix recurrence for the characteristic polynomial is analyzed.

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