There is a revolution sweeping through asymptotics of orthogonal polynomials called the Riemann-Hilbert method. It has enabled researchers such as the present authors and Deift, Kriecherbauer, MacLaughlin and others to obtain very precise (and uniform) asymptotics for orthogonal polynomials for exponential weights, in situations where the classical Bernstein-SzegĂ¶ methods give limited precision. And this paper is the record of one of the first breakthroughs in this exciting development.
Let $$V(z):= gz^4/4+ tz^2/2,$$ where $g>0>t$, so that $V$ is a double-well potential. Let $0<\varepsilon< 1$, and for $n\ge 1$ consider a parameter $N$ satisfying $$\varepsilon>{n\over N}< {t^2\over 4g}- \varepsilon.$$ Let us consider the monic orthogonal polynomials $P_n$ with respect to the varying weight $w:= \exp(-NV)$, so that $$\int^\infty_{-\infty} P_nP_m\exp(- NV)= h_n\delta_{mn},$$ where $h_n> 0$. The authors establish very precise asymptotics for $P_n$ and the associated recurrence coefficients as $n\to\infty$. Then they apply these to establish universality of the local distribution of eigenvalues in the matrix model with quartic potential.
A key point in the analysis is the Fokas-Its-Kitaev Riemann-Hilbert problem, in which the orthogonal polynomials appear explicitly. This is followed by use of an approximate solution to the Riemann-Hilbert problem, and a proof that the approximate solution gives the asymptotic formula. The paper contains an extensive review of related literature; in particular, the context of the results and their motivation is very clearly presented. This paper will be of great use to anyone interested in orthogonal polynomials and their applications.