Strong asymptotics of orthogonal polynomials with respect to exponential weights

We consider asymptotics of orthogonal polynomials with respect to weights w(x)dx = e−Q(x) dx on the real line, where Q(x) = Σ qk xk, q2m > 0, denotes a polynomial of even order with positive leading coefficient. The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [22, 23]. We employ the steepest-descent-type method introduced in [18] and further developed in [17, 19] in order to obtain uniform Plancherel-Rotach-type asymptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients, and the recurrence coefficients of the orthogonal polynomials.