The asymptotic behaviour of recurrence coefficients for orthogonal polynomials with varying exponential weights

We consider orthogonal polynomials {pn,N(x)}n=0∞ on the real line with respect to a weight w(x)=e−NV(x) and in particular the asymptotic behaviour of the coefficients an,Nan,N and bn,Nbn,N in the three-term recurrence xπn,N(x)=πn+1,N(x)+bn,Nπn,N(x)+an,Nπn−1,N(x)xπn,N(x)=πn+1,N(x)+bn,Nπn,N(x)+an,Nπn−1,N(x). For one-cut regular VV we show, using the Deift–Zhou method of steepest descent for Riemann–Hilbert problems, that the diagonal recurrence coefficients an,nan,n and bn,nbn,n have asymptotic expansions as n→∞n→∞ in powers of 1/n21/n2 and powers of 1/n1/n, respectively.