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

We consider orthogonal polynomials { p n , N ( x ) } n = 0 ∞ on the real line with respect to a weight w ( x ) = e − N V ( x ) and in particular the asymptotic behaviour of the coefficients a n , N and b n , N in the three-term recurrence x π n , N ( x ) = π n + 1 , N ( x ) + b n , N π n , N ( x ) + a n , N π n − 1 , N ( x ) . For one-cut regular V we show, using the Deift–Zhou method of steepest descent for Riemann–Hilbert problems, that the diagonal recurrence coefficients a n , n and b n , n have asymptotic expansions as n → ∞ in powers of 1 / n 2 and powers of 1 / n , respectively.