Asymptotics via Steepest Descent for an Operator Riemann-Hilbert Problem

In this paper, we take the first step towards an extension of the nonlinear steepest descent method of Deift, Its and Zhou to the case of operator Riemann-Hilbert problems. In particular, we provide long range asymptotics for a Fredholm determinant arising in the computation of the probability of finding a string of n adjacent parallel spins up in the antiferromagnetic ground state of the spin 1/2 XXX Heisenberg Chain. Such a determinant can be expressed in terms of the solution of an operator Riemann-Hilbert factorization problem. Typeset by A M S-T E X 2 SPYRIDON KAMVISSIS 1. INTRODUCTION-MOTIVATION It has been established by Korepin, Izergin and Bogoliubov [KIB] that many important problems arising in the theory of Quantum Inverse Scattering can be reduced to Classical Inverse Scattering Problems. In particular, the correlation functions for quantum solvable models can be described by solutions of classical integrable equations, or, equivalently, are reducible to the solution of R..