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.