  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.
