  Dirac operator eigenvalues split into two when subjected to two different
external vector sources. In a specific finite-volume scaling regime of gauge
theories with fermions, this problem can be mapped to a chiral Random
Two-Matrix Theory. We derive analytical expressions to leading order in the
associated finite-volume expansion, showing how individual Dirac eigenvalue
distributions and their correlations equivalently can be computed directly from
the effective chiral Lagrangian in the epsilon-regime. Because of its
equivalence to chiral Random Two-Matrix Theory, we use the latter for all
explicit computations. On the mathematical side, we define and determine gap
probabilities and individual eigenvalue distributions in that theory at finite
N, and also derive the relevant scaling limit as N is taken to infinity. In
particular, the gap probability for one Dirac eigenvalue is given in terms of a
new kernel that depends on the external vector source. This expression may give
a new and simple way of determining the pion decay constant F_pi from lattice
gauge theory simulations.
