  We show that in the asymmetric simple exclusion process (ASEP) on a ring,
conditioned on carrying a large flux, the particle experience an effective
long-range potential which in the limit of very large flux takes the simple
form $U= -2\sum_{i\neq j}\log|\sin\pi(n_{i}/L-n_{j}/L)|$, where $n_{1}%
n_{2},\ldots n_{N}$ are the particle positions, similar to the effective
potential between the eigenvalues of the circular unitary ensemble in random
matrices. Effective hopping rates and various quasistationary probabilities
under such a conditioning are found analytically using the Bethe ansatz and
determinantal free fermion techniques. Our asymptotic results extend to the
limit of large current and large activity for a family of reaction-diffusion
processes with on-site exclusion between particles. We point out an intriguing
generic relation between classical stationary probability distributions for
conditioned dynamics and quantum ground state wave functions, in particular, in
the case of exclusion processes, for free fermions.
