Gumbel Laws in the Symmetric Exclusion Process

We consider the symmetric exclusion particle system on ℤ starting from an infinite particle step configuration in which there are no particles to the right of a maximal one. We show that the scaled position X_t/(σ b_t) - a_t of the right-most particle at time t converges to a Gumbel limit law, where b_t = √(t/log t) , a_t = log (t/(√(2π)log t)) , and σ is the standard deviation of the random walk jump probabilities. This work solves a problem left open in Arratia (Ann Probab 11(2):362–373, 1983). Moreover, to investigate the influence of the mass of particles behind the leading one, we consider initial profiles consisting of a block of L particles, where L →∞ as t →∞ . Gumbel limit laws, under appropriate scaling, are obtained for X_t when L diverges in t . In particular, there is a transition when L is of order b_t , above which the displacement of X_t is the same as that under an infinite particle step profile, and below which it is of order √(tlog L) . Proofs are based on recently developed negative dependence properties of the symmetric exclusion system. Remarks are also made on the behavior of the right-most particle starting from a step profile in asymmetric nearest-neighbor exclusion, which complement known results.