We consider the symmetric exclusion particle system on $\mathbb{Z}$ 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 $X_t/(\sigma b_t)
– a_t$ of the right-most particle at time $t$ converges to a Gumbel limit law,
where $b_t = \sqrt{t/\log t}$, $a_t = \log(t/(\sqrt{2\pi}\log t))$, and
$\sigma$ is the standard deviation of the random walk jump probabilities. This
work solves an open problem suggested in Arratia (1983), and it is the first
demonstration of an extreme value limit distribution in exclusion processes.
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 $L \to \infty$ as $t \to \infty$. 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 similar to that under a infinite particle step
profile, and below which it is of order $\sqrt{t\log L}$. 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.
