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.