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

    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.

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