  We study $\gamma$-vectors associated with $h^*$-vectors of symmetric edge
polytopes both from a deterministic and a probabilistic point of view. On the
deterministic side, we prove nonnegativity of $\gamma_2$ for any graph and
completely characterize the case when $\gamma_2 = 0$. The latter also confirms
a conjecture by Lutz and Nevo in the realm of symmetric edge polytopes. On the
probabilistic side, we show that the $\gamma$-vectors of symmetric edge
polytopes of most Erd\H{o}s-R\'enyi random graphs are asymptotically almost
surely nonnegative up to any fixed entry. This proves that Gal's conjecture
holds asymptotically almost surely for arbitrary unimodular triangulations in
this setting.
