On the gamma-vector of symmetric edge polytopes

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.