Spectral expansion of random sum complexes

Let [Formula: see text] be a finite abelian group of order [Formula: see text] and let [Formula: see text] denote the [Formula: see text]-simplex on the vertex set [Formula: see text]. The sum complex [Formula: see text] associated to a subset [Formula: see text] and [Formula: see text], is the [Formula: see text]-dimensional simplicial complex obtained by taking the full [Formula: see text]-skeleton of [Formula: see text] together with all [Formula: see text]-subsets [Formula: see text] that satisfy [Formula: see text]. Let [Formula: see text] denote the space of complex-valued [Formula: see text]-cochains of [Formula: see text]. Let [Formula: see text] denote the reduced [Formula: see text]th Laplacian of [Formula: see text], and let [Formula: see text] be the minimal eigenvalue of [Formula: see text]. It is shown that if [Formula: see text] and [Formula: see text] are fixed, and [Formula: see text] is a random subset of [Formula: see text] of size [Formula: see text], then [Formula: see text]