Spectral expansion of random sum complexes

Let G be a finite abelian group of order n and let δn-1 denote the (n - 1)-simplex on the vertex set G. The sum complex XA,k associated to a subset A C G and k < n, is the k-dimensional simplicial complex obtained by taking the full (k - 1)-skeleton of δn-1 together with all (k + 1)-subsets σ C G that satisfy σx σx A. Let Ck-1(X A,k) denote the space of complex-valued (k - 1)-cochains of XA,k. Let Lk-1: Ck-1(X A,k) → Ck-1(X A,k) denote the reduced (k - 1)th Laplacian of XA,k, and let μk-1(XA,k) be the minimal eigenvalue of Lk-1. It is shown that if k ≥ 1 and > 0 are fixed, and A is a random subset of G of size m = 4k2log n 2, then Pr[μk-1(XA,k) < (1 - )m] = O 1 n.