Independent sets in graph powers are almost contained in juntas

Let G = (V,E) be a simple undirected graph. Define G ⁿ , the n-th power of G, as the graph on the vertex set V ⁿ in which two vertices (u ₁, . . . , u _n ) and (v ₁, . . . , v _n ) are adjacent if and only if u _i is adjacent to v _i in G for every i. We give a characterization of all independent sets in such graphs whenever G is connected and non-bipartite. Consider the stationary measure of the simple random walk on G ⁿ . We show that every independent set is almost contained with respect to this measure in a junta, a cylinder of constant co-dimension. Moreover we show that the projection of that junta defines a nearly independent set, i.e. it spans few edges (this also guarantees that it is not trivially the entire vertex-set). Our approach is based on an analog of Fourier analysis for product spaces combined with spectral techniques and on a powerful invariance principle presented in [MoOO]. This principle has already been shown in [DiMR] to imply that independent sets in such graph products have an influential coordinate. In this work we prove that in fact there is a set of few coordinates and a junta on them that capture the independent set almost completely.