High order random walks: Beyond spectral gap

We study high order random walks on high dimensional expanders on simplicial complexes (i.e., hypergraphs). These walks walk from a k-face (i.e., a k-hyperedge) to a k-face if they are both contained in a k+1-face (i.e, a k+1 hyperedge). This naturally generalizes the random walks on graphs that walk from a vertex (0-face) to a vertex if they are both contained in an edge (1-face). Recent works have studied the spectrum of high order walks operators and deduced fast mixing. However, the spectral gap of high order walks operators is inherently small, due to natural obstructions (called coboundaries) that do not happen for walks on expander graphs. In this work we go beyond spectral gap, and relate the expansion of a function on k-faces (called k-cochain, for k = 0, this is a function on vertices) to its structure. We show a Decomposition Theorem: For every k-cochain defined on high dimensional expander, there exists a decomposition of the cochain into i-cochains such that the square norm of the k-cochain is a sum of the square norms of the i-chains and such that the more weight the k-cochain has on higher levels of the decomposition the better is its expansion, or equivalently, the better is its shrinkage by the high order random walk operator. The following corollaries are implied by the Decomposition Theorem: We characterize highly expanding k-cochains as those whose mass is concentrated on the highest levels of the decomposition that we construct. For example, a function on edges (i.e. a 1-cochain) which is locally thin (i.e. it contains few edges through every vertex) is highly expanding, while a function on edges that contains all edges through a single vertex is not highly expanding. We get optimal mixing for high order random walks on Ramanujan complexes. Ramanujan complexes are recently discovered bounded degree high dimensional expanders. The optimality in their mixing that we prove here, enable us to get from them more efficient Two-Layer-Samplers than those presented by the previous work of Dinur and Kaufman.