High Order Random Walks: Beyond Spectral Gap

We study high order random walks in high dimensional expanders; namely, in complexes which are local spectral expanders. Recent works have studied the spectrum of high order walks and deduced fast mixing. However, the spectral gap of high order walks 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 shrinkage of a k -cochain by the walk operator, to its structure under the assumption of local spectral expansion. A simplicial complex is called a one-sided local spectral expander , if its links have large spectral gaps and a two-sided local spectral expander if its links have large two-sided spectral gaps. We show two Decomposition Theorems (one per one-sided/two-sided local spectral assumption): For every k -cochain ϕ defined on an n -dimensional local spectral expander, there exists a decomposition of ϕ into “orthogonal” parts that are, roughly speaking, the “projections” on the j -dimensional cochains for 0 ≤ j ≤ k . The random walk shrinks each of these parts by a factor of $$\frac{k+1-j}{k+2}$$ k + 1 − j k + 2 plus an error term that depends on the spectral expansion. When assuming one-sided local spectral gap, our Decomposition Theorem yields an optimal mixing for the high order random walk operator. Namely, negative eigenvalues of the links do not matter! This improves over [5] that assumed two-sided spectral gap in the links to get optimal mixing. This improvement is crucial in a recent breakthrough [1] proving a conjecture of Mihail and Vazirani. Additionally, we get an optimal mixing for high order random walks on Ramanujan complexes (whose links are one-sided local spectral expanders). When assuming two-sided local spectral gap, our Decomposition Theorem allows us to describe the whole spectrum of the random walk operator (up to an error term that is determined by the spectral gap) and give an explicit orthogonal decomposition of the spaces of cochains that approximates the decomposition to eigenspaces of the random walk operator.