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 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 an 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 $\phi$ defined on an $n$-dimensional local spectral expander, there exists a decomposition of $\phi$ into `orthogonal' parts that are, roughly speaking, the `projections' on the $j$-dimensional cochains for $0 \leq j \leq k$. The random walk shrinks each of these parts by a factor of $\frac{k+1-j}{k+2}$ plus an error term that depends on the spectral expansion. %Our two Decomposition Theorems differ in their assumptions on the local spectral gaps - we derive different Decomposition Theorems for the cases of one-sided local spectral gap and two-sided local spectral gap.