  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.
