Abstract

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 $$\frac{k+1-j}{k+2}$$ 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.

Highlights

  • 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

  • In this work we study high order random walks on bounded degree simplicial complexes

  • 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)

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Summary

Introduction

In this work we study high order random walks on bounded degree simplicial complexes (i.e., hypergraphs). The focus of previous works [8, 3] was to bound the second largest eigenvalue (in an absolute value) of the high order walk operator in complexes whose links are good spectral expanders. Previous works have shown that in complexes with links that are good spectral expanders, every k-cochain that is orthogonal to the constant functions is shrinked by the k-order random walk operator Mk+. The focus of this work is to relate the structure of a k-cochain φ to its expansion, or equivalently, to the amount of its shrinkage by the random walk operator Mk+, in complexes that are local spectral expanders. We derive an optimal bound on the second eigenvalue (in an absolute value) of the high order random walk operator Mk+ of complexes whose links are good one sided spectral expanders. In the following we discuss the notion of two layers sampler (that was introduced by [3]), and discuss the more efficient two-layers-samplers that we get in this work

More efficient Two-Layer-Samplers implied by this work
On simplicial complexes and localization
A decomposition theorem for high order random walks and its implications
Definitions and notations
Upper and lower random walks
The signless differential
Links and localization
Decomposition theorem for upper random walks
A Proofs of localization results
Full Text
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