There are a lot of recent works on generalizing the spectral theory of graphs and graph partitioning to k-uniform hypergraphs. There have been two broad directions toward this goal. One generalizes the notion of graph conductance to hypergraph conductance [Louis, Makarychev - TOC'16; Chan, Louis, Tang, Zhang - JACM'18]. In the second approach, one can view a hypergraph as a simplicial complex and study its various topological properties [Linial, Meshulam - Combinatorica'06; Meshulam, Wallach - RSA'09; Dotterrer, Kaufman, Wagner - SoCG'16; Parzanchevski, Rosenthal - RSA'17] and spectral properties [Kaufman, Mass - ITCS'17; Dinur, Kaufman - FOCS'17; Kaufman, Openheim - STOC'18; Oppenheim - DCG'18; Kaufman, Openheim - Combinatorica'20].

In this work, we attempt to bridge these two directions of study by relating the spectrum of up-down walks and swap walks on the simplicial complex, a downward closed set system, to hypergraph expansion. More precisely, we study the simplicial complex obtained by downward closing the given hypergraph and random walks between its levels X(l), i.e., the sets of cardinality l. In surprising contrast to random walks on graphs, we show that the spectral gap of swap walks and up-down walks between level m and l with 1 < m ⩽ l cannot be used to infer any bounds on hypergraph conductance. Moreover, we show that the spectral gap of swap walks between X(1) and X(k-1) cannot be used to infer any bounds on hypergraph conductance. In contrast, we give a Cheeger-like inequality relating the spectra of walks between level 1 and l for any l ⩽ k to hypergraph expansion. This is a surprising difference between swaps walks and up-down walks!

Finally, we also give a construction to show that the well-studied notion of link expansion in simplicial complexes cannot be used to bound hypergraph expansion in a Cheeger-like manner.