The cutoff phenomenon was recently confirmed for random walks on Ramanujan graphs by the first author and Peres. In this work, we obtain analogs in higher dimensions, for random walk operators on any Ramanujan complex associated with a simple group G over a local field F. We show that if T is any k-regular G-equivariant operator on the Bruhat-Tits building with a simple combinatorial property (collision-free), the associated random walk on the n-vertex Ramanujan complex has cutoff at time logk n. The high-dimensional case, unlike that of graphs, requires tools from non-commutative harmonic analysis and the infinite-dimensional representation theory of G. Via these, we show that operators T as above on Ramanujan complexes give rise to Ramanujan digraphs with a special property (r-normal), implying cutoff.
Bibliographical noteFunding Information:
Acknowledgments. The authors wish to thank Peter Sarnak for many helpful discussions. E.L. was supported in part by NSF grant DMS-1513403 and BSF grant 2014361. A.L. was supported by the ERC, BSF and NSF. O.P. was supported by ISF grant 1031/17.
© European Mathematical Society 2020
- Cutoff phenomenon
- High dimensional expanders
- Mixing time of random walk
- Ramanujan complexes