Circulant graphs as an example of discrete quantum unique ergodicity

TL;DR

This paper explores quantum ergodicity properties in circulant graphs, revealing perfect equidistribution in some cases and limitations in others, especially for prime order graphs, with implications for spectral multiplicities.

Contribution

It demonstrates the existence of perfectly equidistributed eigenfunction bases in circulant graphs and identifies constraints for prime order 4-regular graphs, including eigenvalue multiplicity bounds.

Findings

Existence of perfectly equidistributed eigenfunction bases in circulant graphs.

No universal equidistribution for all eigenfunctions in prime order 4-regular circulant graphs.

Eigenvalue multiplicity is at most two for large prime order 4-regular circulant graphs.

Abstract

A discrete analog of quantum unique ergodicity was proved for Cayley graphs of quasirandom groups by Magee, Thomas and Zhao. They show that for large graphs there exist real orthonormal basis of eigenfunctions of the adjacency matrix such that quantum probability measures of the eigenfunctions put approximately the correct proportion of their mass on subsets of the vertices that are not too small. We investigate this property for Cayley graphs of cyclic groups (circulant graphs). We observe that there exist sequences of orthonormal eigenfunction bases which are perfectly equidistributed. However, for sequences of 4-regular circulant graphs of prime order, we show that there are no sequences of real orthonormal bases where all sequences of eigenfunctions equidistribute. To obtain this result, we also prove that, for large 4-regular circulant graphs of prime order, the maximum multiplicity of the eigenvalues of the adjacency matrix is two.