Sedentary quantum walks

Let $X$ be a graph with adjacency matrix $A$. The \textsl{continuous quantum walk} on $X$ is determined by the unitary matrices $U(t)=\exp(itA)$. If $X$ is the complete graph $K_n$ and $a\in V(X)$, then \[1-|U(t)_{a,a}|\le2/n. \] In a sense, this means that a quantum walk on a complete graph stay home with high probability. In this paper we consider quantum walks on cones over an $\ell$-regular graph on $n$ vertices. We prove that if $\ell^2/n\to\infty$ as $n$ increases, than a quantum walk that starts on the apex of the cone will remain on it with probability tending to $1$ as $n$ increases. On the other hand, if $\ell\le2$ we prove that there is a time $t$ such that local uniform mixing occurs, i.e., all vertices are equally likely. We investigate when a quantum walk on strongly regular graph has a high probability of "staying at home", producing large families of examples with the stay-at-home property where the valency is small compared to the number of vertices.