Random walks, equidistribution and graphical designs

Let $G=(V,E)$ be a d-regular graph on n vertices and let ${\mu }_0$ be a probability measure on V. The act of moving to a randomly chosen neighbor leads to a sequence of probability measures supported on V given by ${\mu }_{k+1}=A{D}^{-1}{\mu }_k$, where A is the adjacency matrix and D is the diagonal matrix of vertex degrees of G. Ordering the eigenvalues of $A{D}^{-1}$ as $1={\lambda }_1\ge \left|{\lambda }_2\right|\ge \dots \ge \left|{\lambda }_n\right|\ge 0$, it is well-known that the graphs for which $\left|{\lambda }_2\right|$ is small are those in which the random walk process converges quickly to the uniform distribution: for all initial probability measures ${\mu }_0$ and all $k\ge 0$,$\sum _{v\in V}{\left|{\mu }_k\right(v)-\frac{1}{n}|}^2\le {\lambda }_2^{2k}.$ One could wonder whether this rate can be improved for specific initial probability measures ${\mu }_0$. We show that if G is regular, then for any $1\le \ell \le n$, there exists a probability measure ${\mu }_0$ supported on at most ℓ vertices so that$\sum _{v\in V}{\left|{\mu }_k\right(v)-\frac{1}{n}|}^2\le {\lambda }_{\ell +1}^{2k}.$