Covering a graph with independent walks

Let $P$ be an irreducible and reversible transition matrix on a finite state space $V$ with invariant distribution $\pi$. We let $k$ chains start by choosing independent locations distributed according to $\pi$ and then they evolve independently according to $P$. Let $\tau_{\mathrm{cov}}(k)$ be the first time that every vertex of $V$ has been visited at least once by at least one chain and let $t_{\rm{cov}}(k)=\mathbb{E}[\tau_{\mathrm{cov}}(k)]$ with $t_{\rm{cov}}=t_{\rm{cov}}(1)$. We prove that $t_{\rm{cov}}(k)\lesssim t_{\rm{cov}}/k$. When $k\leq t_{\mathrm{cov}}/t_{\rm{rel}}$, where $t_{\rm{rel}}$ is the inverse of the spectral gap, we show that this bound is sharp. For $k\leq t_{\mathrm{cov}}/t_{\rm{mix}}$ with $t_{\rm{mix}}$ the total variation mixing time of $(P+I)/2$ we prove that $k \cdot \max_{x_1,\ldots,x_k}\mathbb{E}_{x_1,\ldots,x_k}[\tau_{\rm{cov}}(k)] \asymp t_{\rm{cov}}$.