Summary: We present results relating mixing times to the intersection time of branching random walk (BRW) in which the logarithm of the expected number of particles grows at rate of the spectral-gap \(\text{gap}\). This is a finite state space analog of a critical branching process. Namely, we show that the maximal expected hitting time of a state by such a BRW is up to a universal constant larger than the \({L_{\infty }}\) mixing-time, whereas under transitivity the same is true for the intersection time of two independent such BRWs.
Using the same methodology, we show that for a sequence of reversible Markov chains, the \({L_{\infty }}\) mixing-times \({t_{\text{mix}}^{(\infty )}}\) are of smaller order than the maximal hitting times \({t_{\text{hit}}}\) iff the product of the spectral-gap and \({t_{\text{hit}}}\) diverges, by establishing the inequality \({t_{\text{mix}}^{(\infty )}}\le \frac{1}{\text{gap}}\log (e{t_{\text{hit}}}\cdot \text{gap})\). This resolves a conjecture of Aldous and Fill “Reversible Markov chains and random walks on graphs” Open Problem 14.12 asserting that under transitivity the condition that \({t_{\text{hit}}}\gg \frac{1}{\text{gap}}\) implies mean-field behavior for the coalescing time of coalescing random walks.