Some inequalities for reversible Markov chains and branching random walks via spectral optimization

Abstract: 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 $\mathrm{gap}$ . 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_{\mathrm{mix}}^{(\infty)} $ are of smaller order than the maximal hitting times $t_{\mathrm{hit}}$ iff the product of the spectral-gap and $t_{\mathrm{hit}}$ diverges, by establishing the inequality $t_{\mathrm{mix}}^{(\infty)} \le \frac{1}{\mathrm{gap}}\log(et_{\mathrm{hit}} \cdot \mathrm{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_{\mathrm{hit}} \gg \frac{1}{\mathrm{gap}} $ implies mean-field behavior for the coalescing time of coalescing random walks.