CUTOFF AT THE ENTROPIC TIME FOR RANDOM WALKS ON COVERED EXPANDER GRAPHS

It is a fact simple to establish that the mixing time of random walk on d-regular graph $G_n$ with n vertices asymptotically bounded from below by $d/ ((d-2)\log (d-1))\log n$. Such bound obtained comparing infinite $d$-regular tree. If one can map another transitive onto $G_n$, then we improve strategy using comparison this (instead regular tree), and obtain lower form $1/h \log n$, where $h$ entropy rate associated graph. We call entropic bound. was recently proved in case tree, sharp when graphs have minimal spectral radius thus exhibit cutoff at time. In paper, provide generalization result providing sufficient condition spectra walks under which applies notably anisotropic $n$-lifts base (including non-reversible walks).