Rate of convergence of the short cycle distribution in random regular graphs generated by pegging (Q1028821)

Summary: The pegging algorithm is a method of generating large random regular graphs beginning with small ones. The \(\epsilon\)-mixing time of the distribution of short cycle counts of these random regular graphs is the time at which the distribution reaches and maintains total variation distance at most \(\epsilon\) from its limiting distribution. We show that this \(\epsilon\)-mixing time is not \(o(\epsilon^{-1})\). This demonstrates that the upper bound \(O(\epsilon^{-1})\) proved recently by the authors is essentially tight.