Rate of Convergence of the Short Cycle Distribution in Random Regular Graphs Generated by Pegging

Abstract

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.