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

The pegging algorithm is a method of generating large random regular graphs beginning with small ones. The ǫ-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 ǫ from its limiting distribution. We show that this ǫ-mixing time is not o(ǫ−1). This demonstrates that the upper bound O(ǫ−1) proved recently by the authors is essentially tight.