Long-Range Percolation Mixing Time

We provide an estimate, sharp up to poly-logarithmic factors, of the asymptotic almost sure mixing time of the graph created by long-range percolation on the cycle of length N ($\Integer/N\Integer$). While it is known that the asymptotic almost sure diameter drops from linear to poly-logarithmic as the exponent s decreases below 2 [4, 9], the asymptotic almost sure mixing time drops from N2 only to Ns-1 (up to poly-logarithmic factors).