Mixing Time Of Near-Critical Random Graphs

Let \(C_1\) be the largest component of the Erd\H{o}s–R\'{e}nyi random graph \(G(n,p)\). The mixing time of random walk on \(C_1\) in the strictly supercritical regime, \(p=c/n\) with fixed \(c>1\), was shown to have order \({log}_2n\) by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald. In the critical window, \(p=(1+\epsilon )/n\) where \(\lambda ={\epsilon }_3n\) is bounded, Nachmias and Peres proved that the mixing time on \(C_1\) is of order \(n\). However, it was unclear how to interpolate between these results, and estimate the mixing time as the giant component emerges from the critical window. Indeed, even the asymptotics of the diameter of \(C_1\) in this regime were only recently obtained by Riordan and Wormald, as well as the present authors and Kim. In this paper, we show that for \(p=(1+\epsilon )/n\) with \(\lambda ={\epsilon }_3n\to \infty\) and \(\lambda =o(n)\), the mixing time on \(C_1\) is with high probability of order \((n/\lambda ){log}_2\lambda\). In addition, we show that this is the order of the largest mixing time over all components, both in the slightly supercritical and in the slightly subcritical regime [i.e., \(p=(1-\epsilon )/n\) with \(\lambda\) as above].