Random walk hitting times and effective resistance in sparsely connected Erdős-Rényi random graphs

We prove expectation and concentration results for the following random variables on an Erd\H{o}s-R\'enyi random graph $\mathcal{G}\left(n,p\right)$ in the sparsely connected regime $\log n + \log\log \log n \leq np < n^{1/10}$: effective resistances, random walk hitting and commute times, the Kirchoff index, cover cost, random target times, the mean hitting time and Kemeny's constant. For the effective resistance between two vertices our concentration result extends further to $np\geq c\log n, \; c>0$. To achieve these results, we show that a strong connectedness property holds with high probability for $\mathcal{G}(n,p)$ in this regime.EPSRC, Grant/Award Number: EP/ HO23364/1; ERC, Grant/Award Numbers: 639046 (RGGC), 679660 (DYNAMIC MARCH