Mixing time of PageRank surfers on sparse random digraphs

We consider the generalized PageRank walk on a digraph G, with refresh probability alpha and resampling distribution lambda. We analyze convergence to stationarity when G is a large sparse random digraph with given degree sequences, in the limit of vanishing alpha. We identify three scenarios: when alpha is much smaller than the inverse of the mixing time of G the relaxation to equilibrium is dominated by the simple random walk and displays a cutoff behavior; when alpha is much larger than the inverse of the mixing time of G on the contrary one has pure exponential decay with rate alpha; when alpha is comparable to the inverse of the mixing time of G there is a mixed behavior interpolating between cutoff and exponential decay. This trichotomy is shown to hold uniformly in the starting point and uniformly in the resampling distribution lambda.