Mixing time of PageRank surfers on sparse random digraphs

We consider the generalized PageRank walk on a digraph G, with refresh probability (Formula presented.) and resampling distribution (Formula presented.). We analyze convergence to stationarity when G is a large sparse random digraph with given degree sequences, in the limit of vanishing (Formula presented.). We identify three scenarios: when (Formula presented.) 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 (Formula presented.) is much larger than the inverse of the mixing time of G on the contrary one has pure exponential decay with rate (Formula presented.); when (Formula presented.) 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 (Formula presented.).