Self-switching random walks on Erdös–Rényi random graphs feel the phase transition

Abstract

We study random walks on Erdös–Rényi random graphs in which, every time the random walk returns to the starting point, first an edge probability is independently sampled according to a priori measure $\mu$, and then an Erdös–Rényi random graph is sampled according to that edge probability. When the edge probability $p$ does not depend on the size of the graph $n$ (dense case), we show that the proportion of time the random walk spends on different values of $p$ – occupation measure – converges to the a priori measure $\mu$ as $n$ goes to infinity. More interestingly, when $p=\lambda /n$ (sparse case), we show that the occupation measure converges to a limiting measure with a density that is a function of the survival probability of a Poisson branching process. This limiting measure is supported on the supercritical values for the Erdös–Rényi random graphs, showing that self-witching random walks can detect the phase transition.