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 $p$ -- {\it 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 supercritial values for the Erdös-Rényi random graphs, showing that self-witching random walks can detect the phase transition.