Understanding random-walk dynamical phase coexistence through waiting times

We study the appearance of first-order dynamical phase transitions (DPTs) as "intermittent" coexisting phases in the fluctuations of random walks on graphs. We show that the diverging timescale leading to critical behavior is the waiting time to jump from one phase to another. This timescale is crucial for observing the system's relaxation to stationarity and demonstrate ergodicity of the system at criticality. We illustrate these results through three analytical examples which provide insights into random walks exploring random graphs.