Analytical results for the distribution of first return times of random walks on random regular graphs

We present analytical results for the distribution of first return (FR) times of random walks (RWs) on random regular graphs (RRGs) consisting of $N$ nodes of degree $c \ge 3$. Starting from a random initial node $i$ at time $t=0$, at each time step $t \ge 1$ an RW hops into a random neighbor of its previous node. We calculate the distribution $P ( T_{\rm FR} = t )$ of first return times to the initial node $i$. We distinguish between first return trajectories in which the RW retrocedes its own steps backwards all the way back to the initial node $i$ and those in which the RW returns to $i$ via a path that does not retrocede its own steps. In the retroceding scenario, each edge that belongs to the RW trajectory is crossed the same number of times in the forward and backward directions. In the non-retroceding scenario the subgraph that consists of the nodes visited by the RW and the edges it has crossed between these nodes includes at least one cycle. In the limit of $N \rightarrow \infty$ the RRG converges towards the Bethe lattice. The Bethe lattice exhibits a tree structure, in which all the first return trajectories belong to the retroceding scenario. Moreover, in the limit of $N \rightarrow \infty$ the trajectories of RWs on RRGs are transient in the sense that they return to the initial node with probability $<1$. In this sense they resemble the trajectories of RWs on regular lattices of dimensions $d \ge 3$. The analytical results are found to be in excellent agreement with the results obtained from computer simulations.