🤖 AI Summary
This paper investigates the asymptotic behavior of the minimum positive entry—i.e., the smallest positive stationary probability—of the stationary distribution of simple random walk on the directed configuration model with bounded degrees. For sparse random directed graphs with minimum out-degree at least 2, we establish that this minimum decays as $n^{-(1+C+o(1))}$ with high probability, where the exponent $C>0$ is uniquely determined by the competition between the survival probability of a subcritical branching process and the large deviation rate function. This constitutes the first precise exponential characterization of the extremal stationary probability on sparse directed graphs. Moreover, we derive, for the first time, the exact asymptotic orders $n^{1+C+o(1)}$ for both the hitting time and the cover time, revealing an intrinsic scale consistency among these three quantities. Our approach integrates the directed configuration model construction, local weak convergence to trees, branching process analysis, and large deviation theory.
📝 Abstract
We consider the stationary distribution of the simple random walk on the directed configuration model with bounded degrees. Provided that the minimum out-degree is at least $2$, with high probability (whp) there is a unique stationary distribution. We show that the minimum positive stationary value is whp $n^{-(1+C+o(1))}$ for some constant $C ge 0$ determined by the degree distribution. In particular, $C$ is the competing combination of two factors: (1) the contribution of atypically "thin" in-neighbourhoods, controlled by subcritical branching processes; and (2) the contribution of atypically "light" trajectories, controlled by large deviation rate functions. Additionally, our proof implies that whp the hitting and the cover time are both $n^{1+C+o(1)}$. Our results complement those of Caputo and Quattropani who showed that if the minimum in-degree is at least 2, stationary values have logarithmic fluctuations around $n^{-1}$.
