Year: 2026
Type: article
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 (uniqueness regime). We show that the minimum positive stationary value is whp n−(1+C+o(1)) for some constant C≥0 determined by the degree distribution, answering a question raised by Bordenave, Caputo and Salez (Probab. Theory Related Fields 170 (2018) 933–960). 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, we give estimates for the expected lower tail of the empirical stationary distribution. As a by-product of our proof, we obtain that the hitting and the cover time are both n1+C+o(1) whp. Our results are in sharp contrast to those of Caputo and Quattropani (Probab. Theory Related Fields 178 (2020) 1011–1066) who showed that under the additional condition of minimum in-degree at least 2 (ergodicity regime), stationary values only have logarithmic fluctuations around n−1.
DOI: 10.1214/25-aap2247
Authors: Xing Shi Cai, Guillem Perarnau
Institutions: Duke Kunshan University, Universitat Politècnica de Catalunya
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Agencia Estatal de Investigación
OpenAlex ID: W7130353370