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Minimum stationary values of sparse random directed graphs

Xing Shi Cai & Guillem Córdoba Perarnau

Annals of Applied Probability2026https://doi.org/10.1214/25-aap2247article
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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.

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https://doi.org/https://doi.org/10.1214/25-aap2247

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@article{xing2026,
  title        = {{Minimum stationary values of sparse random directed graphs}},
  author       = {Xing Shi Cai & Guillem Córdoba Perarnau},
  journal      = {Annals of Applied Probability},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1214/25-aap2247},
}

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