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Self-switching random walks on Erdös–Rényi random graphs feel the phase transition

Author

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  • Iacobelli, G.
  • Ost, G.
  • Takahashi, D.Y.

Abstract

We study random walks on Erdös–Rényi random graphs in which, every time the random walk returns to the starting point, first an edge probability is independently sampled according to a priori measure μ, and then an Erdös–Rényi random graph is sampled according to that edge probability. When the edge probability p does not depend on the size of the graph n (dense case), we show that the proportion of time the random walk spends on different values of p – occupation measure – converges to the a priori measure μ as n goes to infinity. More interestingly, when p=λ/n (sparse case), we show that the occupation measure converges to a limiting measure with a density that is a function of the survival probability of a Poisson branching process. This limiting measure is supported on the supercritical values for the Erdös–Rényi random graphs, showing that self-witching random walks can detect the phase transition.

Suggested Citation

  • Iacobelli, G. & Ost, G. & Takahashi, D.Y., 2025. "Self-switching random walks on Erdös–Rényi random graphs feel the phase transition," Stochastic Processes and their Applications, Elsevier, vol. 183(C).
    • Handle: RePEc:eee:spapps:v:183:y:2025:i:c:s0304414925000304
    DOI: 10.1016/j.spa.2025.104589
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    References listed on IDEAS

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    1. Garcia, Nancy Lopes & Palacios, José Luis, 2001. "On inverse moments of nonnegative random variables," Statistics & Probability Letters, Elsevier, vol. 53(3), pages 235-239, June.
    2. Matthias Löwe & Sara Terveer, 2023. "A Central Limit Theorem for the Mean Starting Hitting Time for a Random Walk on a Random Graph," Journal of Theoretical Probability, Springer, vol. 36(2), pages 779-810, June.
    3. Gallo, S. & Iacobelli, G. & Ost, G. & Takahashi, D.Y., 2022. "Self-Switching Markov Chains: Emerging dominance phenomena," Stochastic Processes and their Applications, Elsevier, vol. 143(C), pages 254-284.
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