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On the Variance of the Particle Number of a Supercritical Branching Random Walk on Periodic Graphs

We study the asymptotic behavior of the variance of the local particle number for a supercritical branching random walk with branching sources that are located periodically on Zd.

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  1. 1.

    G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Amer. Math. Soc. (2013).

  2. 2.

    M. Bessonov, S. Molchanov, and J. Whitmeyer, “A multi-class extension of the mean field Bolker–Pacala population model,” Random Oper. Stoch. Equ., 26, 163–174 (2018).

    MathSciNet  Article  Google Scholar 

  3. 3.

    D. M. Cvetkovic, M. Doob, and M. Sachs, Spectra of Graphs, Academic Press, New York (1980).

    MATH  Google Scholar 

  4. 4.

    J. Gärtner and S. A. Molchanov, “Parabolic problems for the Anderson model. I. Intermittency and related topics,” Comm. Math. Phys., 132, 613–655 (1990).

    MathSciNet  Article  Google Scholar 

  5. 5.

    Y. Higuchi and Y. Nomura, “Spectral structure of the Laplacian on a covering graph,” Eur. J. Combin., 30, 570–585 (2009).

    MathSciNet  Article  Google Scholar 

  6. 6.

    I. I. Khristolyubov and E. B. Yarovaya, “A limit theorem for a supercritical branching walk with sources of varying intensity,” Teor. Veroyatn. Primen., 64, 456–480 (2019).

    MathSciNet  Article  Google Scholar 

  7. 7.

    M. Kimmel and D. E. Axelrod, Branching Processed in Biology, Interdisciplinary Applied Mathematics (2002).

  8. 8.

    E. Korotyaev and N. Saburova, “Schrödinger operators on periodic discrete graphs,” J. Math. Anal. Appl., 420, 576–611 (2014).

    MathSciNet  Article  Google Scholar 

  9. 9.

    B. G. Malkiel and K. McCue, A Random Walk Down Wall Street, Norton, New York (1985).

    Google Scholar 

  10. 10.

    J. J. Molitierno, Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs, Chapman and Hall/CRC (2016).

  11. 11.

    M. Reed and B. Simon, Analysis of Operators, Vol. IV, Elsevier (1978).

  12. 12.

    V. Topchii, V. Vatutin, and E. Yarovaya, “Catalytic branching random walk and queueing systems with random number of independent ververs,” Theory Probab. Math. Statist., 69, 1–15 (2004).

    Google Scholar 

  13. 13.

    L. V. Bogachev and E. B. Yarovaya, “Moment analysis of a branching random walk on a lattice with a single source,” Dokl. Ros. Akad. Nauk, 363, 439–442 (1998).

    MathSciNet  MATH  Google Scholar 

  14. 14.

    E. Vl. Bulinskaya, “Catalytic branding random walk on two-dimensional lattice,” Teor. Veroyatn. Primen., 55, 142–148 (2010).

  15. 15.

    E. Vl. Bulinskaya, “Complete classification of catalytic branching processes,” Teor. Veroyatn. Primen., 59, 639–666 (2014).

  16. 16.

    V. A. Vatutin and V. A. Topchii, “A limit theorem for critical catalytic branching random walks,” Teor. Veroyatn. Primen., 49, 461–484 (2004).

    MathSciNet  Article  Google Scholar 

  17. 17.

    V. A. Vatutin and V. A. Topchii, “Catalytic branching random walks on ℤd with branching at the origin,” Mat. Tr., 14, 28–72 (2011).

    MATH  Google Scholar 

  18. 18.

    V. A. Vatutin, V. A. Topchii, and Y. Khu, “A branching random walk in the lattice ℤ4 with brancting at only the origin,” Teor. Veroyatn. Primen., 56, 224–247 (2011).

    Article  Google Scholar 

  19. 19.

    I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes [in Russian], Moscow (1977).

  20. 20.

    M. V. Platonova and K. S. Ryadovkin, “Branching random walks on ℤd with periodically distributed branching sources,” Teor. Veroyatn. Primen., 64, 283–307 (2019).

    MathSciNet  Article  Google Scholar 

  21. 21.

    M. V. Fedoryuk, The Saddle-point Method [in Russian], Moscow (1977).

  22. 22.

    E. B. Yarovaya, Branching Random Walks in Intromogeneous Media [in Russian], Moscow Univ. Publ. (2007).

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Correspondence to M. V. Platonova.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 486, 2019, pp. 233–253.

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Platonova, M.V., Ryadovkin, K.S. On the Variance of the Particle Number of a Supercritical Branching Random Walk on Periodic Graphs. J Math Sci 258, 897–911 (2021).

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