Journal of Mathematical Sciences

, Volume 238, Issue 4, pp 348–365 | Cite as

Large Deviations for Level Sets of a Branching Brownian Motion and Gaussian Free Fields

  • E. AïdékonEmail author
  • Yueyun Hu
  • Zhan Shi

We study deviation probabilities for the number of high positioned particles in branching Brownian motion and confirm a conjecture of Derrida and Shi. We also solve the corresponding problem for the two-dimensional discrete Gaussian free field. Our method relies on an elementary inequality for inhomogeneous Galton–Watson processes.


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  1. 1.
    J. D. Biggins, “The growth and spread of the general branching random walk,” Ann. Appl. Probab., 5, 1008–1024 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    M. Biskup and O. Louidor, “On intermediate level sets of the two-dimensional discrete Gaussian free field” (2016) arXiv:1612.01424.Google Scholar
  3. 3.
    E. Bolthausen, J.-D. Deuschel, and G. Giacomin, “Entropic repulsion and the maximum of the two-dimensional harmonic crystal,” Ann. Probab., 29, 1670–1692 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A. Bovier, Gaussian Processes on Trees. From Spin Glasses to Branching Brownian Motion, Cambridge Univ. Press, Cambridge (2017).Google Scholar
  5. 5.
    M. D. Bramson, “Maximal displacement of branching Brownian motion,” Comm. Pure Appl. Math., 31, 531–581 (1978).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    M. D. Bramson, “Convergence of solutions of the Kolmogorov equation to travelling waves,” Mem. Amer. Math. Soc., 44, No. 285, (1983).Google Scholar
  7. 7.
    M. Bramson, J. Ding, and O. Zeitouni, “Convergence in law of the maximum of the two-dimensional discrete Gaussian free field,” Commun. Pure Appl. Math., 69, 62–123 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    B. Chauvin and A. Rouault, “KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees,” Probab. Theory Related Fields, 80, 299–314 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    O. Daviaud, “Extremes of the discrete two-dimensional Gaussian free field,” Ann. Probab., 34, 962–986 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    B. Derrida and Z. Shi, “Large deviations for the branching Brownian motion in presence of selection or coalescence,” J. Statist. Phys., 163, 1285–1311 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    R. A. Fisher, “The wave of advance of advantageous genes,” Ann. Human Genetics, 7, 355–369 (1937).zbMATHGoogle Scholar
  12. 12.
    A. N. Kolmogorov, I. Petrovskii, and N. Piskunov, “Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique,” Bull. Univ. Moscou Série internationale, Section A, Mathématiques et mécanique, 1, 1–25 (1937).Google Scholar
  13. 13.
    G. F. Lawler, Intersections of Random Walks, Birkhäuser, Boston (1991).CrossRefzbMATHGoogle Scholar
  14. 14.
    H. P. McKean, “Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov,” Comm. Pure Appl. Math., 28, 323–331 (1975). Erratum: 29, 553–554.Google Scholar
  15. 15.
    A. Rouault, “Large deviations and branching processes,” in: Proc. 9th International Summer School on Probability Theory and Mathematical Statistics (Sozopol, 1997), Pliska Studia Mathematica Bulgarica, 13, 15–38.Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.LPMA, Université Pierre et Marie CurieParisFrance
  2. 2.LAGA, Université Paris XIIIVilletaneuseFrance

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