The Annals of Probability

Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees

Yueyun Hu and Zhan Shi

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We establish a second-order almost sure limit theorem for the minimal position in a one-dimensional super-critical branching random walk, and also prove a martingale convergence theorem which answers a question of Biggins and Kyprianou [Electron. J. Probab. 10 (2005) 609–631]. Our method applies, furthermore, to the study of directed polymers on a disordered tree. In particular, we give a rigorous proof of a phase transition phenomenon for the partition function (from the point of view of convergence in probability), already described by Derrida and Spohn [J. Statist. Phys. 51 (1988) 817–840]. Surprisingly, this phase transition phenomenon disappears in the sense of upper almost sure limits.

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Ann. Probab., Volume 37, Number 2 (2009), 742-789.

First available in Project Euclid: 30 April 2009

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Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Branching random walk minimal position martingale convergence spine marked tree directed polymer on a tree


Hu, Yueyun; Shi, Zhan. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37 (2009), no. 2, 742--789. doi:10.1214/08-AOP419.

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  • [1] Addario-Berry, L. (2006). Ballot theorems and the heights of trees. Ph.D. thesis, McGill Univ.
  • [2] Aldous, D. J. and Bandyopadhyay, A. (2005). A survey of max-type recursive distributional equations. Ann. Appl. Probab. 15 1047–1110.
  • [3] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Die Grundlehren der mathematischen Wissenschaften 196. Springer, New York.
  • [4] Bachmann, M. (2000). Limit theorems for the minimal position in a branching random walk with independent logconcave displacements. Adv. in Appl. Probab. 32 159–176.
  • [5] Biggins, J. D. (1976). The first- and last-birth problems for a multitype age-dependent branching process. Adv. in Appl. Probab. 8 446–459.
  • [6] Biggins, J. D. and Grey, D. R. (1979). Continuity of limit random variables in the branching random walk. J. Appl. Probab. 16 740–749.
  • [7] Biggins, J. D. and Kyprianou, A. E. (1997). Seneta-Heyde norming in the branching random walk. Ann. Probab. 25 337–360.
  • [8] Biggins, J. D. and Kyprianou, A. E. (2004). Measure change in multitype branching. Adv. in Appl. Probab. 36 544–581.
  • [9] Biggins, J. D. and Kyprianou, A. E. (2005). Fixed points of the smoothing transform: The boundary case. Electron. J. Probab. 10 609–631 (electronic).
  • [10] Bingham, N. H. (1973). Limit theorems in fluctuation theory. Adv. in Appl. Probab. 5 554–569.
  • [11] Bolthausen, E. (1989). A central limit theorem for two-dimensional random walks in random sceneries. Ann. Probab. 17 108–115.
  • [12] Bramson, M. D. (1978). Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31 531–581.
  • [13] Bramson, M. D. (1978). Minimal displacement of branching random walk. Z. Wahrsch. Verw. Gebiete 45 89–108.
  • [14] Bramson, M. D. and Zeitouni, O. (2009). Tightness for a family of recursion equations. Ann. Probab. 37 615–653.
  • [15] Chauvin, B., Rouault, A. and Wakolbinger, A. (1991). Growing conditioned trees. Stochastic Process. Appl. 39 117–130.
  • [16] Dekking, F. M. and Host, B. (1991). Limit distributions for minimal displacement of branching random walks. Probab. Theory Related Fields 90 403–426.
  • [17] Derrida, B. and Spohn, H. (1988). Polymers on disordered trees, spin glasses, and traveling waves. J. Statist. Phys. 51 817–840.
  • [18] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. Wiley, New York.
  • [19] Hammersley, J. M. (1974). Postulates for subadditive processes. Ann. Probab. 2 652–680.
  • [20] Hardy, R. and Harris, S. C. (2004). A new formulation of the spine approach to branching diffusions. Mathematics Preprint, Univ. Bath.
  • [21] Harris, T. E. (1963). The Theory of Branching Processes. Die Grundlehren der Mathematischen Wissenschaften 119. Springer, Berlin.
  • [22] Heyde, C. C. (1970). Extension of a result of Seneta for the super-critical Galton–Watson process. Ann. Math. Statist. 41 739–742.
  • [23] Kingman, J. F. C. (1975). The first birth problem for an age-dependent branching process. Ann. Probab. 3 790–801.
  • [24] Kozlov, M. V. (1976). The asymptotic behavior of the probability of non-extinction of critical branching processes in a random environment. Teor. Verojatnost. i Primenen. 21 813–825.
  • [25] Kyprianou, A. E. (1998). Slow variation and uniqueness of solutions to the functional equation in the branching random walk. J. Appl. Probab. 35 795–801.
  • [26] Lifshits, M. A. (2007). Some limit theorems on binary trees. (In preparation.)
  • [27] Liu, Q. S. (2000). On generalized multiplicative cascades. Stochastic Process. Appl. 86 263–286.
  • [28] Liu, Q. S. (2001). Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stochastic Process. Appl. 95 83–107.
  • [29] Lyons, R. (1997). A simple path to Biggins’ martingale convergence for branching random walk. In Classical and Modern Branching Processes (Minneapolis, MN, 1994). IMA Vol. Math. Appl. 84 217–221. Springer, New York.
  • [30] Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of L log L criteria for mean behavior of branching processes. Ann. Probab. 23 1125–1138.
  • [31] McDiarmid, C. (1995). Minimal positions in a branching random walk. Ann. Appl. Probab. 5 128–139.
  • [32] Neveu, J. (1986). Arbres et processus de Galton–Watson. Ann. Inst. H. Poincaré Probab. Statist. 22 199–207.
  • [33] Neveu, J. (1988). Multiplicative martingales for spatial branching processes. In Seminar on Stochastic Processes, 1987 (Princeton, NJ, 1987). Progr. Probab. Statist. 15 223–242. Birkhäuser Boston, Boston.
  • [34] Petrov, V. V. (1995). Limit Theorems of Probability Theory: Sequences of independent random variables. Oxford Studies in Probability 4. Clarendon, Oxford.
  • [35] Seneta, E. (1968). On recent theorems concerning the supercritical Galton–Watson process. Ann. Math. Statist. 39 2098–2102.