×

zbMATH — the first resource for mathematics

Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. (English) Zbl 1169.60021
Authors’ abstract: 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 J.D. Biggins and A.E. Kyprianou [Electron. J. Probab. 10, Paper No. 17, 609–631, electronic only (2005; Zbl 1110.60081)]. 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 B. Derrida and H. Spohn [J. Statist. Phys. 51, No. 5-6, 817–840 (1988; Zbl 1036.82522)]. Surprisingly, this phase transition phenomenon disappears in the sense of upper almost sure limits.

MSC:
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[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. · Zbl 1105.60012 · doi:10.1214/105051605000000142
[3] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Die Grundlehren der mathematischen Wissenschaften 196 . Springer, New York. · Zbl 0259.60002
[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. · Zbl 0973.60098 · doi:10.1239/aap/1013540028
[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. JSTOR: · Zbl 0339.60074 · doi:10.2307/1426138 · links.jstor.org
[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. JSTOR: · Zbl 0425.60069 · doi:10.2307/3213141 · links.jstor.org
[7] Biggins, J. D. and Kyprianou, A. E. (1997). Seneta-Heyde norming in the branching random walk. Ann. Probab. 25 337-360. · Zbl 0873.60062 · doi:10.1214/aop/1024404291
[8] Biggins, J. D. and Kyprianou, A. E. (2004). Measure change in multitype branching. Adv. in Appl. Probab. 36 544-581. · Zbl 1056.60082 · doi:10.1239/aap/1086957585
[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). · Zbl 1110.60081 · eudml:127254
[10] Bingham, N. H. (1973). Limit theorems in fluctuation theory. Adv. in Appl. Probab. 5 554-569. JSTOR: · Zbl 0273.60066 · doi:10.2307/1425834 · links.jstor.org
[11] Bolthausen, E. (1989). A central limit theorem for two-dimensional random walks in random sceneries. Ann. Probab. 17 108-115. · Zbl 0679.60028 · doi:10.1214/aop/1176991497
[12] Bramson, M. D. (1978). Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31 531-581. · Zbl 0361.60052 · doi:10.1002/cpa.3160310502
[13] Bramson, M. D. (1978). Minimal displacement of branching random walk. Z. Wahrsch. Verw. Gebiete 45 89-108. · Zbl 0373.60089 · doi:10.1007/BF00715186
[14] Bramson, M. D. and Zeitouni, O. (2009). Tightness for a family of recursion equations. Ann. Probab. 37 615-653. · Zbl 1169.60020 · doi:10.1214/08-AOP414
[15] Chauvin, B., Rouault, A. and Wakolbinger, A. (1991). Growing conditioned trees. Stochastic Process. Appl. 39 117-130. · Zbl 0747.60077 · doi:10.1016/0304-4149(91)90036-C
[16] Dekking, F. M. and Host, B. (1991). Limit distributions for minimal displacement of branching random walks. Probab. Theory Related Fields 90 403-426. · Zbl 0734.60074 · doi:10.1007/BF01193752
[17] Derrida, B. and Spohn, H. (1988). Polymers on disordered trees, spin glasses, and traveling waves. J. Statist. Phys. 51 817-840. · Zbl 1036.82522 · doi:10.1007/BF01014886
[18] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II , 2nd ed. Wiley, New York. · Zbl 0219.60003
[19] Hammersley, J. M. (1974). Postulates for subadditive processes. Ann. Probab. 2 652-680. · Zbl 0303.60044 · doi:10.1214/aop/1176996611
[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. · Zbl 0117.13002
[22] Heyde, C. C. (1970). Extension of a result of Seneta for the super-critical Galton-Watson process. Ann. Math. Statist. 41 739-742. · Zbl 0195.19201 · doi:10.1214/aoms/1177697127
[23] Kingman, J. F. C. (1975). The first birth problem for an age-dependent branching process. Ann. Probab. 3 790-801. · Zbl 0325.60079 · doi:10.1214/aop/1176996266
[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. · Zbl 0384.60058 · doi:10.1137/1121091
[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. · Zbl 0930.60066 · doi:10.1239/jap/1032438375
[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. · Zbl 1028.60087 · doi:10.1016/S0304-4149(99)00097-6
[28] Liu, Q. S. (2001). Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stochastic Process. Appl. 95 83-107. · Zbl 1058.60068 · doi:10.1016/S0304-4149(01)00092-8
[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. · Zbl 0897.60086
[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. · Zbl 0840.60077 · doi:10.1214/aop/1176988176
[31] McDiarmid, C. (1995). Minimal positions in a branching random walk. Ann. Appl. Probab. 5 128-139. · Zbl 0836.60089 · doi:10.1214/aoap/1177004832
[32] Neveu, J. (1986). Arbres et processus de Galton-Watson. Ann. Inst. H. Poincaré Probab. Statist. 22 199-207. · Zbl 0601.60082 · numdam:AIHPB_1986__22_2_199_0 · eudml:77276
[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. · Zbl 0652.60089
[34] Petrov, V. V. (1995). Limit Theorems of Probability Theory : Sequences of independent random variables. Oxford Studies in Probability 4 . Clarendon, Oxford. · Zbl 0826.60001
[35] Seneta, E. (1968). On recent theorems concerning the supercritical Galton-Watson process. Ann. Math. Statist. 39 2098-2102. · Zbl 0176.47603 · doi:10.1214/aoms/1177698037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.