Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation

Probability Surveys

Volumn 9, Issue 1, 2012, Pages 103-252

  Janson, Svante ^a  

^a NONE

Author keywords

Balls in boxes; Galton Watson trees; Random allocations; Random forests; Random trees; Simply generated trees; Size biased Galton Watson tree

Indexed keywords

EID: 84877762771     PISSN: None     EISSN: 15495787     Source Type: Journal    
DOI: 10.1214/11-PS188     Document Type: Article

References (109)

1. L. Addario-Berry, L. Devroye & S. Janson, Sub-Gaussian tail bounds for the width and height of conditioned Galton-Watson trees. Ann. Probab., to appear. arXiv:1011.4121
2. D. Aldous, Asymptotic fringe distributions for general families of random trees. Ann. Appl. Probab. 1 (1991), no. 2, 228-266. MR1102319
3. D. Aldous, The continuum random tree I. Ann. Probab. 19 (1991), no. 1, 1-28. MR1085326
4. D. Aldous, The continuum random tree II: an overview. Stochastic Anal- ysis (Durham, 1990), 23-70, London Math. Soc. Lecture Note Ser. 167, Cambridge Univ. Press, Cambridge, 1991. MR1166406
5. D. Aldous, The continuum random tree III. Ann. Probab. 21 (1993), no. 1, 248-289. MR1207226
6. D. Aldous & J. Pitman, Tree-valued Markov chains derived from Galton-Watson processes. Ann. Inst. H. Poincaré Probab. Statist. 34 (1998), no. 5, 637-686. MR1641670 Simply generated trees and random allocations 247
7. R. Arratia, A. D. Barbour & S. Tavaré, Logarithmic Combinatorial Structures: a Probabilistic Approach, EMS, Zürich, 2003. MR2032426
8. K. B. Athreya & P. E. Ney, Branching Processes. Springer-Verlag, Berlin, 1972. MR0373040
9. M. T. Barlow & T. Kumagai, Random walk on the incipient infinite cluster on trees. Illinois J. Math. 50 (2006), no. 1-4, 33-65. MR2247823
10. D. Beihoffer, J. Hendry, A. Nijenhuis & S. Wagon, Faster algorithms for Frobenius numbers. Electron. J. Combin. 12 (2005), R27. MR2156681
11. J. Bennies & G. Kersting, A random walk approach to Galton-Watson trees. J. Theoret. Probab. 13 (2000), no. 3, 777-803. MR1785529
12. E. S. Bernikovich & Yu. L. Pavlov, On the maximum size of a tree in a random unlabelled unrooted forest. Diskret. Mat. 23 (2011), no. 1, 3-20 (Russian). English transl.: Discrete Math. Appl. 21 (2011), no. 1, 1-21. MR2830693
13. P. Bialas & Z. Burda, Phase transition in fluctuating branched geometry. Physics Letters B 384 (1996), 75-80. MR1410422
14. P. Bialas, Z. Burda & D. Johnston, Condensation in the backgammon model. Nuclear Physics 493 (1997), 505-516.
15. P. Billingsley, Convergence of Probability Measures. Wiley, New York, 1968. MR0233396
16. N. H. Bingham, C. M. Goldie & J. L. Teugels, Regular Variation. Cambridge Univ. Press, Cambridge, 1987. MR0898871
17. C. W. Borchardt, Ueber eine der Interpolation entsprechende Darstellung der Eliminations-Resultante. J. reine und angewandte Mathematik 57 (1860), 111-121.
18. é. Borel, Sur l'emploi du théoréme de Bernoulli pour faciliter le calcul d'une infinité de coefficients. Application au probléme de l'attente á un guichet. C. R. Acad. Sci. Paris 214 (1942), 452-456. MR0008126
19. A. V. Boyd, Formal power series and the total progeny in a branching process. J. Math. Anal. Appl. 34 (1971), 565-566. MR0281270
20. V. E. Britikov, Asymptotic number of forests from unrooted trees. Mat. Zametki 43 (1988), no. 5, 672-684, 703 (Russian). English transl.: Math. Notes 43 (1988), no. 5-6, 387-394. MR0954351
21. R. Carr, W. M. Y. Goh & E. Schmutz, The maximum degree in a random tree and related problems. Random Struct. Alg. 5 (1994), no. 1, 13-24. MR1248172
22. A. Cayley, A theorem on trees. Quart. J. Math. 23 (1889), 376-378.
23. P. Chassaing & B. Durhuus, Local limit of labeled trees and expected volume growth in a random quadrangulation. Ann. Probab. 34 (2006), no. 3, 879-917. MR2243873
24. P. Chassaing & G. Louchard, Phase transition for parking blocks, Brownian excursion and coalescence. Random Struct. Alg. 21 (2002), no. 1, 76-119. MR1913079 248 S. Janson
25. P. Chassaing, J.-F. Marckert & M. Yor, The height and width of simple trees. Mathematics and Computer Science (Versailles, 2000), 17-30, Trends Math., Birkhäuser, Basel, 2000. MR1798284
26. R. M. Corless, G. H. Gonnet, D. E. Hare, D. J. Jeffrey & D. E. Knuth, On the Lambert W function. Adv. Comput. Math. 5 (1996), no. 4, 329-359. MR1414285
27. H. Cramér, Sur un noveau théoréme-limite de la théorie des probabilit és. Les sommes et les fonctions de variables aléatoires, Actualités Scientifiques et Industrielles 736, Hermann, Paris, 1938, pp. 5-23.
28. D. Croydon, Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008), no. 6, 987-1019. MR2469332
29. D. Croydon, Scaling limits for simple random walks on random ordered graph trees. Adv. Appl. Probab. 42 (2010), no. 2, 528-558. MR2675115
30. D. Croydon & T. Kumagai, Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive. Elec- tron. J. Probab. 13 (2008), no. 51, 1419-1441. MR2438812
31. A. Dembo & O. Zeitouni, Large Deviations Techniques and Applica- tions. 2nd ed., Springer, New York, 1998. MR1619036
32. L. Devroye, Branching processes and their applications in the analysis of tree structures and tree algorithms. Probabilistic Methods for Algorithmic Discrete Mathematics, eds. M. Habib, C. McDiarmid, J. Ramirez and B. Reed, Springer, Berlin, 1998, pp. 249-314. MR1678582
33. M. Drmota, Random Trees, Springer, Vienna, 2009. MR2484382
34. T. Duquesne, A limit theorem for the contour process of conditioned Galton-Watson trees. Ann. Probab. 31 (2003), no. 2, 996-1027. MR1964956
35. B. Durhuus, T. Jonsson & J. F. Wheater, The spectral dimension of generic trees. J. Stat. Phys. 128 (2007), 1237-1260. MR2348795
36. M. Dwass, The total progeny in a branching process and a related random walk. J. Appl. Probab. 6 (1969), 682-686. MR0253433
37. F. Eggenberger & G. Pólya, Über die Statistik verketteter Vorgänge. Zeitschrift Angew. Math. Mech. 3 (1923), 279-289.
38. W. Feller, An Introduction to Probability Theory and its Applications, Volume I, 2nd ed., Wiley, New York, 1957. MR0088081
39. W. Feller, An Introduction to Probability Theory and its Applications, Volume II, 2nd ed., Wiley, New York, 1971. MR0270403
40. P. Flajolet & R. Sedgewick, Analytic Combinatorics. Cambridge Univ. Press, Cambridge, UK, 2009. MR2483235
41. S. Franz & F. Ritort, Dynamical solution of a model without energy barriers. Europhysics Letters 31 (1995), 507-512
42. S. Franz & F. Ritort, Glassy mean-field dynamics of the backgammon model. J. Stat. Phys. 85 (1996), 131-150.
43. I. Fujii & T. Kumagai, Heat kernel estimates on the incipient infinite cluster for critical branching processes. Proceedings of German-Japanese Symposium in Kyoto 2006, RIMS Kôkyûroku Bessatsu B6 (2008), pp. 8-95. MR2407556
44. J. Geiger, Elementary new proofs of classical limit theorems for Galton- Watson processes. J. Appl. Probab. 36 (1999), no. 2, 301-309.MR1724856
45. J. Geiger & L. Kauffmann, The shape of large Galton-Watson trees with possibly infinite variance. Random Struct. Alg. 25 (2004), no. 3, 311-335. MR2086163
46. B. V. Gnedenko & A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1949 (Russian). English transl.: Addison-Wesley, Cambridge, Mass., 1954. MR0062975
47. G. R. Grimmett, Random labelled trees and their branching networks. J. Austral. Math. Soc. Ser. A 30 (1980/81), no. 2, 229-237. MR0607933
48. G. R. Grimmett, The Random-Cluster Model, Springer, Berlin, 2006. MR2243761
49. A. Gut, Probability: A Graduate Course. Springer, New York, 2005. MR2125120
50. G. H. Hardy, J. E. Littlewood & G. Pólya, Inequalities. 2nd ed., Cambridge, at the University Press, 1952. MR0046395
51. T. E. Harris, A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc. 56 (1960), 13-20. MR0115221
52. L. Holst, Two conditional limit theorems with applications. Ann. Statist. 7 (1979), no. 3, 551-557. MR0527490
53. L. Holst, A unified approach to limit theorems for urn models. J. Appl. Probab. 16 (1979), 154-162. MR0520945
54. I. A. Ibragimov & Yu. V. Linnik, Independent and Stationary Se- quences of Random Variables. Nauka, Moscow, 1965 (Russian). English transl.: Wolters-Noordhoff Publishing, Groningen, 1971. MR0322926
55. S. Janson, Moment convergence in conditional limit theorems. J. Appl. Probab. 38 (2001), no. 2, 421-437. MR1834751
56. S. Janson, Asymptotic distribution for the cost of linear probing hashing. Random Struct. Alg. 19 (2001), no. 3-4, 438-471. MR1871562
57. S. Janson, Cycles and unicyclic components in random graphs. Combin. Probab. Comput. 12 (2003), 27-52. MR1967484
58. S. Janson, Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl. 110 (2004), no. 2, 177-245. MR2040966
59. S. Janson, Random cutting and records in deterministic and random trees. Random Struct. Alg. 29 (2006), no. 2, 139-179. MR2245498
60. S. Janson, Rounding of continuous random variables and oscillatory asymptotics. Ann. Probab. 34 (2006), no. 5, 1807-1826. MR2271483
61. S. Janson, On the asymptotic joint distribution of height and width in random trees, Studia Sci. Math. Hungar. 45 (2008), no. 4, 451-467. MR2641443 250 S. Janson
62. S. Janson, Probability asymptotics: notes on notation. Institute Mittag- Leffler Report 12, 2009 spring. arXiv:1108.3924 [63] S. Janson, Stable distributions. Unpublished notes, 2011. arXiv:1112.0220
63. S. Janson, Stable distributions. Unpublished notes, 2011. arXiv:1112.0220
64. S. Janson, T. Jonsson & S. Ö. Stefánsson, Random trees with superexponential branching weights. J. Phys. A: Math. Theor. 44 (2011), 485002.
65. S. Janson, T. Luczak & A. Ruciński, Random Graphs. Wiley, New York, 2000. MR1782847
66. N. L. Johnson & S. Kotz, Urn Models and their Application. Wiley, New York, 1977. MR0488211
67. T. Jonsson & S. Ö. Stefánsson, Condensation in nongeneric trees. J. Stat. Phys. 142 (2011), no. 2, 277-313. MR2764126
68. O. Kallenberg, Random Measures. Akademie-Verlag, Berlin, 1983. MR0818219
69. O. Kallenberg, Foundations of Modern Probability. 2nd ed., Springer, New York, 2002. MR1876169
70. N. I. Kazimirov, On some conditions for absence of a giant component in the generalized allocation scheme. Diskret. Mat. 14 (2002), no. 2, 107-118 (Russian). English transl.: Discrete Math. Appl. 12 (2002), no. 3, 291-302. MR1937012
71. N. I. Kazimirov, Emergence of a giant component in a random permutation with a given number of cycles. Diskret. Mat. 15 (2003), no. 3, 145-159 (Russian). English transl.: Discrete Math. Appl. 13 (2003), no. 5, 523-535. MR2021211
72. N. I. Kazimirov & Yu. L. Pavlov, A remark on the Galton-Watson forests. Diskret. Mat. 12 (2000), no. 1, 47-59 (Russian). English transl.: Discrete Math. Appl. 10 (2000), no. 1, 49-62. MR1778765
73. D. P. Kennedy, The Galton-Watson process conditioned on the total progeny. J. Appl. Probab. 12 (1975), 800-806. MR0386042
74. H. Kesten, Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 4, 425-487. MR0871905
75. D. E. Knuth, The Art of Computer Programming. Vol. 3: Sorting and Searching. 2nd ed., Addison-Wesley, Reading, Mass., 1998. MR0445948
76. V. F. Kolchin, Random Mappings. Nauka, Moscow, 1984 (Russian). English transl.: Optimization Software, New York, 1986. MR0865130 [77] V. F. Kolchin, B. A. Sevast'yanov & V. P. Chistyakov, Random Allocations. Nauka, Moscow, 1976 (Russian). English transl.: Winston, Washington, D.C., 1978. MR0471016
77. V. F. Kolchin, B. A. Sevast'yanov & V. P. Chistyakov, Random Allocations. Nauka, Moscow, 1976 (Russian). English transl.: Winston, Washington, D.C., 1978. MR0471016
78. T. Kurtz, R. Lyons, R. Pemantle & Y. Peres, A conceptual proof of the Kesten-Stigum Theorem for multi-type branching processes. Classi- cal and Modern Branching Processes (Minneapolis, MN, 1994), IMA Vol. Math. Appl., 84, Springer, New York, 1997, pp. 181-185. MR1601737 Simply generated trees and random allocations 251
79. J.-L. Lagrange, Nouvelle méthode pour résoudre les équations littérales par le moyen des séries. Mémoires de l'Académie royale des Sciences et Belles-Lettres de Berlin, XXIV (1770), 5-73.
80. J.-F. Le Gall, Random trees and applications. Probab. Surveys 2 (2005), 245-311. MR2203728
81. J.-F. Le Gall, Random real trees. Ann. Fac. Sci. Toulouse Math. (6) 15 (2006), no. 1, 35-62. MR2225746
82. M. R. Leadbetter, G. Lindgren & H. Rootzén, Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York, 1983. MR0691492
83. T. Luczak & B. Pittel, Components of random forests. Combin. Probab. Comput. 1 (1992), no. 1, 35-52. MR1167294
84. R. Lyons, R. Pemantle & Y. Peres, Conceptual proofs of L logL criteria for mean behavior of branching processes. Ann. Probab. 23 (1995), no. 3, 1125-1138. MR1349164
85. A. Meir & J. W. Moon, On the altitude of nodes in random trees. Canad. J. Math., 30 (1978), 997-1015. MR0506256
86. A. Meir & J. W. Moon, On the maximum out-degree in random trees. Australas. J. Combin. 2 (1990), 147-156. MR1126991
87. A. Meir & J. W. Moon, On nodes of large out-degree in random trees. Congr. Numer. 82 (1991), 3-13. MR1152053
88. A. Meir & J. W. Moon, A note on trees with concentrated maximum degrees. Utilitas Math. 42 (1992), 61-64. Coorigendum: Utilitas Math. 43 (1993), 253. MR1199088
89. N. Minami, On the number of vertices with a given degree in a Galton- Watson tree. Adv. Appl. Probab. 37 (2005), no. 1, 229-264. MR2135161
90. J. W. Moon, On the maximum degree in a random tree. Michigan Math. J. 15 (1968), 429-432. MR0233729
91. J. Neveu, Arbres et processus de Galton-Watson. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 2, 199-207. MR0850756
92. R. Otter, The number of trees. Ann. of Math. (2) 49 (1948), 583-599. MR0025715
93. R. Otter, The multiplicative process. Ann. Math. Statistics 20 (1949), 206-224. MR0030716
94. Yu. L. Pavlov, The asymptotic distribution of maximum tree size in a random forest. Teor. Verojatnost. i Primenen. 22 (1977), no. 3, 523-533 (Russian). English transl.: Th. Probab. Appl. 22 (1977), no. 3, 509-520. MR0461619
95. Yu. L. Pavlov, The limit distributions of the maximum size of a tree in a random forest. Diskret. Mat. 7 (1995), no. 3, 19-32 (Russian). English transl.: Discrete Math. Appl. 5 (1995), no. 4, 301-315. MR1361491
96. Yu. L. Pavlov, Random Forests. Karelian Centre Russian Acad. Sci., Petrozavodsk, 1996 (Russian). English transl.: VSP, Zeist, The Netherlands, 2000. MR1651128 252 S. Janson
97. Yu. L. Pavlov, Limit theorems on sizes of trees in a random unlabelled forest. Diskret. Mat. 17 (2005), no. 2, 70-86 (Russian). English transl.: Discrete Math. Appl. 15 (2005), no. 2, 153-170. MR2167801
98. Yu. L. Pavlov & E. A. Loseva, Limit distributions of the maximum size of a tree in a random recursive forest. Diskret. Mat. 14 (2002), no. 1, 60-74 (Russian). English transl.: Discrete Math. Appl. 12 (2002), no. 1, 45-59. MR1919856
99. J. Pitman, Enumerations of trees and forests related to branching processes and random walks. Microsurveys in Discrete Probability (Princeton, NJ, 1997), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 41, Amer. Math. Soc., Providence, RI, 1998, pp. 163-180. MR1630413
100. F. Ritort, Glassiness in a model without energy barriers. Physical Re- view Letters 75 (1995), 1190-1193.
101. W. Rudin, Real and Complex Analysis. McGraw-Hill, London, 1970
102. S. Sagitov & M. C. Serra, Multitype Bienaymé-Galton-Watson processes escaping extinction. Adv. Appl. Probab. 41 (2009), no. 1, 225-246. MR2514952
103. R. P. Stanley, Enumerative Combinatorics, Volume 2. Cambridge Univ. Press, Cambridge, 1999. MR1676282
104. J. J. Sylvester, On the change of systems of independent variables, Quart J. Math. 1 (1857), 42-56.
105. L. Takács, A generalization of the ballot problem and its application in the theory of queues. J. Amer. Statist. Assoc. 57 (1962), 327-337. MR0138139
106. L. Takács, Ballots, queues and random graphs. J. Appl. Probab. 26 (1989), no. 1, 103-112. MR0981255
107. J.C. Tanner, A derivation of the Borel distribution. Biometrika 48 (1961), 222-224. MR0125648
108. J. G. Wendel, Left-continuous random walk and the Lagrange expansion. Amer. Math. Monthly 82 (1975), 494-499. MR0381000
109. Herbert S. Wilf, generatingfunctionology. 2nd ed., Academic Press, 1994. MR1277813