Hostname: page-component-6bf8c574d5-7jkgd Total loading time: 0 Render date: 2025-02-17T13:35:10.090Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuous phase transitions on Galton–Watson trees

Part of: Equilibrium statistical mechanics Markov processes Special processes

Published online by Cambridge University Press:  06 July 2021

Tobias Johnson
Tobias Johnson*
Affiliation:
Department of Mathematics, College of Staten Island, Staten Island, NY 10314, USA
*
Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Distinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton–Watson trees. For example, let $\mathcal{T}_1$ be the event that a Galton–Watson tree is infinite and let $\mathcal{T}_2$ be the event that it contains an infinite binary tree starting from its root. These events satisfy similar recursive properties: $\mathcal{T}_1$ holds if and only if $\mathcal{T}_1$ holds for at least one of the trees initiated by children of the root, and $\mathcal{T}_2$ holds if and only if $\mathcal{T}_2$ holds for at least two of these trees. The probability of $\mathcal{T}_1$ has a continuous phase transition, increasing from 0 when the mean of the child distribution increases above 1. On the other hand, the probability of $\mathcal{T}_2$ has a first-order phase transition, jumping discontinuously to a non-zero value at criticality. Given the recursive property satisfied by the event, we describe the critical child distributions where a continuous phase transition takes place. In many cases, we also characterise the event undergoing the phase transition.

Keywords

Galton–Watson treephase transitionfirst-order

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B26: Phase transitions (general)
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 31 , Issue 2 , March 2022 , pp. 198 - 228
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

The author received support from NSF grant DMS-1811952 and PSC-CUNY Award #62628-00 50.

References

Auffinger, A. and Cable, D. (2017) Pemantle’s min-plus binary tree, available at arXiv:1709.07849.Google Scholar
Alon, N. and Spencer, J. H. (2016) The Probabilistic Method, 4th edn., Wiley Series in Discrete Mathematics and Optimization. John Wiley & Sons, Inc., Hoboken, NJ.Google Scholar
Boccaletti, S., Almendral, J. A., Guan, S., Leyva, I., Liu, Z., Sendia-Nadal, I., Wang, Z. and Zou, Y. (2016) Explosive transitions in complex networks structure and dynamics: percolation and synchronization. Phys. Rep. 660 1–94. Explosive transitions in complex networks structure and dynamics: Percolation and synchronization.CrossRefGoogle Scholar
Bollobás, B. and Riordan, O. (2006) A short proof of the Harris-Kesten theorem. Bull. London Math. Soc. 38(3) 470–484.CrossRefGoogle Scholar
Duminil-Copin, H. (2019) Sharp threshold phenomena in statistical physics. Jpn. J. Math. 14(1) 1–25.CrossRefGoogle Scholar
Dekking, F. M. (1991) Branching processes that grow faster than binary splitting. Am. Math. Monthly 98(8) 728–731.CrossRefGoogle Scholar
Encinas, J. M., Harunari, P. E., de Oliveira, M. M. and Fiore, C. E. (2018) Fundamental ingredients for discontinuous phase transitions in the inertial majority vote model. Sci. Rep. 8(9338).10.1038/s41598-018-27240-4CrossRefGoogle Scholar
Friedgut, E. and Kalai, G. (1996) Every monotone graph property has a sharp threshold. Proc. Am. Math. Soc. 124(10) 2993–3002.CrossRefGoogle Scholar
Friedgut, E. (1999) Sharp thresholds of graph properties, and the k-sat problem. J. Am. Math. Soc. 12(4), 1017–1054. With an appendix by Jean Bourgain.CrossRefGoogle Scholar
Fitzner, R. and van der Hofstad, R. (2017) Mean-field behavior for nearest-neighbor percolation in $d>10$ . Electr. J. Prob. 22, Paper No. 43, 65.10$+.+Electr.+J.+Prob.+22,+Paper+No.+43,+65.>Google Scholar
Garban, C. and Steif, J. E. (2015) Noise Sensitivity of Boolean Functions and Percolation, Institute of Mathematical Statistics Textbooks. Cambridge University Press, New York.10.1017/CBO9781139924160CrossRefGoogle Scholar
Holroyd, A. E. and Martin, J. B. (2019) Galton–Watson games, available at https://onlinelibrary.wiley.com/doi/full/10.1002/rsa.21008.Google Scholar
Hara, T. and Slade, G. (1990) Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys. 128(2) 333–391.CrossRefGoogle Scholar
Johnson, T., Podder, M. and Skerman, F. (2020) Random tree recursions: which fixed points correspond to tangible sets of trees?. Random Struct. Algorithms 56(3) 796–837.10.1002/rsa.20895CrossRefGoogle Scholar
Łuczak, T. and Spencer, J. (1991) When does the zero-one law hold? J. Am. Math. Soc. 4(3) 451–468.CrossRefGoogle Scholar
Podder, M. and Spencer, J. (2017) First order probabilities for Galton-Watson trees. In A Journey through Discrete Mathematics, Springer, Cham, pp. 711–734.CrossRefGoogle Scholar
Podder, M. and Spencer, J. (2017) Galton-Watson probability contraction. Electr. Commun. Prob. 22, Paper No. 20, 16.Google Scholar
Pittel, B., Spencer, J. and Wormald, N. (1996) Sudden emergence of a giant k-core in a random graph. J. Comb. Theory Ser. B 67(1) 111–151.CrossRefGoogle Scholar
Riordan, O. (2008) The k-core and branching processes. Comb. Prob. Comput. 17(1) 111–136.10.1017/S0963548307008589CrossRefGoogle Scholar
Rudin, W. (1976) Principles of Mathematical Analysis, 3rd edn., International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York-Auckland-Düsseldorf.Google Scholar
Shelah, S. and Spencer, J. (1988) Zero-one laws for sparse random graphs. J. Am. Math. Soc. 1(1) 97–115.CrossRefGoogle Scholar