Electronic Journal of Probability

Rigid representations of the multiplicative coalescent with linear deletion

James B. Martin and Balázs Ráth

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We introduce the multiplicative coalescent with linear deletion, a continuous-time Markov process describing the evolution of a collection of blocks. Any two blocks of sizes $x$ and $y$ merge at rate $xy$, and any block of size $x$ is deleted with rate $\lambda x$ (where $\lambda \geq 0$ is a fixed parameter). This process arises for example in connection with a variety of random-graph models which exhibit self-organised criticality. We focus on results describing states of the process in terms of collections of excursion lengths of random functions. For the case $\lambda =0$ (the coalescent without deletion) we revisit and generalise previous works by authors including Aldous, Limic, Armendariz, Uribe Bravo, and Broutin and Marckert, in which the coalescence is related to a “tilt” of a random function, which increases with time; for $\lambda >0$ we find a novel representation in which this tilt is complemented by a “shift” mechanism which produces the deletion of blocks. We describe and illustrate other representations which, like the tilt-and-shift representation, are “rigid”, in the sense that the coalescent process is constructed as a projection of some process which has all of its randomness in its initial state. We explain some applications of these constructions to models including mean-field forest-fire and frozen-percolation processes.

Article information

Electron. J. Probab. Volume 22 (2017), paper no. 83, 47 pp.

Received: 14 October 2016
Accepted: 31 August 2017
First available in Project Euclid: 14 October 2017

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Zentralblatt MATH identifier

Primary: 60J99: None of the above, but in this section 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 05C80: Random graphs [See also 60B20]

multiplicative coalescent Erdős-Rényi random graph frozen percolation

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Martin, James B.; Ráth, Balázs. Rigid representations of the multiplicative coalescent with linear deletion. Electron. J. Probab. 22 (2017), paper no. 83, 47 pp. doi:10.1214/17-EJP100. https://projecteuclid.org/euclid.ejp/1507946758

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  • [1] L. Addario-Berry, N. Broutin, and C. Goldschmidt,The continuum limit of critical random graphs, Probab. Theory Related Fields152(2012), no. 3-4, 367–406.
  • [2] David Aldous,Brownian excursions, critical random graphs and the multiplicative coalescent, Ann. Probab.25(1997), no. 2, 812–854.
  • [3] David Aldous,Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists, Bernoulli5(1999), no. 1, 3–48.
  • [4] David Aldous,The percolation process on a tree where infinite clusters are frozen, Math. Proc. Cambridge Philos. Soc.128(2000), no. 3, 465–477.
  • [5] David Aldous and Vlada Limic,The entrance boundary of the multiplicative coalescent, Electron. J. Probab.3(1998), no. 3, 59 pp.
  • [6] David Aldous and Jim Pitman,The standard additive coalescent, Ann. Probab.26(1998), no. 4, 1703–1726.
  • [7] Ines Armendáriz,Brownian excursions and coalescing particle systems, PhD thesis, New York University, 2001.
  • [8] Ines Armendáriz,Dual fragmentation and multiplicative coagulation, Unpublished preprint, 2005.
  • [9] Jean Bertoin,The inviscid Burgers equation with Brownian initial velocity, Communications in Mathematical Physics193(1998), no. 2, 397–406.
  • [10] Jean Bertoin,A fragmentation process connected to Brownian motion, Probab. Theory Related Fields117(2000), no. 2, 289–301.
  • [11] S. Bhamidi, N. Broutin, S. Sen, and X. Wang,Scaling limits of random graph models at criticality: Universality and the basin of attraction of the Erdős-Rényi random graph,arXiv:1411.3417.
  • [12] S. Bhamidi and S. Sen,Geometry of the vacant set left by random walk on random graphs, Wright’s constants, and critical random graphs with prescribed degrees,arXiv:1608.07153.
  • [13] Shankar Bhamidi, Amarjit Budhiraja, and Xuan Wang,The augmented multiplicative coalescent, bounded size rules and critical dynamics of random graphs, Probab. Theory Related Fields160(2014), no. 3-4, 733–796.
  • [14] Shankar Bhamidi, Sanchayan Sen, and Xuan Wang,Continuum limit of critical inhomogeneous random graphs, Probability Theory and Related Fields (2016).
  • [15] Shankar Bhamidi, Remco van der Hofstad, and Sanchayan Sen,The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs, Probability Theory and Related Fields (2017).
  • [16] Shankar Bhamidi, Remco van der Hofstad, and Johan van Leeuwaarden,Scaling limits for critical inhomogeneous random graphs with finite third moments, Electron. J. Probab.15(2010), no. 54, 1682–1703.
  • [17] Shankar Bhamidi, Remco van der Hofstad, and Johan van Leeuwaarden,Novel scaling limits for critical inhomogeneous random graphs, Ann. Probab.40(2012), no. 6, 2299–2361.
  • [18] Patrick Billingsley,Convergence of probability measures, second ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999.
  • [19] Nicolas Broutin and Jean-François Marckert,A new encoding of coalescent processes: applications to the additive and multiplicative cases, Probab. Theory Related Fields166(2016), no. 1-2, 515–552.
  • [20] Laurent Carraro and Jean Duchon,Équation de Burgers avec conditions initiales à accroissements indépendants et homogènes, Annales de l’Institut Henri Poincare (C) Non Linear Analysis15(1998), no. 4, 431–458.
  • [21] Edward Crane, Nic Freeman, and Bálint Tóth,Cluster growth in the dynamical Erdős-Rényi process with forest fires, Electron. J. Probab.20(2015), no. 101, 33 pp.
  • [22] A. Dembo, A. Levit, and S. Vadlamani,Brownian excursions and critical quantum random graphs,arXiv:1404.5705.
  • [23] Nicolas Fournier and Philippe Laurençot,Marcus-Lushnikov processes, Smoluchowski’s and Flory’s models, Stochastic Process. Appl.119(2009), no. 1, 167–189.
  • [24] Adrien Joseph,The component sizes of a critical random graph with given degree sequence, Ann. Appl. Probab.24(2014), no. 6, 2560–2594.
  • [25] David C. Kaspar and Fraydoun Rezakhanlou,Scalar conservation laws with monotone pure-jump Markov initial conditions, Probability Theory and Related Fields165(2016), no. 3, 867–899.
  • [26] Vlada Limic,Extreme eternal multiplicative coalescent is encoded by its Lévy-type processes,arXiv:1601.01325.
  • [27] James Martin and Balázs Ráth,Scaling limits of mean-field frozen percolation and forest fire processes, In preparation.
  • [28] James Martin and Dominic Yeo,A scaling limit for the component sizes of critical random forests, In preparation.
  • [29] Govind Menon and Robert L. Pego,Universality classes in burgers turbulence, Communications in Mathematical Physics273(2007), no. 1, 177–202.
  • [30] M. Merle and R. Normand,Self-organized criticality in a discrete model for Smoluchowski’s equation,arXiv:1410.8338.
  • [31] M. Merle and R. Normand,Self-organized criticality in a discrete model for Smoluchowski’s equation with limited aggregations,arXiv:1509.00934.
  • [32] Peter Mörters and Yuval Peres,Brownian motion, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2010, With an appendix by Oded Schramm and Wendelin Werner.
  • [33] Asaf Nachmias and Yuval Peres,Critical percolation on random regular graphs, Random Structures Algorithms36(2010), no. 2, 111–148.
  • [34] Balázs Ráth,Feller property of the multiplicative coalescent with linear deletion,(accepted for publication at Bernoulli Journal),arXiv:1610.00021.
  • [35] Balázs Ráth,Mean field frozen percolation, J. Stat. Phys.137(2009), no. 3, 459–499.
  • [36] Balázs Ráth and Bálint Tóth,Erdős-Rényi random graphs $+$ forest fires $=$ self-organized criticality, Electron. J. Probab.14(2009), no. 45, 1290–1327.
  • [37] Oliver Riordan,The phase transition in the configuration model, Combin. Probab. Comput.21(2012), no. 1-2, 265–299.
  • [38] Geronimo Uribe Bravo,Coding multiplicative coalescence by an inhomogeneous random walk, Work in progress.
  • [39] Geronimo Uribe Bravo,Markovian bridges, Brownian excursions, and stochastic fragmentation and coalescence, PhD thesis, UNAM, 2007.