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Convergence of blanket times for sequences of random walks on critical random graphs

Part of: Graph theory Special processes Limit theorems Markov processes

Published online by Cambridge University Press:  09 January 2023

George Andriopoulos
George Andriopoulos*
Affiliation:
NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, China.
*
Email: gandriopoulos91@gmail.com
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Abstract

Under the assumption that sequences of graphs equipped with resistances, associated measures, walks and local times converge in a suitable Gromov-Hausdorff topology, we establish asymptotic bounds on the distribution of the $\varepsilon$ -blanket times of the random walks in the sequence. The precise nature of these bounds ensures convergence of the $\varepsilon$ -blanket times of the random walks if the $\varepsilon$ -blanket time of the limiting diffusion is continuous at $\varepsilon$ with probability 1. This result enables us to prove annealed convergence in various examples of critical random graphs, including critical Galton-Watson trees and the Erdős-Rényi random graph in the critical window. We highlight that proving continuity of the $\varepsilon$ -blanket time of the limiting diffusion relies on the scale invariance of a finite measure that gives rise to realizations of the limiting compact random metric space, and therefore we expect our results to hold for other examples of random graphs with a similar scale invariance property.

Keywords

random walk in random environmentblanket timeGromov-Hausdorff convergenceGalton-Watson treeErdős-Rényi random graph

MSC classification

Primary: 60K37: Processes in random environments 60F17: Functional limit theorems; invariance principles 05C81: Random walks on graphs
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 05C80: Random graphs 60J25: Continuous-time Markov processes on general state spaces
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 32 , Issue 3 , May 2023 , pp. 478 - 515
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Abraham, R., Delmas, J.-F. and Hoscheit, P. (2013) A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces. Electron. J. Probab. 18(14) 1–21.CrossRefGoogle Scholar
Addario-Berry, L., Broutin, N. and Goldschmidt, C. (2012) The continuum limit of critical random graphs. Probab. Theory Related Fields 152(3-4) 367–406.Google Scholar
Aldous, D. (1991) Random walk covering of some special trees. J. Math. Anal. Appl. 157(1) 271–283.CrossRefGoogle Scholar
Aldous, D. (1993) The continuum random tree. III. Ann. Probab. 21(1) 248–289.Google Scholar
Aldous, D. (1997) Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25(2) 812–854.CrossRefGoogle Scholar
Athreya, S., Eckhoff, M. and Winter, A. (2013) Brownian motion on $\Bbb{R}$ -trees. Trans. Amer. Math. Soc. 365(6) 3115–3150.CrossRefGoogle Scholar
Barlow, M. T., Ding, J., Nachmias, A. and Peres, Y. (2011) The evolution of the cover time. Combin. Probab. Comput. 20(3) 331–345.CrossRefGoogle Scholar
Bhamidi, S., Broutin, N., Sen, S. and Wang, X. Scaling limits of random graph models at criticality: Universality and the basin of attraction of the Erdős-Rényi random graph, Preprint available at arXiv: 1411.3417.Google Scholar
Bhamidi, S. and Sen, S. (2020) Geometry of the vacant set left by random walk on random graphs, Wright’s constants, and critical random graphs with prescribed degrees. Random Struct. Algor. 56(3) 676–721, Preprint available at arXiv: 1608.07153.CrossRefGoogle Scholar
Bhamidi, S., Sen, S. and Wang, X. (2017) Continuum limit of critical inhomogeneous random graphs. Probab. Theory Related Fields 169(1-2) 565–641.CrossRefGoogle Scholar
Billingsley, P. (1999) Convergence of Probability Measures. Wiley Series in Probability and Statistics: Probability and Statistics, A Wiley-Interscience Publication. second ed. John Wiley & Sons, Inc.CrossRefGoogle Scholar
Burago, D., Burago, Y. and Ivanov, S. (2001) A course in metric geometry, Graduate Studies in Mathematics, vol. 33. American Mathematical Society.Google Scholar
Croydon, D. A. (2008) Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree. Ann. Inst. Henri Poincaré Probab. Stat. 44(6) 987–1019.CrossRefGoogle Scholar
Croydon, D. A. (2009) Hausdorff measure of arcs and Brownian motion on Brownian spatial trees. Ann. Probab. 37(3) 946–978.CrossRefGoogle Scholar
Croydon, D. A. (2010) Scaling limits for simple random walks on random ordered graph trees. Adv. Appl. Probab. 42(2) 528–558.CrossRefGoogle Scholar
Croydon, D. A. (2012) Scaling limit for the random walk on the largest connected component of the critical random graph. Publ. Res. Inst. Math. Sci. 48(2) 279–338.CrossRefGoogle Scholar
Croydon, D. A. (2015) Moduli of continuity of local times of random walks on graphs in terms of the resistance metric. Trans. London Math. Soc. 2(1) 57–79.CrossRefGoogle Scholar
Croydon, D. A. (2018) Scaling limits of stochastic processes associated with resistance forms. Ann. Inst. Henri Poincaré Probab. Stat. 54(4) 1939–1968.CrossRefGoogle Scholar
Croydon, D. A., Hambly, B. M. and Kumagai, T. (2012) Convergence of mixing times for sequences of random walks on finite graphs. Electron. J. Probab. 17(3) 1–32.CrossRefGoogle Scholar
Croydon, D. A., Hambly, B. M. and Kumagai, T. (2017) Time-changes of stochastic processes associated with resistance forms. Electron. J. Probab. 22(82) 1–41.CrossRefGoogle Scholar
Curien, N. and Kortchemski, I. (2014) Random stable looptrees. Electron. J. Probab. 19(108) 1–35.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (2008) An Introduction to the Theory of Point Processes, Probability and its Applications (New York). General Theory and Structure. second ed. Springer . Google Scholar
Ding, J., Lee, J. R. and Peres, Y. (2012) Cover times, blanket times, and majorizing measures. Ann.Math. 175(3) 1409–1471.CrossRefGoogle Scholar
Duquesne, T. and Gall, J.-F. Le (2005) Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131(4) 553–603.CrossRefGoogle Scholar
Erdős, P. and Rényi, A. (1960) On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 17–61.Google Scholar
Fukushima, M., Oshima, Y. and Takeda, M. (2011) Dirichlet Forms and Symmetric Markov Processes, De Gruyter Studies in Mathematics, 19, extended ed. Walter de Gruyter & Co.Google Scholar
Gittenberger, B. (2003) State spaces of the snake and its tour—convergence of the discrete snake. [J. Theoret. Probab. 16(4), 1015–1046; mr2033196] by J.-F. Marckert and A. Mokkadem. J. Theoret. Probab. 16(4) 1063–1067 (2004).Google Scholar
Goldschmidt, C. (2016) A short introduction to random trees. Mongolian Math. J. 20 53–72.Google Scholar
Janson, S. and Marckert, J.-F. (2005) Convergence of discrete snakes. J. Theoret. Probab. 18(3) 615–647.CrossRefGoogle Scholar
Kigami, J. (1995) Harmonic calculus on limits of networks and its application to dendrites. J. Funct. Anal. 128(1) 48–86.CrossRefGoogle Scholar
Kigami, J. (2012) Resistance forms, quasisymmetric maps and heat kernel estimates. Mem. Amer. Math. Soc. 216(1015) vi+132.Google Scholar
Gall, J.-F. (1993) The uniform random tree in a Brownian excursion. Probab. Theory Related Fields 96(3) 369–383.CrossRefGoogle Scholar
Gall, J.-F. (2006) Random real trees. Ann. Fac. Sci. Toulouse Math. 15(1) 35–62.CrossRefGoogle Scholar
Levin, D. A. and Peres, Y. (2017) Markov Chains and Mixing Times. American Mathematical SocietySecond edition of [ MR2466937], With contributions by Elizabeth L. Wilmer, With a chapter on “Coupling from the past” by James G. Propp and David B. Wilson.Google Scholar
Marckert, J.-F. and Mokkadem, A. (2003) The depth first processes of Galton-Watson trees converge to the same Brownian excursion. Ann. Probab. 31(3) 1655–1678.CrossRefGoogle Scholar
Marcus, M. B. and Rosen, J. (1992) Sample path properties of the local times of strongly symmetric Markov processes via Gaussian processes. Ann. Probab. 20(4) 1603–1684.Google Scholar
Marzouk, C. (2020) Scaling limits of discrete snakes with stable branching. Ann. Inst. Henri Poincaré Probab. Stat. 56(1) 502–523.CrossRefGoogle Scholar
Revuz, D. and Yor, M. (1999) Continuous martingales and Brownian motion, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293, third ed. Springer-Verlag.Google Scholar
Winkler, P. and Zuckerman, D. (1996) Multiple cover time. Random Struct. Algor. 9(4) 403–411.3.0.CO;2-0>CrossRefGoogle Scholar