Croydon, David A.. (2011) Scaling limit for the random walk on the largest connected component of the critical random graph. Research Institute for Mathematical Sciences. Publications, Vol.48 (No.2). pp. 279-338. ISSN 1663-4926

Preview

PDF WRAP_Croydon_crg_revise.pdf - Submitted Version - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (363Kb)

Abstract

In this article, a scaling limit for the simple random walk on the largest connected component of the Erdos-Rényi random graph G(n,p) in the critical window, p = n−1+λn−4/3, is deduced. The limiting diffusion is constructed using resistance form techniques, and is shown to satisfy the same quenched short-time heat kernel asymptotics as the Brownian motion on the continuum random tree.

[error in script] [error in script]

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Random walks (Mathematics)
Journal or Publication Title:	Research Institute for Mathematical Sciences. Publications
Publisher:	European Mathematical Society Publishing House
ISSN:	1663-4926
Date:	November 2011
Volume:	Vol.48
Number:	No.2
Page Range:	pp. 279-338
Identification Number:	10.2977/PRIMS/70
Status:	Not Peer Reviewed
Publication Status:	Published
References:	[1] L. Addario-Berry, N. Broutin, and C. Goldschmidt, The continuum limit of critical random graphs, Preprint. [2] D. Aldous, The continuum random tree. II. An overview, Stochastic analysis (Durham, 1990), London Math. Soc. Lecture Note Ser., vol. 167, Cambridge Univ. Press, Cambridge, 1991, pp. 23–70. [3] , The continuum random tree. III, Ann. Probab. 21 (1993), no. 1, 248–289. [4] , Brownian excursions, critical random graphs and the multiplicative coalescent, Ann. Probab. 25 (1997), no. 2, 812–854. [5] M. T. Barlow, Diffusions on fractals, Lectures on probability theory and statistics (Saint-Flour, 1995), Lecture Notes in Math., vol. 1690, Springer, Berlin, 1998, pp. 1– 121. [6] P. Billingsley, Convergence of probability measures, second ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons Inc., New York, 1999, A Wiley-Interscience Publication. [7] E. Bolthausen, On the global asymptotic behavior of Brownian local time on the circle, Trans. Amer. Math. Soc. 253 (1979), 317–328. [8] A. N. Borodin, The asymptotic behavior of local times of recurrent random walks with finite variance, Teor. Veroyatnost. i Primenen. 26 (1981), no. 4, 769–783. [9] D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. [10] D. A. Croydon, Heat kernel fluctuations for a resistance form with non-uniform volume growth, Proc. Lond. Math. Soc. (3) 94 (2007), no. 3, 672–694. [11] , Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree, Ann. Inst. Henri Poincar´e Probab. Stat. 44 (2008), no. 6, 987–1019. [12] , Volume growth and heat kernel estimates for the continuum random tree, Probab. Theory Related Fields 140 (2008), no. 1-2, 207–238. [13] , Hausdorff measure of arcs and Brownian motion on Brownian spatial trees, Ann. Probab. 37 (2009), no. 3, 946–978. [14] , Scaling limits for simple random walks on random ordered graph trees, Adv. in Appl. Probab. 42 (2010), no. 2, 528–558. [15] P. G. Doyle and J. L. Snell, Random walks and electric networks, Carus Mathematical Monographs, vol. 22, Mathematical Association of America, Washington, DC, 1984. [16] T. Duquesne and J.-F. Le Gall, Random trees, L´evy processes and spatial branching processes, Ast´erisque (2002), no. 281, vi+147. [17] , Probabilistic and fractal aspects of L´evy trees, Probab. Theory Related Fields 131 (2005), no. 4, 553–603. [18] P. Erd˝os and A. R´enyi, On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutat´o Int. K¨ozl. 5 (1960), 17–61. [19] M. Fukushima, Y. ¯ Oshima, and M. Takeda, Dirichlet forms and symmetric Markov processes, de Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1994. [20] B. M. Hambly and V. Metz, The homogenization problem for the Vicsek set, Stochastic Process. Appl. 76 (1998), no. 2, 167–190. [21] O. Kallenberg, Foundations of modern probability, second ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002. [22] J. Kigami, Resistance forms, quasisymmetric maps and heat kernel estimates, Preprint. [23] , Harmonic calculus on limits of networks and its application to dendrites, J. Funct. Anal. 128 (1995), no. 1, 48–86. [24] , Analysis on fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. [25] G. Kozma and A. Nachmias, The Alexander-Orbach conjecture holds in high dimensions, Invent. Math. 178 (2009), no. 3, 635–654. [26] T. Kumagai, Heat kernel estimates and parabolic Harnack inequalities on graphs and resistance forms, Publ. Res. Inst. Math. Sci. 40 (2004), no. 3, 793–818. [27] R. Lyons and Y. Peres, Probability on trees and networks, In preparation. [28] M. B. Marcus and J. Rosen, Sample path properties of the local times of strongly symmetric Markov processes via Gaussian processes, Ann. Probab. 20 (1992), no. 4, 1603–1684. [29] U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal. 123 (1994), no. 2, 368–421. [30] A. Nachmias and Y. Peres, Critical random graphs: diameter and mixing time, Ann. Probab. 36 (2008), no. 4, 1267–1286. [31] G. Slade, The lace expansion and its applications, Lecture Notes in Mathematics, vol. 1879, Springer-Verlag, Berlin, 2006, Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004, Edited and with a foreword by Jean Picard.
URI:	http://wrap.warwick.ac.uk/id/eprint/39469

Request changes to a record

Actions (login required)

View Item