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Scaling limit for the random walk on the largest connected component of the critical random graph
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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. 279338. ISSN 16634926

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Official URL: http://dx.doi.org/10.2977/PRIMS/70
Abstract
In this article, a scaling limit for the simple random walk on the largest connected component of the ErdosRényi 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 shorttime 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:  16634926 
Date:  November 2011 
Volume:  Vol.48 
Number:  No.2 
Page Range:  pp. 279338 
Identification Number:  10.2977/PRIMS/70 
Status:  Not Peer Reviewed 
Publication Status:  Published 
References:  [1] L. AddarioBerry, 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 (SaintFlour, 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 WileyInterscience 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 nonuniform 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. 12, 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), SpringerVerlag, 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 AlexanderOrbach 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, SpringerVerlag, Berlin, 2006, Lectures from the 34th Summer School on Probability Theory held in SaintFlour, July 6–24, 2004, Edited and with a foreword by Jean Picard. 
URI:  http://wrap.warwick.ac.uk/id/eprint/39469 
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