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Local Limit Theorems for Sequences of Simple Random Walks on Graphs

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Abstract

In this article, local limit theorems for sequences of simple random walks on graphs are established. The results formulated are motivated by a variety of random graph models, and explanations are provided as to how they apply to supercritical percolation clusters, graph trees converging to the continuum random tree and the homogenisation problem for nested fractals. A subsequential local limit theorem for the simple random walks on generalised Sierpinski carpet graphs is also presented.

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References

  1. Aldous, D.: The continuum random tree. II. An overview. In: Stochastic Analysis (Durham, 1990). London Math. Soc. Lecture Note Ser., vol. 167, pp. 23–70. Cambridge Univ. Press, Cambridge (1991)

    Google Scholar 

  2. Aldous, D.: The continuum random tree. III. Ann. Probab. 21(1), 248–289 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  3. Barlow, M.T.: Diffusions on fractals. In: Lectures on Probability Theory and Statistics (Saint-Flour, 1995). Lecture Notes in Math., vol. 1690, pp. 1–121. Springer, Berlin (1998)

    Chapter  Google Scholar 

  4. Barlow, M.T., Bass, R.F.: The construction of Brownian motion on the Sierpiński carpet. Ann. Inst. H. Poincaré Probab. Statist. 25(3), 225–257 (1989)

    MATH  MathSciNet  Google Scholar 

  5. Barlow, M.T., Bass, R.F.: Transition densities for Brownian motion on the Sierpiński carpet. Probab. Theory Related Fields 91(3–4), 307–330 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. Barlow, M.T., Bass, R.F.: Brownian motion and harmonic analysis on Sierpinski carpets. Canad. J. Math. 51(4), 673–744 (1999)

    MATH  MathSciNet  Google Scholar 

  7. Barlow, M.T., Bass, R.F.: Random walks on graphical Sierpinski carpets. In: Random Walks and Discrete Potential Theory (Cortona, 1997). Sympos. Math., vol. XXXIX, pp. 26–55. Cambridge Univ. Press, Cambridge (1999)

    Google Scholar 

  8. Barlow, M.T., Coulhon, T., Kumagai, T.: Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs. Comm. Pure Appl. Math. 58(12), 1642–1677 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Barlow, M.T., Hambly, B.M.: Parabolic Harnack inequality and local limit theorem for percolation clusters. Available at arXiv.org/abs/0810.2467 (preprint)

  10. Billingsley, P.: Convergence of Probability Measures, 2nd edn. In: Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. Wiley, New York (1999)

    Google Scholar 

  11. Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry. In: Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence (2001)

    Google Scholar 

  12. Cristea, L.L.: A geometric property of the Sierpiński carpet. Quaestiones Math. 28(2), 251–262 (2005)

    MATH  MathSciNet  Google Scholar 

  13. Croydon, D.A.: Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree. Ann. Inst. H. Poincaré Probab. Statist. (2008, in press)

  14. Croydon, D.A.: Volume growth and heat kernel estimates for the continuum random tree. Probab. Theory Related Fields 140(1–2), 207–238 (2008)

  15. Duquesne, T., Le Gall, J.-F.: Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131(4), 553–603 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Falconer, K.J.: Fractal Geometry. Mathematical Foundations and Applications. Wiley, Chichester (1990)

    MATH  Google Scholar 

  17. Feller, W.: An Introduction to Probability Theory and its Applications, vol. II, 2nd edn. Wiley, New York (1971)

    MATH  Google Scholar 

  18. Fitzsimmons, P.J., Hambly, B.M., Kumagai, T.: Transition density estimates for Brownian motion on affine nested fractals. Comm. Math. Phys. 165(3), 595–620 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  19. Grigor’yan, A., Telcs, A.: Harnack inequalities and sub-Gaussian estimates for random walks. Math. Ann. 324(3), 521–556 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hambly, B.M., Kumagai, T.: Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries. In: Fractal Geometry and Applications: a Jubilee of Benoît Mandelbrot, Part 2, Proc. Sympos. Pure Math., vol. 72, pp. 233–259. American Mathematical Society, Providence (2004)

    Google Scholar 

  21. Hambly, B.M., Metz, V.: The homogenization problem for the Vicsek set. Stochastic Process Appl. 76(2), 167–190 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Probability and its Applications (New York). Springer, New York (2002)

    MATH  Google Scholar 

  23. Kigami, J.: Harmonic calculus on limits of networks and its application to dendrites. J. Funct. Anal. 128(1), 48–86 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kigami, J.: Analysis on fractals. In: Cambridge Tracts in Mathematics, vol. 143. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  25. Kumagai, T.: Heat kernel estimates and parabolic Harnack inequalities on graphs and resistance forms. Publ. Res. Inst. Math. Sci. 40(3), 793–818 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kumagai, T.: Homogenization on finitely ramified fractals. In: Stochastic Analysis and Related Topics in Kyoto. Adv. Stud. Pure Math., vol. 41, pp. 189–207. Mathematical Society, Tokyo (2004)

    Google Scholar 

  27. Kumagai, T., Kusuoka, S.: Homogenization on nested fractals. Probab. Theory Related Fields 104(3), 375–398 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  28. Le Gall, J.-F.: Random real trees. Ann. Fac. Sci. Toulouse Math. (6) 15(1), 35–62 (2006)

    MATH  MathSciNet  Google Scholar 

  29. Osada, H.: Isoperimetric constants and estimates of heat kernels of pre Sierpiński carpets. Probab. Theory Related Fields 86(4), 469–490 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  30. Sabot, C.: Existence and uniqueness of diffusions on finitely ramified self-similar fractals. Ann. Sci. École Norm. Sup. (4) 30(5), 605–673 (1997)

    MATH  MathSciNet  Google Scholar 

  31. Telcs, A.: The Einstein relation for random walks on graphs. J. Statist. Phys. 122(4), 617–645 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK

    D. A. Croydon

  2. Mathematical Institute, 24-29 St. Giles’, Oxford, OX1 3LB, UK

    B. M. Hambly

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  1. D. A. Croydon
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  2. B. M. Hambly
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Correspondence to D. A. Croydon.

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Croydon, D.A., Hambly, B.M. Local Limit Theorems for Sequences of Simple Random Walks on Graphs. Potential Anal 29, 351–389 (2008). https://doi.org/10.1007/s11118-008-9101-9

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