The Average Displacement of the Simple Random Walk on the Sierpiński Graph

We investigate the asymptotic behaviour of the average displacement of the simple random walk on the Sierpinski graph. The existence of an oscillating factor in this asymptotics is shown rigorously. The proof depends mainly on the analysis of the corresponding generating function. Using a functional equation and techniques from complex analysis we obtain the desired properties of this generating function.

[1]  O. Kallenberg.  Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[2]  B Kr¨on,et al.  Green Functions on Self-similar Graphs and Bounds for the Spectrum of the Laplacian , 2002 .

[3]  Martin T. Barlow,et al.  Diffusions on fractals , 1998 .

[4]  W. Woess.  Random walks on infinite graphs and groups, by Wolfgang Woess, Cambridge Tracts , 2001 .

[5]  S. Goldstein.  Random Walks and Diffusions on Fractals , 1987 .

[6]  Jun Kigami,et al.  Weyl's problem for the spectral distribution of Laplacians on P.C.F. self-similar fractals , 1993 .

[7]  Peter J. Grabner,et al.  Functional iterations and stopping times for Brownian motion on the Sierpiński gasket , 1997 .

[8]  Owen Dafydd Jones,et al.  Transition probabilities for the simple random walk on the Sierpinski graph , 1996 .

[9]  Bruijn de Ng.  An asymptotic problem on iterated functions , 1979 .

[10]  Gérard Ben Arous,et al.  Large deviations for Brownian motion on the Sierpinski gasket , 2000 .

[11]  A. Odlyzko.  Periodic oscillations of coefficients of power series that satisfy functional equations , 1982 .

[12]  Philippe Flajolet,et al.  Singularity Analysis of Generating Functions , 1990, SIAM J. Discret. Math..

[13]  Tadashi Shima,et al.  On a spectral analysis for the Sierpinski gasket , 1992 .

[14]  Bernhard Kron,et al.  Green functions on self-similar graphs and bounds for the spectrum of the Laplacian , 2002, math/0202172.

[15]  Joseph G. Conlon.  Even cycles in graphs , 2004 .

[16]  Martin T. Barlow,et al.  Brownian motion on the Sierpinski gasket , 1988 .

[17]  J. Kigami,et al.  Analysis on Fractals , 2001 .

[18]  B. Krön.  Growth of Self-Similar Graphs , 2004, J. Graph Theory.

[19]  Wolfgang Woess,et al.  Functional iterations and periodic oscillations for simple random walk on the Sierpiński graph , 1997 .