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2006 Asymptotic Behaviour of the Simple Random Walk on the 2-dimensional Comb
Daniela Bertacchi
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Electron. J. Probab. 11: 1184-1203 (2006). DOI: 10.1214/EJP.v11-377


We analyze the differences between the horizontal and the vertical component of the simple random walk on the 2-dimensional comb. In particular we evaluate by combinatorial methods the asymptotic behaviour of the expected value of the distance from the origin, the maximal deviation and the maximal span in $n$ steps, proving that for all these quantities the order is $n^{1/4}$ for the horizontal projection and $n^{1/2}$ for the vertical one (the exact constants are determined). Then we rescale the two projections of the random walk dividing by $n^{1/4}$ and $n^{1/2}$ the horizontal and vertical ones, respectively. The limit process is obtained. With similar techniques the walk dimension is determined, showing that the Einstein relation between the fractal, spectral and walk dimensions does not hold on the comb.


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Daniela Bertacchi. "Asymptotic Behaviour of the Simple Random Walk on the 2-dimensional Comb." Electron. J. Probab. 11 1184 - 1203, 2006.


Accepted: 7 December 2006; Published: 2006
First available in Project Euclid: 31 May 2016

zbMATH: 1135.60045
MathSciNet: MR2268542
Digital Object Identifier: 10.1214/EJP.v11-377

Primary: 60J10
Secondary: 05A15 , 60J65

Keywords: Brownian motion , comb , generating function , Maximal Excursion , Random walk

Vol.11 • 2006
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