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

Bertacchi, D. (2006). Asymptotic behaviour of the simple random walk on the 2-dimensional comb. ELECTRONIC JOURNAL OF PROBABILITY, 11, 1184-1203 [10.1214/EJP.v11-377].

### Asymptotic behaviour of the simple random walk on the 2-dimensional comb

#### Abstract

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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Articolo in rivista - Articolo scientifico
random walk; maximal excursion; generating function; comb; Brownian motion
English
2006
11
1184
1203
open
Bertacchi, D. (2006). Asymptotic behaviour of the simple random walk on the 2-dimensional comb. ELECTRONIC JOURNAL OF PROBABILITY, 11, 1184-1203 [10.1214/EJP.v11-377].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10281/1040`