## Asymptotic behaviour of the simple random walk on the 2-dimensional comb.(English)Zbl 1135.60045

The 2-dimensional comb is an inhomogeneous graph obtained from $$\mathbb Z^d$$ by removing all horizontal edges off the $$x$$-axis, so that the nearest neighbors of $$(k,n)$$ are $$(k,n-1)$$, $$(k,n+1)$$, $$(k-1,n)$$, $$(k+1,n)$$ if $$n=0$$ and $$(k,n-1)$$, $$(k,n+1)$$ otherwise. In the article vertical and horizontal components of the simple random walk on the 2-dimensional comb are analyzed by combinatorial methods. In particular asymptotic behaviour of the expected value of the distance from the origin, the maximal deviation and the maximal span in $$n$$ steps are evaluated, 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).
Infinite graphs are usually characterized by the spectral dimension $$\delta_s$$, the fractal dimension $$\delta_f$$ and the walk dimension $$\delta_w$$ which in typical cases are linked by Einstein relation $$\delta_s\delta_w=2\delta_f$$. It is proved that Einstein relation does not hold for the comb.
Then the limit continuous time interpolation of the process is obtained dividing by $$n^{1/4}$$ and by $$n^{1/2}$$ respectively the horizontal and the vertical projections of the position after $$n$$ steps. The limit of the vertical component is the Brownian motion, while the limit of the horizontal component is a Brownian motion indexed by the local time at 0 of the vertical component.

### MSC:

 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 05A15 Exact enumeration problems, generating functions 60J65 Brownian motion 60G50 Sums of independent random variables; random walks
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