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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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