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Random walks on lattices. II. (English) Zbl 1342.60067
The random walks considered here take place on lattices of points in \(d\)-dimensional space or on the \(d\)-dimensional torus. The authors write down the characteristic function of the random walk and generating functions of several probabilities, such as the probability that a point is visited for the \(r\)th time at time \(n\). The Fourier inversion formula and Tauberian theorems yield asymptotic formulae for these probabilities, as well as expectations of related random variables such as first passage times and the number of points visited exactly \(r\) times in the first \(n\) steps.
For Part I see [E. W. Montroll, in: Proc. Sympos. Appl. Math. 16, 193–220 (1964; Zbl 0139.34901)].

MSC:
60G50 Sums of independent random variables; random walks
05C81 Random walks on graphs
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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References:
[1] DOI: 10.1007/BF01458701 · JFM 48.0603.01
[2] DOI: 10.1090/psapm/016/0161378
[3] DOI: 10.1063/1.1704049 · Zbl 0119.45503
[4] DOI: 10.1007/BF02020631 · Zbl 0091.13303
[5] DOI: 10.1093/qmath/os-10.1.266 · Zbl 0022.33202
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