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  • 1
    Electronic Resource
    Electronic Resource
    Random walks on generalized lattices (1996)
    Springer
    Monatshefte für Mathematik 121 (1996), S. 145-161 
    Details
    ISSN: 1436-5081
    Keywords: 60J15 ; 05C25 ; Random walks ; graphs and groups
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A generalized lattice is a graph on which the groupZ d acts almost transitively. The relations among various features of random walks on generalized lattices are studied. In particular we relate the mean displacement, the drift-freeness of the random walk and the existence of “linear” harmonic functions. Applications to recurrence criteria are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01299644
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  • 2
    Electronic Resource
    Electronic Resource
    On the norms of group-invariant transition operators on graphs (1992)
    Springer
    Journal of theoretical probability 5 (1992), S. 563-576 
    Details
    ISSN: 1572-9230
    Keywords: Random walks ; infinite graph ; amenable group
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we consider reversible random walks on an infinite grapin, invariant under the action of a closed subgroup of automorphisms which acts with a finite number of orbits on the vertex-set. Thel 2-norm (spectral radius) of the simple random walk is equal to one if and only if the group is both amenable and unimodular, and this also holds for arbitrary random walks with bounded invariant measure. In general, the norm is bounded above by the Perron-Frobenius eigenvalue of a finite matrix, and this bound is attained if and only if the group is both amenable and unimodular.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01060436
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