685 research outputs found
Phase Transitions for Random Walk Asymptotics on Free Products of Groups
Suppose we are given finitely generated groups
equipped with irreducible random walks. Thereby we assume that the expansions
of the corresponding Green functions at their radii of convergence contain only
logarithmic or algebraic terms as singular terms up to sufficiently large order
(except for some degenerate cases). We consider transient random walks on the
free product {} and give a complete
classification of the possible asymptotic behaviour of the corresponding
-step return probabilities. They either inherit a law of the form
from one of the free
factors or obey a -law, where
is the corresponding spectral radius and is the period of
the random walk. In addition, we determine the full range of the asymptotic
behaviour in the case of nearest neighbour random walks on free products of the
form . Moreover, we characterize the possible
phase transitions of the non-exponential types
in the case .Comment: 32 page
Spectral computations on lamplighter groups and Diestel-Leader graphs
The Diestel-Leader graph DL(q,r) is the horocyclic product of the homogeneous
trees with respective degrees q+1 and r+1. When q=r, it is the Cayley graph of
the lamplighter group (wreath product of the cyclic group of order q with the
infinite cyclic group) with respect to a natural generating set. For the
"Simple random walk" (SRW) operator on the latter group, Grigorchuk & Zuk and
Dicks & Schick have determined the spectrum and the (on-diagonal) spectral
measure (Plancherel measure). Here, we show that thanks to the geometric
realization, these results can be obtained for all DL-graphs by directly
computing an l^2-complete orthonormal system of finitely supported
eigenfunctions of the SRW. This allows computation of all matrix elements of
the spectral resolution, including the Plancherel measure. As one application,
we determine the sharp asymptotic behaviour of the N-step return probabilities
of SRW. The spectral computations involve a natural approximating sequence of
finite subgraphs, and we study the question whether the cumulative spectral
distributions of the latter converge weakly to the Plancherel measure. To this
end, we provide a general result regarding Foelner approximations; in the
specific case of DL(q,r), the answer is positive only when r=q
Polyharmonic functions for finite graphs and Markov chains
On a finite graph with a chosen partition of the vertex set into interior and
boundary vertices, a -polyharmonic function is a complex function
on the vertex set which satisfies at each
interior vertex. Here, may be the normalised adjaceny matrix, but more
generally, we consider the transition matrix of an arbitrary Markov chain
to which the (oriented) graph structure is adapted. After describing these
`global' polyharmonic functions, we turn to solving the Riquier problem, where
boundary functions are preassigned and a corresponding `tower' of
successive Dirichlet type problems are solved. The resulting unique solution
will be polyharmonic only at those points which have distance at least from
the boundary. Finally, we compare these results with those concerning infinite
trees with the end boundary, as studied by Cohen, Colonnna, Gowrisankaran and
Singman, and more recently, by Picardello and Woess
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