J. Aust. Math. Soc.  75 (2003), 325-353
Uniform asymptotic estimates of transition probabilities on combs

Daniela Bertacchi
  Università  di Milano-Bicocca
  Dipartimento di Matematica e Applicazioni
  Via Bicocca degli Arcimboldi 8
  20126 Milano
  Italy
  bertacchi@matapp.unimib.it
and
Fabio Zucca
  Politecnico di Milano
  Dipartimento di Matematica
  Piazza Leonardo da Vinci 32
  20133 Milano
  Italy
  zucca@mate.polimi.it


Abstract
We investigate the asymptotical behaviour of the transition probabilities of the simple random walk on the 2-comb. In particular, we obtain space-time uniform asymptotical estimates which show the lack of symmetry of this walk better than local limit estimates. Our results also point out the impossibility of getting sub-Gaussian estimates involving the spectral and walk dimensions of the graph.
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