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.
Bertacchi, D., Zucca, F. (2003). Uniform asymptotic estimates of transition probabilities on combs. JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 75(3), 325-353 [10.1017/s1446788700008144].
Uniform asymptotic estimates of transition probabilities on combs
BERTACCHI, DANIELA;
2003
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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