Mathematics – Probability
Scientific paper
2010-03-18
Stochastic Processes and their Applications, Vol. 121 (2011), Issue 5, pages 925-956
Mathematics
Probability
33 pages, 2 figures, revised arguments, results unchanged
Scientific paper
A small world is obtained from the $d$-dimensional torus of size 2L adding randomly chosen connections between sites, in a way such that each site has exactly one random neighbour in addition to its deterministic neighbours. We study the asymptotic behaviour of the meeting time $T_L$ of two random walks moving on this small world and compare it with the result on the torus. On the torus, in order to have convergence, we have to rescale $T_L$ by a factor $C_1L^2$ if $d=1$, by $C_2L^2\log L$ if $d=2$ and $C_dL^d$ if $d\ge3$. We prove that on the small world the rescaling factor is $C^\prime_dL^d$ and identify the constant $C^\prime_d$, proving that the walks always meet faster on the small world than on the torus if $d\le2$, while if $d\ge3$ this depends on the probability of moving along the random connection. As an application, we obtain results on the hitting time to the origin of a single walk and on the convergence of coalescing random walk systems on the small world.
Bertacchi Daniela
Borrello Davide
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