# The small world effect on the coalescing time of random walks - Mathematics > Probability

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Abstract: A small world is obtained from the $d$-dimensional torus of size 2L addingrandomly chosen connections between sites, in a way such that each site hasexactly one random neighbour in addition to its deterministic neighbours. Westudy the asymptotic behaviour of the meeting time $T L$ of two random walksmoving on this small world and compare it with the result on the torus. On thetorus, 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 provethat on the small world the rescaling factor is $C^\prime dL^d$ and identifythe constant $C^\prime d$, proving that the walks always meet faster on thesmall world than on the torus if $d\le2$, while if $d\ge3$ this depends on theprobability of moving along the random connection. As an application, we obtainresults on the hitting time to the origin of a single walk and on theconvergence of coalescing random walk systems on the small world.

Autor: Daniela Bertacchi, Davide Borrello

Fuente: https://arxiv.org/