Logarithmic components of the vacant set for random walk on a discrete torus

Details Logarithmic components of the vacant set for random walk on a discrete torus Logarithmic components of the vacant set for random walk on a discrete torus

2008-04-25

arxiv.org/abs/0804.4097v1

Electronic Journal of Probability 2008, Vol. 13, 880-897

Mathematics

Probability

17 pages, accepted for publication in the Electronic Journal of Probability

Scientific paper

This work continues the investigation, initiated in a recent work by Benjamini and Sznitman, of percolative properties of the set of points not visited by a random walk on the discrete torus (Z/NZ)^d up to time uN^d in high dimension d. If u>0 is chosen sufficiently small it has been shown that with overwhelming probability this vacant set contains a unique giant component containing segments of length c_0 log N for some constant c_0 > 0, and this component occupies a non-degenerate fraction of the total volume as N tends to infinity. Within the same setup, we investigate here the complement of the giant component in the vacant set and show that some components consist of segments of logarithmic size. In particular, this shows that the choice of a sufficiently large constant c_0 > 0 is crucial in the definition of the giant component.

Affiliated with

Also associated with

No associations

Rating

Logarithmic components of the vacant set for random walk on a discrete torus does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Logarithmic components of the vacant set for random walk on a discrete torus, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Logarithmic components of the vacant set for random walk on a discrete torus will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-728142