an:05255325
Zbl 1141.60057
Benjamini, Itai; Sznitman, Alain-Sol
Giant component and vacant set for random walk on a discrete torus
EN
J. Eur. Math. Soc. (JEMS) 10, No. 1, 133-172 (2008).
00213137
2008
j
60K35 60G50 82B41
probability theory; random walk; percolation; stochastic processes; coupon collector
The authors consider symmetric nearest neighbour random walk \(X\) on the \(d\)-dimensional (\(d\geq3\)) integer lattice torus \(E:=\big(\mathbb{Z}/(N\mathbb{Z})\big)^d\) of side-length \(N\). It is well-known that the cover time is of order \(N^d\log N\) if \(d\geq3\). In this paper, Benjamini and Sznitman investigate the percolative structure of the set \(V\subset E\) of sites that are \textit{not} visited by \(X\) up to time \(uN^d\), where \(u>0\) is assumed to be small. First of all they show (Corollary 4.5) that there are constants \(c=c(d)\) and \(c'=c'(d)\) such that
\[
\lim_{N\to\infty}\mathbb{P}[e^{-cu}\leq \# V/N^d\leq e^{-c'u}]=1.
\]
In Theorem 1.2 it is shown for \(d\geq 4\) and for any \(\beta\in(0,1)\) and \(K>0\) that if \(u>0\) is small enough, then with probability tending to one (as \(N\to\infty\)), with probability tending to one, every point \(x\in E\) is in distance at most \(N^\beta\) to some point in \(V\) that is in a straight line segment in \(V\) of length at least \(K\log N\). The next results hold for \(d\) larger than some \(d_0\) (and which the reviewer computed to be actually \(d_0=123\)). In Corollary 2.6 it is shown for \(d\geq d_0\) that if \(u>0\) is small enough, then with probability tending to one (as \(N\to\infty\)) there is a unique connected component \(O\subset V\) that contains straight line segments (in any of the \(d\) directions) of size \(c_0\log N\) (where \(c_0\) is a dimension dependent constant). Moreover, in Corollary 4.6 it is shown that for \(d\geq d_0\), for any \(\gamma\in(0,1)\) and \(u=u(\gamma)>0\) sufficiently small, with probability tending to one, the cardinality of \(O\) is a least \(\gamma N^d\). That is, \(O\) contains a substantial fraction (depending on \(u\)) of points of \(E\). However, it remains open if \(V\) contains more connected components of substantial size.
Achim Klenke (Mainz)