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Giant component and vacant set for random walk on a discrete torus. (English) Zbl 1141.60057
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 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.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G50 Sums of independent random variables; random walks 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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##### References:
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