... time c accross those slices that are perpendicularto direction k, and let C cτ(η)=dk=1 C k c (η). (These random variables do notdepend on N because we consider crossings of all the slices.) ... (c )|−E c ,ε(|V (c )|)≥ δ= −∞.(2.11)Proof. The crucial step is to apply a concentration inequality of Talagrandtwice, as follows. First note that, conditioned on Xi ,c ,ε, Wi (c ) ... define C cτ(Sjη,N)=number of η-crossings in Sjη,Nup to time c ,(2.54)V c (Sjη,N)=Vaτ−1/(d−2) (c ) ∩ Sjη,N.Because the copies in (2.43) cover Rdexactly d times, on the event C cτ,M,η∩V c defined...