Statement of the results

We will also have use for another intermediate event between A and ˜ ˜ A: ¯ A I I ′ j, σ n, N , for which we only impose the landing areas I I ′ of the j arms. We do not ask a priori the sub-intervals to be η-separated either, just to be disjoint. Note however that if they are ηη ′ -separated, then the extremities of the different crossings will be ηη ′ -separated too. To summarize: A j, σ n, N = { j arms ∂ S n ∂ S N , color σ} separated at scale ηη ′ + small extensions hh hh hh h tthh hh hh hh landing areas I I ′ U U U U U U U U U U U U U U U ˜ A ηη ′ j, σ n, N landing areas I I ′ U U U U U U U U U U U U U U ¯ A I I ′ j, σ n, N small extensions if I I ′ are ηη ′ -separated ii ii ii ii ttii ii ii ˜ ˜ A η,Iη ′ ,I ′ j, σ n, N Remark 10. If we take for instance alternating colors ¯ σ = BW BW , and as landing areas ¯I 1 , . . . , ¯I 4 the resp. right, top, left and bottom sides of ∂ S N , the 4-arm event ¯ A . ¯I 4, ¯ σ 0, N the “.” meaning that we do not put any condition on the internal boundary is then the event that 0 is pivotal for the existence of a left-right crossing of S N .

4.3 Statement of the results

Main result Our main separation result is the following: Theorem 11. Fix an integer j ≥ 1, some color sequence σ ∈ ˜ S j and η , η ′ ∈ 0, 1. Then we have ˆ P ˜ ˜ A η,Iη ′ ,I ′ j, σ n, N ≍ ˆP A j, σ n, N 4.8 uniformly in all landing sequences I I ′ of size ηη ′ , with η ≥ η and η ′ ≥ η ′ , p, ˆ P between P p and P 1 −p , n ≤ N ≤ Lp. First relations Before turning to the proof of this theorem, we list some direct consequences of the RSW estimates that will be needed. Proposition 12. Fix j ≥ 1, σ ∈ ˜ S j and η , η ′ ∈ 0, 1. 1. “Extendability”: We have ˆ P ˜ ˜ A η,I ˜ η ′ ,˜I ′ j, σ n, 2N , ˆ P ˜ ˜ A ˜ η,˜Iη ′ ,I ′ j, σ n2, N ≍ ˆP ˜˜ A η,Iη ′ ,I ′ j, σ n, N uniformly in p, ˆ P between P p and P 1 −p , n ≤ N ≤ Lp, and all landing sequences II ′ resp. ˜I˜I ′ of size ηη ′ resp. ˜ η ˜ η ′ larger than η η ′ . In other words: “once well-separated, the arms can easily be extended”. 1577 2. “Quasi-multiplicativity”: We have for some C = C η , η ′ ˆ P A j, σ n 1 , n 3 ≥ C ˆP ˜ ˜ A . η,I η j, σ n 1 , n 2 4ˆ P ˜ ˜ A η ′ ,I η′ . j, σ n 2 , n 3 uniformly in p, ˆ P between P p and P 1 −p , n j ≤ n 1 n 2 n 3 ≤ Lp with n 2 ≥ 4n 1 , and all landing sequences I I ′ of size ηη ′ larger than η η ′ . 3. For any η, η ′ 0, there exists a constant C = Cη, η ′ 0 with the following property: for any p, ˆ P between P p and P 1 −p , n ≤ N ≤ Lp, there exist two landing sequences I and I ′ of size η and η ′ that may depend on all the parameters mentioned such that ˆ P ˜ ˜ A η,Iη ′ ,I ′ j, σ n, N ≥ C ˆP ˜ A η,η ′ j, σ n, N . Proof. The proof relies on gluing arguments based on RSW constructions. However, the events considered are not monotone when σ is non-constant there is at least one black arm and one white arm. We will thus need a slight generalization of the FKG inequality for events “locally monotone”. Lemma 13. Consider A + , ˜ A + two increasing events, and A − , ˜ A − two decreasing events. Assume that there exist three disjoint finite sets of vertices A , A + and A − such that A + , A − , ˜ A + and ˜ A − depend only on the sites in, respectively, A ∪ A + , A ∪ A − , A + and A − . Then we have ˆ P ˜ A + ∩ ˜ A − |A + ∩ A − ≥ ˆP ˜ A + ˆ P ˜ A − 4.9 for any product measure ˆ P . Proof. Conditionally on the configuration ω A in A , the events A + ∩ ˜ A + and A − ∩ ˜ A − are independent, so that ˆ P A + ∩ ˜ A + ∩ A − ∩ ˜ A − |ω A = ˆ P A + ∩ ˜ A + |ω A ˆ P A − ∩ ˜ A − |ω A . The FKG inequality implies that ˆ P A + ∩ ˜ A + |ω A ≥ ˆPA + |ω A ˆ P ˜ A + |ω A = ˆ P A + |ω A ˆ P ˜ A + and similarly with A − and ˜ A − . Hence, ˆ P A + ∩ ˜ A + ∩ A − ∩ ˜ A − |ω A ≥ ˆPA + |ω A ˆ P ˜ A + ˆ P A − |ω A ˆ P ˜ A − = ˆ P A + ∩ A − |ω A ˆ P ˜ A + ˆ P ˜ A − . The conclusion follows by summing over all configurations ω A . Once this lemma at our disposal, items 1. and 2. are straightforward. For item 3., we consider a covering of ∂ S n resp. ∂ S N with at most 8 η −1 resp. 8 η ′−1 intervals {I} of length η resp. I ′ of length η ′ . Then for some I, I ′ , ˆ P ˜ ˜ A η,Iη ′ ,I ′ j, σ n, N ≥ 8η −1 −1 8η ′−1 −1 ˆ P ˜ A η,η ′ j, σ n, N . 1578 We also have the following a-priori bounds for the arm events: Proposition 14. Fix some j ≥ 1, σ ∈ ˜ S j and η , η ′ ∈ 0, 1. Then there exist some exponents α j , α ′ ∞, as well as constants 0 C j , C ′ ∞, such that C j n N α j ≤ ˆP ˜ ˜ A η,Iη ′ ,I ′ j, σ n, N ≤ C ′ n N α ′ 4.10 uniformly in p, ˆ P between P p and P 1 −p , n ≤ N ≤ Lp, and all landing sequences II ′ of size ηη ′ larger than η η ′ . The lower bound comes from iterating item 1. The upper bound can be obtained by using concentric annuli: in each of them, RSW implies that there is a probability bounded away from zero to observe a black circuit, preventing the existence of a white arm consider a white circuit instead if σ = BB . . . B.

4.4 Proof of the main result