A percolation formula

Let $A$ be an arc on the boundary of the unit disk $U$. We prove an asymptotic formula for the probability that there is a percolation cluster $K$ for critical site percolation on the triangular grid in $U$ which intersects $A$ and such that 0 is surrounded by $K\cup A$.


Motivated by questions raised by Langlands et al
and by M. Aizenman, J. Cardy [Car92,Car] derived a formula for the asymptotic probability for the existence of a crossing of a rectangle by a critical percolation cluster. Recently, S. Smirnov [Smi1] proved Cardy's formula and established the conformal invariance of critical site percolation on the triangular grid. The paper [LSW] has a generalization of Cardy's formula. Another percolation formula, which is still unproven, was derived by G. M. T. Watts [Wat96]. The current paper will state and prove yet another such formula. A short discussion elaborating on the general context of these results appears at the end of the paper.
Consider site percolation on a triangular lattice in C with small mesh δ > 0, where each site is declared open with probability 1/2, independently. (See [Gri89,Kes82] for background on percolation.) It is convenient to represent a percolation configuration by coloring the corresponding hexagonal faces of the dual grid; black for an open site, white for a closed site. Let B denote the union of the black hexagons, intersected with the closed unit disk U, and for θ ∈ (0, 2π) let A = A(θ) be the event that there is a connected component K of B which intersects the arc and such that 0 is in a bounded component of C \ A θ ∪ K . Figure 1 shows the two distinct topological ways in which this could happen.
There is a second interpretation of the Theorem. Suppose that 0 is not on the boundary of any hexagon. Let C 1 be the cluster of either black or white hexagons which contains 0. Let C 2 be the cluster of the opposite color which surrounds C 1 and is adjacent to it. Inductively, let C n+1 be the cluster surrounding and adjacent to C n . Let m be the least integer such that C m is not contained in U, and let C ′ m be the component of U ∩ C m which surrounds 0. Let X : Theorem 1 will be proved by utilizing the relation between the scaling limit of percolation and Stochastic Loewner evolution with parameter κ = 6 (a.k.a. SLE 6 ), which was conjectured in [Sch00] and proven by S. Smirnov [Smi1].
We now very briefly review the definition and the relevant properties of chordal SLE. For a thorough treatment, see [RS]. Let κ ≥ 0, let B(t) be Brownian motion on R starting from B(0) = 0, and set W (t) =

√ κ B(t).
For z in the upper half plane H consider the time flow g t (z) given by (1) The process t → g t is called Stochastic Loewner evolution with parameter κ, or SLE κ . In [RS] it was proven that at least for κ = 8 a.s. there is a uniquely defined continuous path γ : [0, ∞) → H, called the trace of the SLE, such that for where z tends to W (t) from within H. Additionally, it was shown that γ is a.s. transient, namely lim t→∞ γ(t) = ∞, and that when κ ∈ (0, 8) for every Then we may ask if γ passes to the right or to the left of z 0 , topologically. (Formally, this should be defined in terms of winding numbers, as follows. Let β t be the path from γ(t) to 0 which follows the arc |γ(t)|∂U clockwise from γ(t) to |γ(t)| and then takes the straight line segment in R to 0. Then γ passes to the left of z 0 if the winding number of γ[0, t] ∪ β t around z 0 is 1 for all large t.) Theorem 1 will be established by applying the following with κ = 6: Theorem 2. Let κ ∈ [0, 8), and let z 0 = x 0 + i y 0 ∈ H. Then the trace γ of chordal SLE κ satisfies P γ passes to the left of z 0 = 1 2 When κ = 2, 8/3, 4 and 8 the right hand side simplifies to 1+ x 0 y 0 π |z 0 | 2 − arg z 0 π , 1 2 + x 0 |z 0 | , 1 − arg z 0 π and 1 2 , respectively. Let x t := Re g t (z 0 ) − W (t), y t := Im g t (z 0 ), and w t := x t /y t . Proof. Suppose first that κ ∈ [0, 4]. In that case, a.s. γ is a simple path and τ (z 0 ) = ∞, by [RS]. Let r > 0 be much larger than |z 0 |, and let τ r be the first time t such that |γ(t)| = r. Let D + (r) ⊂ H be the bounded domain whose boundary consists of [0, r] ∪ γ[0, τ r ] and an arc on the circle r∂U, and let D − (r) ⊂ H be the bounded domain whose boundary consists of [−r, 0] ∪ γ[0, τ r ] and another arc on the circle r∂U. Given γ we start a planar Brownian motion B from z 0 . With high probability, B will hit γ[0, τ r ] ∪ R before exiting the disk r∂U, provided r is large. This means that if z 0 ∈ D + (r), then B is likely to hit γ[0, τ r ] from within D + (r), or hit [0, r]. By conformal invariance of harmonic measure, this means that the harmonic measure in H of [W (τ r ), ∞) from g τr (z 0 ) is close to 1 if z 0 ∈ D + (r) and close to zero if z 0 ∈ D − (r). Hence, the harmonic measure in H of [0, ∞) from x τr + i y τr is close to 1 if z 0 ∈ D + (r) and close to zero if z 0 ∈ D − (r). Therefore, w τr is close to ±∞ depending on whether z 0 ∈ D ± (r). This proves the lemma in the case κ ∈ [0, 4].
For κ ∈ (4, 8), the analysis is similar. The difference is that a.s. γ is not a simple path, τ (z 0 ) < ∞, and z 0 is in a bounded component of R ∪ γ 0, τ (z 0 ) (see [RS]). Clearly, z 0 is not in a bounded component of R ∪ γ[0, t] when t < τ (z 0 ). Hence, at time τ (z 0 ) the path γ closes a loop around z 0 . The issue then is whether this is a clockwise or counter-clockwise loop. An argument as above shows that this is determined by whether w t → ±∞ as t ↑ τ (z 0 ).
Proof of Theorem 2. Writing (1) in terms of the real and imaginary parts gives, Itô's formula then gives, Make the time change and setW Note thatW / √ κ is Brownian motion as a function of u. From (2), we now get dw = −dW + 4 w du w 2 + 1 . ( We got rid of x t and y t , and are left with a single variable diffusion process w(u). (This is no mystery, but a simple consequence of scale invariance.) Given a starting pointŵ ∈ R for the diffusion (3), and given a, b ∈ R with a <ŵ < b, we are interested in the probability h(ŵ) = h a,b (ŵ) that w will hit b before hitting a. Note that h(w u ) is a local martingale. Therefore, assuming for the moment that h is smooth, by Itô's formula, h satisfies By the maximum principle, these equations have a unique solution, and therefore we find that where f (w) := F 2,1 (1/2, 4/κ, 3/2, −w 2 ) w .
We may now dispose of the assumption that h is smooth, because Itô's formula implies that the right hand side in (4) is a martingale, and it easily follows that it must equal h. By [EMOT53, 2.10.(3)] and our assumption κ < 8 it follows that In particular, the limit is finite, which shows that lim b→∞ h a,b (w) > 0 for all w > a. Hence, the diffusion process (3) is transient. Moreover, An appeal to the lemma now completes the proof.
Proof of Theorem 1. For simplicity, assume that the points 0, 1 and e iθ are not on the boundary of any hexagon in the grid. As above, let B be the intersection of the union of the black hexagons with U, and letB be the union of B and the set S := r e is : r ≥ 1, s ∈ [0, θ] . Let β be the boundary of the intersection of U with the component ofB containing S. Then β is a path in U from 1 to e iθ . It is immediate that the event A is equivalent to the event that 0 appears to the right of the path β; that is, that the winding number of the concatenation of β with the arc A θ with the clockwise orientation around 0 is 1. S. Smirnov [Smi1] has shown that as δ ↓ 0 the law of β tends weakly to the law of the image of the chordal SLE 6 trace γ under any fixed conformal map φ : H → U satisfying φ(0) = 1 and φ(∞) = e iθ . (See also [Smi2].) We may take φ(z) = e iθ z + cot θ 2 − i z + cot θ 2 + i .
The theorem now follows by setting κ = 6 in Theorem 2.
Discussion. According to J. Cardy (private communication, 2001), presently, the conformal field theory methods used by him to derive his formula do not seem to supply even a heuristic derivation of Theorem 1. On the other hand, it seems that, in principle, probabilities for "reasonable" events involving critical percolation can be expressed as solutions of boundary-value PDE problems, via SLE 6 . But this is not always easy. In particular, it would be nice to obtain a proof of Watts' formula [Wat96]. The event A studied here was chosen because the corresponding proof is particularly simple, and because the PDE can be solved explicitly.