A CLT for winding angles of the arms for critical planar percolation

Consider critical percolation in two dimensions. Under the condition that there are k disjoint alternating black and white arms crossing the annulus A(l,n), we prove a central limit theorem and variance estimates for the winding angles of the arms (as n\rightarrow \infty, l fixed). This result confirms a prediction of Beffara and Nolin (Ann. Probab. 39: 1286--1304, 2011). Using this theorem, we also get a CLT for the multiple-armed incipient infinite cluster (IIC) measures.


Introduction
Percolation is a central model of statistical physics. Recall that performing a site percolation with parameter p on a lattice means that each site is chosen independently to be black (open) with probability p and white (closed) with probability 1 − p. It is wellknown that site percolation on the regular triangular lattice exhibits a phase transition at a critical point p c = 1/2: when p ≤ p c there is almost surely no infinite black connected component, whereas when p > p c there is almost surely a unique infinite black connected component.
Consider percolation on a planar lattice. In the literature, given an annulus in the lattice, the arm events are referred to the existence of some number of disjoint paths (arms, see below for a formal definition) crossing the annulus, the color of each path (black or white) being prescribed. These events are very useful for studying critical and near-critical percolation, the so-called arm exponents can be used to describe some fractal properties of critical percolation (see [1,21,27]).
In this paper, we investigate the winding angles of the arms. The motivation mainly came from [2]. In that paper, Beffara and Nolin proved the existence of the monochromatic exponents, and the monochromatic j-arm exponent is strictly between the polychromatic j-arm and (j + 1)-arm exponents. Their proof relied on analyzing the winding angles of the monochromatic and polychromatic arms. They believed that a central limit theorem should hold on the winding angles but did not give the proof.
The winding angles are also interesting in their own rights. In fact, the winding angles for several different planar models have been studied in the literature (e.g., random walk [3], Brownian motion [20,25], loop-erased random walk (LERW) [14], self-avoiding walk (SAW) [8], radial Schramm-Loewner evolution (SLE) [7,23,26], see also Remark 1.4). In these models, from a macroscopic view, the conformal invariance properties were extensively used to derive the winding angle variance and CLT.
We focus on site percolation on the triangular lattice at criticality. We will realize the triangular lattice with site (or vertex) set Z 2 . For a given (x, y) ∈ Z 2 , its neighbors are defined as (x±1, y), (x, y±1), (x+1, y−1) and (x−1, y+1). Edges (bonds) between neighboring or adjacent sites therefore correspond to vertical or horizontal displacements of one unit, or diagonal displacements between two nearest sites along a line making an angle of 135 • with the positive x-axis. Let each site of Z 2 be black or white with probability 1/2 independently of each other, and denote P = P 1/2 the corresponding product probability measure on the set of configurations. We also represent the measure as a (black or white) random coloring of the faces of the dual hexagonal lattice. Let us mention that the results in this paper also hold for critical bond percolation on Z 2 .
A path is a sequence of distinct sites connected by nearest neighbor bonds. The event that two sets of sites X 1 , X 2 ⊂ Z 2 are connected by a black path is denoted by X 1 ↔ X 2 , and X 1 , X 2 are connected by a black or white path is denoted by X 1 ↔ 1 X 2 . Given a set X of sites, let ∂X denote the boundary of X which contains sites in X that are adjacent to some site not in X. A circuit is a path which starts and ends at the same site and does not visit the same site twice, except for the starting site. For a circuit C, define C := C ∪ interior sites of C.
Let σ = (σ i ) be a sequence of colors. Given two circuits C 1 , C 2 such that C 1 ⊂ C 2 , we say that C 1 is σ-connected to C 2 , if there exist |σ| disjoint paths (arms) connecting C 1 and C 2 , ordered counterclockwise in a cyclic way, and the color of the i-th path is σ i . Denote this event by C 1 ↔ σ C 2 . Define ||x|| ∞ := max{|x|, |y|} for x = (x, y) ∈ Z 2 . For any r ≥ 0, define the square box of sites B(r) := {x ∈ Z 2 : ||x|| ∞ ≤ r}. For 0 < n < m, define the annulus A(n, m) := B(m)\B(n).
For a crossing arm γ in an annulus A(n, m), we often consider γ as a continuous curve by connecting the neighbor sites with line segments and assume the direction of γ is from ∂B(n) to ∂B(m). The winding angle of γ is the overall (algebraic) variation of the argument along it and is denoted by θ(γ).
For a polychromatic configuration in the annulus A(n, m) (i.e., with at least one arm of each color), it is easy to see that the winding angles of the arms differ by at most 2π. In the following, we fix a deterministic way to choose a unique arm γ n,m and focus on θ(γ n,m ), since there is essentially a unique winding angle from a macroscopic point of view.
For two positive functions f and g, the notation f g means that f and g remain of the same order of magnitude, in other words that there exist two positive and finite constants c 1 and c 2 such that c 1 g ≤ f ≤ c 2 g. Now we give our main result in Theorem 1.1, from which we see that a crossing arm of a polychromatic configuration in a long annulus looks like a random logarithmic spiral. Theorem 1.1. Assume that σ is alternating and |σ| is even. Let l be the minimal number such that |∂B(l)| ≥ |σ|. We condition on the event ∂B(l) ↔ σ ∂B(n), n > l. Let θ n := θ(γ l,n ), and a n := V ar[θ n ]. Then we have a n log n, n > l, and under the conditional measure P (·|∂B(l) ↔ σ ∂B(n)) θ n a n → d N (0, 1).
which might be proved by conformal invariance and SLE approach. Heuristically, one can decompose a typical arm into a short path near the origin and for which the winding angle contribution is of a smaller order than √ log n and a long path far from the origin and for which the winding angle contribution can be approximated by the winding angle of multiple (mutually-avoiding) SLE paths (for multiple SLE paths, see Remark 1.4). However, it is still not clear how to prove it rigorously. We will actually use another sequence h n ∼ a n instead of a n , for the expressions for h n , see (3.27).

Remark 1.3.
Our proof mainly relies on the Strong Separation Lemma and the coupling argument in [10]. These two ingredients may be extended to the following more general case without too much work: σ is polychromatic and σ either does not contain neighboring white colors or does not contain neighboring black colors (here we take the first and last elements of σ to be neighbors). See Remark 7 in [5] and subsection 5.4 in [10]. Thus Theorem 1.1 can also be extended to this case.

Remark 1.4.
There exist some analogous results on the winding angles of various random paths. For the classic results on random walk and Brownian motion, the interested reader is referred to a short survey [4]. We address some results concerning SLE as follows. For radial SLE κ , Schramm [23] showed that the variance of the winding angle of the radial SLE κ path truncated at distance ε from the origin grows like (κ+o(1)) log(1/ε) (see also [24]), a CLT was proved simultaneously. However, as the authors said in [26], conditioned there are k disjoint random paths in a long annulus, there are few results about the windings compared with the one path case. Conditioned on the event that there are k mutually-avoiding SLE κ paths crossing the annulus A(1, R) of R 2 , Wieland and Wilson [26] made the conjecture that the winding angle variance of the paths is In [14], conditioned on the annulus A(1, R) of δZ 2 has 2 (resp. 3) disjoint LERWs, Kenyon showed that the winding angle of the paths has variance tending to ( 1 2 +o(1)) log R (resp. ( 2 9 + o(1)) log R) as R → ∞ while δ → 0 (see "Remarks on LERW" in [26] about Kenyon's incorrect values). This confirms the formula (1.1) in the cases of κ = 2 and k = 2, 3, since LERW converges to SLE 2 [18]. We also note that in section 8 and subsection 10.6 in [6], using the method from quantum gravity, Duplantier showed the formula 1.1. See also [7] for the proof from Coulomb gas method. Idea of the proof. The strategy of the proof is similar to [17]. In that paper, Kesten and Zhang constructed a sequence of black circuits surrounding the origin in a Markovian way (the circuits could be thought of as stopping times). Using these circuits, they got a martingale structure on the maximal number of disjoint black circuits in a large box, and then they applied McLeish's CLT [19] for the martingales. However, for our setting clearly we can not use Markovian black circuits to get a martingale structure of the winding angle, thanks to [10], we can use faces instead. As introduced in Section 3 of [10], the faces are some type of circuits which are composed of alternating color paths. With some conditions added to the faces, we construct a sequence of good faces to get a martingale structure of the winding angle. See Fig. 1. Since we are considering conditional measure, it is hard to estimate some events and check the conditions in McLeish's CLT. Thanks to the coupling argument in [10], we can get some weak dependence of the faces and carry out the method from [17]. In the proof of Theorem 1.6 in [5], the authors used black circuits with defects, we note that these circuits may not adapt to the proof of our setting.
Let us give a direct corollary of Theorem 1.1 in the following. First we introduce some definitions. In the celebrated paper [15], Kesten gave the mathematical rigorous definition of the incipient infinite cluster, which describes large critical percolation clusters from the microscopic (lattice scale) perspective [13] and the configuration at a "typical exceptional time" of dynamical critical percolation [11]. More precisely, let 0 denote the origin, it is shown in [15] that the limit exists for any event E that depends on the state of finitely many sites in Z 2 . The unique extension of ν to a probability measure on configurations of Z 2 exists and we call ν the incipient infinite cluster (IIC) measure or one-armed IIC measure. Following Kesten's spirit, Damron and Sapozhnikov introduced multiple-armed IIC measures in [5]. Suppose that σ is alternating and let be the minimal number such that |∂B(l)| ≥ |σ|. For every cylinder event E, it is shown in Theorem 1.6 in [5] the limit exists. The unique extension of ν σ to a probability measure on the configurations of Z 2 exists. We call ν σ the σ-IIC measure. Corollary 1.5. Suppose that σ is alternating and let l be the minimal number such that |∂B(l)| ≥ |σ|. Suppose a n , n > l is the sequence defined in Theorem 1.1. Under P (·|∂B(l) ↔ ∂B(m)), m > n and ν σ , we define θ n similarly as in Theorem 1.1. Under the above measures, we have θ n a n → d N (0, 1). Remark 1.6. Under P (·|∂B(l) ↔ ∂B(m)) and ν σ , using Lemma 2.1 and the coupling result in Lemma 2.3, it is not hard to check that V ar(θ n ) = (1 + o(1))a n . We shall not give the proof here, though.

Remark 1.7.
Following the spirit of [13,26], choosing a typical site from the boundary (or external perimeter) of a large cluster or the pivotal sites of a crossing event in a large box uniformly, we can consider how the arms wind around the chosen site. Since it is expected that the local measure viewed from the typical site converges to the corresponding σ-IIC measure, one would expect a CLT from this corollary. (For the existence of the limiting measures, see Remark of Theorem 1 in [13]. In the 4-arm case, see also Remark 1.7 in [9] for the analog of the tightness result in Theorem 8 in [13].) For a monochromatic σ, there are many ways to select the arms and the winding angles of these arms may differ a lot. Let j = |σ|. We denote by I j,n the set of all the winding angles of the arms in the annulus A( , n), where |∂B(l)| ≥ j. Let θ max,n = θ max,j,n := max{α, α ∈ I j,n } and θ min,n := min{α, α ∈ I j,n }. It is easy to show (see [2]) that θ max,n and θ min,n are of order ± log n. Furthermore, by Proposition 7 in [2], if one sorts the elements of I j,n in increasing order: α 1 < α 2 < · · · < α |Ij,n| , then for every In the 1-arm case, one can also get central limit theorems for θ max,n and θ max,n − θ min,n by similar methods for the proof of Theorem 1.1. Using good black circuits and the coupling argument for the 1-arm case in [10], the proof is similar and simpler, we leave it to the reader and just give the following statements for θ max,n .
Under the conditional measure P (·|0 ↔ σ ∂B(n)) and the IIC measure ν we both have E[θ max,n ] log n, V ar[θ max,n ] log n, In this paper, we only prove the alternating four arm case, since the proof for this case applies to all cases that σ is alternating, with no essential changes. In general, we assume σ = (black, white, black, white) in the following. Throughout this paper, c, c 1 , c 2 , . . . denote positive finite constants that may change from line to line or page to page according to the context.

Preliminary results
As remarked above, we focus on the alternating four arm case. Firstly, following the terminology of [10], let us introduce some definitions. Suppose Γ is the set of percolation interfaces which cross the annulus A = A(m, n). If there are p ≥ 2 interfaces crossing A and if x 1 , . . . , x p denote the endpoints of these interfaces on ∂B(n), define the quality where | · | denotes Euclidean distance. If Γ = ∅, we define Q(Γ) = 0. Let x 1 , . . . , x 4 be four midpoints of four distinct bonds in ∂B(n). We will adopt here cyclic notation, i.e., for any i, j ∈ Z, we have x j = x i if j ≡ i mod 4. For any i ∈ Z, let γ i be a simple path of hexagons joining x i to x i+1 and γ i ⊂ B(n) (here we see γ as a sequence of sites). Assume γ i is black if i is odd and white otherwise. Then we call the circuit Θ which is composed of these four paths a configuration of (interior) faces, and say Θ are faces of ∂B(n). Define the quality of a configuration of faces Q(Θ) to be the least distance between the endpoints (i.e., x 1 , . . . , x 4 ) , normalized by n. Similar to the definition of faces, we call a circuit around ∂B(n) exterior faces, if the circuit is composed of four alternating color paths contained in (Z 2 \B(n))∪∂B(n) with endpoints on ∂B(n). Note that our definition of exterior faces is exactly the same as the definition of faces in [10]. Similar to the quality of faces, we can also define quality of exterior faces.
The following properties of arm events are well-known, see [16,21]. We assume that the reader is familiar with the FKG-inequality, the Russo-Seymour-Welsh (RSW) technology. See [12,27]. Using FKG, RSW and Theorem 11 in [21], the statements related to faces can be easily obtained from the classic results, the proof is omitted here.
Note that for general alternating color sequence σ with even |σ|, the corresponding notion of faces can be defined, and analogous results hold in this more general case.
Proof. Since the proof is basically the same as for Proposition 3.1 in [10] with little modifications for our setting, we only sketch the proof and omit some details. First let us prove inequality (2.3).
Without loss of generality, let N = t/8 ≥ 1. For 0 ≤ i ≤ N , let r i = 8 i r. Now we sample under P (·|Θ ↔ ∂B(R)) the set of interfaces Γ(r i ) which start from the four endpoints of Θ until they reach radius r i . We proceed by induction on the scale r i , i ≥ 0.
By the Strong Separation Lemma (Lemma 3.3 in [10], Lemma 6.2 in [5] is another version), we have It can be checked that Θ plus Γ(2r i ) induce configurations of faces of ∂B(2r i ), which is denoted by Θ i . For each A(4r i , r i+1 ), define Let S i be the union of all sites whose color is determined by the crossing interfaces according to the measure P (·|Θ i , Θ i ↔ σ ∂B(R)). Let S i be a possible value for S i such that R i holds. If Q(Γ(2r i )) > 1/4, by Lemma 2.2, using a gluing technique, it can be showed (the same to the proof of (3.1) in [10]) there is a universal constant c > 0 such Now let us prove the coupling result. Sampling the interfaces for the measures P (·|Θ 1 ↔ σ ∂B(R)) and P (·|Θ 2 ↔ σ ∂B(R)) by induction similarly to the above argument, using the Strong Separation Lemma and (2.5), one can show the coupling result with the strategy very similar to the proof of Proposition 3.1 in [10]. We omitted the details here and refer the reader to "Proof of Proposition 3.1, continued." in [10]. To couple P (·|∂B(1) ↔ σ ∂B(R)) and P (·|Θ 2 ↔ σ ∂B(R)), one can use an argument analogous to the proof of Proposition 3.6 in [10], the details are also omitted here.

Remark 2.4. The coupling argument was introduced in [10]
, which is a very useful tool to gain weak independence of events. The coupling argument is based on the Strong Separation Lemma, which was first proposed in [5] (see a broad overview for the strong separation phenomenon in many planar statistical physics models in Appendix A of [10]), and is an extension of Kesten's arm separation lemma [16,21].

Proof of theorem
As remarked in the Introduction, we focus on the alternating 4-arm case throughout this section. Let Θ p denote the good faces in A(m(p)), 1 ≤ p ≤ q. Recall Θ 0 = ∂B(1).
Let F 0 be the trivial σ-field.
It will also be necessary to introduce a copy (Ω , B , P q,Θp(ω) ) of (Ω , B , P q,Θp(ω) ). The element of Ω is denoted by ω and expectation with respect to P q,Θp(ω) is denoted by E q,Θp(ω) . The following lemma is the analog of (2.11) in [17]. However, we can not get the analogous results corresponding to (2.12) and Lemma 2 in [17]: in [17], exact independence can be revealed, which is not possible here due to the "global" conditioning.
Proof. The proof is similar to the proof of (2.11) in [17]. Fix a configuration ω. By (3.1) and the definition of ∆ p in (3.2), we have the conclusion follows immediately.
Using the symmetric property of the polychromatic setting of the windings and coupling argument, one may give a short proof of (3.6). However, the following method we used to prove (3.6) can be modified easily to prove the analogous result of the 1-arm case.
By Lemma 3.1, for 1 ≤ p ≤ q, For fixed ω, now we will prove there exists a constant c 10 > 0 such that Remark 3.3. When p = 1, we just let P q,Θp−1(ω) = P q , E q,Θp−1(ω) = E q , the arguments in the following also adapt to this case.
To obtain inequality (3.11), let us consider the two cases in the following.
Let us note first that, We now estimate the four terms separately.
By (3.10) and (3.11), we can choose an appropriate constant c 33 such that Now we bound the two terms in the r.h.s of above inequality. For the first term, by (3.5) we get For the second term, if 2 ≤ p ≤ q, by (3.12), there exist c 34 , c 35 > 0 such that if p = 1, we can bound the second term directly by (3.12) and Note 3.3. Thus (3.6) is concluded. Using (3.6), we conclude (3.7),(3.8) and the second inequality in (3.9) immediately. Now let us prove the first inequality in (3.9). By Lemma 3.1 and (3.11), we have ∆ p ≥ θ(Θ p−1 , Θ p ) − c 10 (m(p) − p + 1).
Applying (2.10) and inequality in the above, gives
By (3.12), it is easy to see that E q,n θ(∂B(1), Θ q,n ) exists. Since the winding angles of the arms between ∂B(1) and Θ q,n differ at most 2π, hence E q,n θ max , E q,n θ min also exist. By the symmetry of the lattice and the definition of Θ q,n , it is obvious that E q,n θ max = −E q,n θ min . Hence E q,n [θ max + θ min ] = 0. Then we conclude |E q,n θ(∂B(1), Θ q,n )| ≤ 2π by |E n θ n | ≤ 2π can be proved similarly.  Let us now check the three conditions of Theorem 2.3 in [19]. First we set X p,q,n := ∆ p,q,n q p=1 E q,n ∆ 2 p,q,n 1/2 , then we can write θ(∂B(1), Θ q,n ) − E q,n θ(∂B(1), Θ q,n ) q p=1 E q,n ∆ 2 p,q,n 1/2 = q p=1 X p,q,n .