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

Transcription

1 A CLT for winding angles of the paths for critical planar percolation Changlong Yao Peking University May 26, 2012 Changlong Yao (Peking University) Winding angles for critical percolation May / 20

2 Definition of winding angle What is the winding angle of a path (curve) γ? Definition Let γ(0) be the starting point of γ. We denote by θ(t) = arg(γ(t)) arg(γ(0)) the winding angle of the path γ([0, t]) around 0, with arg chosen continuous along γ. Changlong Yao (Peking University) Winding angles for critical percolation May / 20

3 Classic results on Brownian motion and random walk Brownian motion (Spitzer, 1958). ( ) 2θ(t) x lim P t log t x dy = π(1 + y 2 ) (Cauchy distribution). Random walk (Bélisle, 1989). ( ) 2θ(n) x lim P t log n x = 1 ( πy ) 2 sech dy 2 (hyperbolic secant). Remark: The asymptotic distribution of the big winding angle of Brownian motion is also hyperbolic secant (Messulam and Yor, 1982). Changlong Yao (Peking University) Winding angles for critical percolation May / 20

4 Percolation Consider critical site percolation on the triangular lattice (p c = 1 2 ). Let us mention that our results also hold for critical bond percolation on Z 2. Changlong Yao (Peking University) Winding angles for critical percolation May / 20

5 Arm events Suppose there exist k disjoint paths connecting B(m) and B(n) with a prescribed color sequence σ, we call these paths arms and denote this arm event by B(m) σ B(n). The arm events play a central role in critical percolation. The following graph illustrates a 4-arm event B(1) σ B(R), where σ = (black,white,black,white). Changlong Yao (Peking University) Winding angles for critical percolation May / 20

6 Arm exponents Thanks to the two major breakthroughs due to Schramm (2000) and Smirnov (2001), a lot of arm exponents for critical percolation on the triangular lattice can be computed rigorously by SLE approach and conformal invariance. For example, in the case of polychromatic 4-arm events (Lawler,Schramm and Werner, 2001), Example P( B(1) σ B(R)) = R 5 4 +o(1). Changlong Yao (Peking University) Winding angles for critical percolation May / 20

7 Motivation Beffara and Nolin (2011) got some properties of the monochromatic exponents by analyzing the winding angles of the monochromatic and polychromatic arms. They believed that a central limit theorem hold on the winding angles. Beffara, V. and Nolin, P. (2011). On monochromatic arm exponents for 2D critical percolation. Ann. Probab. 39, Changlong Yao (Peking University) Winding angles for critical percolation May / 20

8 Motivation: related models Radial SLE κ (Schramm, 2000). Var[θ R ] = (κ + o(1)) log(r). Loop-erased random walk, uniform spanning tree (Kenyon, 2000). Two disjoint paths: Var[θ R ] = ( o(1)) log R. Three disjoint paths: Var[θ R ] = ( o(1)) log R. Uniform spanning tree (figure from Kenyon (2000)) Changlong Yao (Peking University) Winding angles for critical percolation May / 20

9 Motivation: related models Self-avoiding walk (Duplantier, 1988, Coulomb gas method) No rigorous result for the winding angle distribution. Multiple SLE paths (Wieland and Wilson, 2003). Conjecture: Conditioned on the event that there are k mutually-avoiding SLE κ paths crossing the annulus A(1, R) of R 2, the winding angle variance of the paths is ( κ k 2 + o(1) ) log R as R. Note: Duplantier showed this conjecture by the methods from quantum gravity (2006) and Coulomb gas (2008). Changlong Yao (Peking University) Winding angles for critical percolation May / 20

10 Main result: CLT for the winding angles Theorem Suppose that σ is alternating and let M be the minimal number such that B(M) σ. Suppose B(M) σ B(n), n > M. Let θ n denote the winding angle of some arm between B(M) and B(n). Let a n := Var[θ n ]. Then we have a n log n, and under the conditional measure P( B(M) σ B(n)) θ n a n d N(0, 1). Note: The arm can be seen as a type of random logarithmic spiral. Changlong Yao (Peking University) Winding angles for critical percolation May / 20

11 Main idea Quasi-multiplicativity and priori bounds for arm events. A martingale method from first passage percolation (Kesten and Zhang, 1997), which was based on McLeish s CLT. Strong separation lemma (Damron and Sapozhnikov, 2011) and coupling arguments (Garban, Pete, and Schramm, 2010, preprint). Some estimates from Beffara and Nolin (2011). Roughly speaking, we construct a sequence of circuits surrounding the origin in a Markovian way. Using these circuits, we can get a martingale structure for the winding angle. Then we check the conditions of the CLT for martingales. Thanks to the coupling arguments, we can get some weak independence of the model. Changlong Yao (Peking University) Winding angles for critical percolation May / 20

12 Main idea Changlong Yao (Peking University) Winding angles for critical percolation May / 20

13 Remark 1: conjecture Following the conjecture of Weiland and Wilson (2003), it is expected that ( ) 6 Var[θ n ] = σ 2 + o(1) log n as n, which may be proved by conformal invariance and SLE approach. Changlong Yao (Peking University) Winding angles for critical percolation May / 20

14 Remark 2: general case By the arguments in Remark 7 in Damron and Sapozhnikov (2011) and subsection 5.4 in Garban, Pete and Schramm (2010), the theorem can 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). In particular, one can prove the CLT for all cases such that σ = 3. Changlong Yao (Peking University) Winding angles for critical percolation May / 20

15 Incipient infinite cluster Following Kesten s spirit (1986), Damron and Sapozhnikov (2011) introduced multiple-armed incipient infinite cluster (IIC) as follows. Definition Suppose that σ is alternating and let M be the minimal number such that B(M) σ. For every cylinder event E, the limit ν σ (E) := lim n P(E B(M) σ B(n)) 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. Changlong Yao (Peking University) Winding angles for critical percolation May / 20

16 Corollary Corollary Under the σ-iic measure ν σ, we have θ n Varθn d N(0, 1) and a 2 n Varθ n 1. Changlong Yao (Peking University) Winding angles for critical percolation May / 20

17 Remark 3: boundary, external perimeter and pivotal sites Choose a typical site from the boundary or external perimeter of a large cluster in a large box uniformly (or choose a typical pivotal site of a crossing event), one 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 can conclude a CLT from our corollary. Changlong Yao (Peking University) Winding angles for critical percolation May / 20

18 An easy extension Our method adapts easily to the 1-arm case. Using a similar but simpler argument, one has the following result. Suppose 0 B(n), let θ max,n (resp. θ min,n ) be the maximum (resp. minimum) winding angle of the arm configuration in B(n). Under P( 0 σ B(n)) and Kesten s IIC measure ν we both have E[θ max,n ] log n, Var[θ max,n ] log n, and θ max,n E[θ max,n ] Var[θmax,n ] d N(0, 1). Remark: One can get analogous results for θ max,n θ min,n. Changlong Yao (Peking University) Winding angles for critical percolation May / 20

19 References Beffara, V. and Nolin, P. (2011). On monochromatic arm exponents for 2D critical percolation. Ann. Probab. 39, Damron, M. and Sapozhnikov, A. (2011). Outlets of 2D invasion percolation and multiple-armed incipient infinite clusters. Probab. Theor. Relat. Fields. 150, Garban, C., Pete, G. and Schramm, O. Pivotal, cluster and interface measures for critical planar percolation. Preprint, arxiv: v2, Járai, A.A. (2003). Incipient infinite percolation clusters in 2D. Ann. Prob. 31, Kesten, H. and Zhang, Y. (1997). A central limit theorem for critical first-passage percolation in two-dimensions. Probab. Theor. Relat. Fields. 107, Weiland, B. and Wilson, D. B. (2003). Winding angle variance of Fortuin-Kasteleyn contours. Phys. Rev. E 68, Changlong Yao (Peking University) Winding angles for critical percolation May / 20

20 Thank you! Changlong Yao (Peking University) Winding angles for critical percolation May / 20