Two-curve Greens function for 2 -SLE Dapeng Zhan Michigan State - PowerPoint PPT Presentation

Two-curve Green’s function for 2 -SLE Dapeng Zhan Michigan State University Random Conformal Geometry and Related Fields June 18-22, 2018, KIAS 1 / 37

2 -SLE A multiple 2-SLE κ ( κ ∈ (0 , 8)) is a pair of random curves ( η 1 , η 2 ) in a simply connected domain D connecting two pairs of boundary points ( a 1 , b 1 ; a 2 , b 2 ) such that conditioning on any curve, the other is a chordal SLE κ curve in a complement domain. If κ ∈ (0 , 4], η 1 and η 2 are disjoint; if κ ∈ (4 , 8), η 1 and η 2 may or may not intersect. A 2-SLE arises naturally as interacting flow lines in imaginary geometry, as scaling limit of some lattice model with alternating boundary conditions, and as two exploration curves of a CLE. It is known that a 2-SLE κ exists for any κ ∈ (0 , 8) and any admissible connection pattern ( D ; a 1 ↔ b 1 , a 2 ↔ b 2 ), and its law is unique. Moreover, the marginal law of either η 1 or η 2 is that of an hSLE κ , i.e., hypergeometric SLE κ with the other pair of points as force points. 2 / 37

Two-curve Green’s Function A two-curve Green’s function for a 2-SLE κ : ( η 1 , η 2 ) at z 0 ∈ D is the limit r ↓ 0 r − α P [dist( z 0 , η j ) < r, j = 1 , 2] lim for some suitable α . We need to find the correct exponent α , prove the convergence of the limit, and find the explicit formula for the limit. We can ask the similar question for a point z 1 ∈ ∂D \ { a 1 , b 1 , a 2 , b 2 } assuming that ∂D is smooth near z 1 . 3 / 37

Works on one-curve Green’s Function Lawler, ’09: conformal radius version of Green’s function for chordal SLE. Lawler-Rezaei, ’15: Euclidean distance Green’s function for chordal SLE. Lawler, ’15: boundary point Green’s function for chordal SLE. Alberts-Kozdron-Lawler, ’12: Green’s function for radial SLE. Lenells-Viklund, ’17: Green’s function for SLE κ ( ρ ) and hSLE. Lawler-Werness, ’13: two-point Green’s function for chordal SLE. Rezaei-Zhan, ’16: multi-point Green’s function for chordal SLE. Mackey-Zhan, ’17: multi-point estimate for radial SLE. 4 / 37

Main Results Throughout, we fix κ ∈ (0 , 8). Sometimes we require that κ ∈ (4 , 8). A constant depends only on κ . Define two exponents: α 0 = (12 − κ )( κ + 4) α 1 = 2 κ (12 − κ ) . , 8 κ The α 0 appeared in the work [Miller-Wu], where 2 − α 0 was shown to be the Hausdorff dimension of the double points of SLE κ for κ ∈ (4 , 8). Let F be the hypergeometric function 2 F 1 ( α, β ; γ, · ) with α = 4 κ , β = 1 − 4 κ , γ = 8 κ , defined by ∞ � ( α ) n ( β ) n x n , F ( x ) = n !( γ ) n n =0 where ( α ) 0 = 1 and ( α ) n = α ( α + 1) · · · ( α + n − 1) if n ≥ 1. The series has radius 1. With these particular parameters α, β, γ , F extends continuously to x = 1, and is positive on [0 , 1]. Such F was used to define hSLE κ . 5 / 37

Interior Point Green’s Function Let D be a simply connected domain with four distinct boundary points (prime ends): a 1 , b 1 , a 2 , b 2 such that b 1 and b 2 do not lie on the same connected component of ∂D \ { a 1 , a 2 } . Define G 0 D ; a 1 ,b 1 ; a 2 ,b 2 on D by the following. If D = D = {| z | < 1 } and z 0 = 0, then 8 4 G 0 κ − 1 ( | a 1 − a 2 || b 1 − b 2 | ) D ; a 1 ,b 1 ; a 2 ,b 2 (0) =( | a 1 − b 1 || a 2 − b 2 | ) κ � | a 1 − b 2 || a 2 − b 1 | � − 1 × F . | a 1 − a 2 || b 1 − b 2 | In general, if f maps D conformally onto D and takes z 0 to 0, then G 0 D ; a 1 ,b 1 ; a 2 ,b 2 ( z 0 ) = | f ′ ( z 0 ) | α 0 G 0 D ; f ( a 1 ) ,f ( b 1 ); f ( a 2 ) ,f ( b 2 ) (0) . 6 / 37

Main Theorems Theorem Let ( η 1 , η 2 ) be a 2 -SLE κ with connection pattern ( D ; a 1 ↔ b 1 , a 2 ↔ b 2 ) . There exist constants C 0 , β 0 > 0 such that for any z 0 ∈ D , with R := dist( z 0 , ∂D ) , P [dist( z 0 , η j ) < r, j = 1 , 2] = r α 0 C 0 G 0 D ; a 1 ,b 1 ; a 2 ,b 2 ( z 0 )(1 + O ( r/R ) β 0 ) . This implies that � r � α 0 . P [dist( z 0 , η j ) < r, j = 1 , 2] � R If κ ∈ (4 , 8) , then there is a constant C ′ 0 > 0 such that P [dist( z 0 , η 1 ∩ η 2 ) < r ] = r α 0 C ′ 0 G 0 D ; a 1 ,b 1 ; a 2 ,b 2 ( z )(1 + O ( r/R ) β 0 ) . 7 / 37

Boundary Point Green’s Function Define another function G 1 D ; a 1 ,b 1 ; a 2 ,b 2 on the analytic part of ∂D \ { a 1 , b 1 , a 2 , b 2 } by the following. If D = H = { Im z > 0 } , z 1 = 0 and a 1 , b 1 , a 2 , b 2 ∈ R \ { 0 } , then 8 4 G 1 κ − 1 ( | a 1 − a 2 || b 1 − b 2 | ) H ; a 1 ,b 1 ; a 2 ,b 2 (0) =( | a 1 − b 1 || a 2 − b 2 | ) κ � | a 1 − b 2 || a 2 − b 1 | � − 1 × | a 1 b 1 a 2 b 2 | 1 − 12 κ F . | a 1 − a 2 || b 1 − b 2 | In general, if f maps D conformally onto H and takes z 1 to 0, then G 1 D ; a 1 ,b 1 ; a 2 ,b 2 ( z 1 ) = | f ′ ( z 1 ) | α 1 G 1 H ; f ( a 1 ) ,f ( b 1 ); f ( a 2 ) ,f ( b 2 ) (0) . 8 / 37

Theorem There exist constants C 1 , C ′ 1 , β 1 > 0 such that if D = H , z 1 ∈ R \ { a 1 , b 1 , a 2 , b 2 } , then with R := dist( z 1 , { a 1 , b 1 , a 2 , b 2 } ) , P [dist( z 1 , η j ) < r, j = 1 , 2] = r α 1 C 1 G 1 H ; a 1 ,b 1 ; a 2 ,b 2 ( z 1 )(1 + O ( r/R ) β 1 ); if κ ∈ (4 , 8) , then P [dist( z 1 , η 1 ∩ η 2 ) < r ] = r α 1 C ′ 1 G 1 H ; a 1 ,b 1 ; a 2 ,b 2 ( z 1 )(1 + O ( r/R ) β 1 ) . For a general D and an analytic point z 1 ∈ ∂D \ { a 1 , b 1 , a 2 , b 2 } , we have r ↓ 0 r − α 1 P [dist( z 1 , η j ) < r, j = 1 , 2] = C 1 G 1 lim D ; a 1 ,b 1 ; a 2 ,b 2 ( z 1 ); and when κ ∈ (4 , 8) , r ↓ 0 r − α 1 P [dist( z 1 , η 1 ∩ η 2 ) < r ] = C ′ 1 G 1 lim D ; a 1 ,b 1 ; a 2 ,b 2 ( z 1 ) . 9 / 37

Commuting hSLEs We focus on the interior case, and suppose D = D and z 0 = 0. Because of the reversibility of SLE κ , we may assume that η j grows from a j to b j , j = 1 , 2. If η j disconnects 0 from b j at some time T j , then it will not get closer to 0 after T j . So it suffices to consider the portions of η 1 and η 2 before separating 0 from b 1 and b 2 . We may parametrize these portions of η 1 and η 2 using radial parametrization (viewed from 0). Then they become two radial Loewner curves such that η 1 is an hSLE κ in D from a 1 to b 1 with force points a 2 and b 2 ; and η 2 is likewise. η 1 and η 2 commute with each other in the sense that if τ 2 is a stopping time for η 2 that happens before T 2 , then conditional on η 2 | [0 ,τ 2 ] , η 1 up to hitting η 2 [0 , τ 2 ] is an hSLE κ from a 1 to b 1 in a complement domain of η 2 [0 , τ 2 ] in D with force points η 2 ( τ 2 ) and b 2 ; and η 2 is likewise. 10 / 37

Commuting radial SLE κ (2 , 2 , 2) There are other pairs of random curves in D starting from a 1 and a 2 that satisfy similar commutation relations. One of them is the commuting pair of radial SLE κ (2 , 2 , 2) curves. More specifically, there is a pair ( η 1 , η 2 ) such that η 1 is a radial SLE κ (2 , 2 , 2) curve in D from a 1 to 0 with force points b 1 , a 2 , b 2 ; and η 2 is likewise. If τ 2 is a stopping time for η 2 , then conditional on η 2 | [0 ,τ 2 ] , η 1 is a radial SLE κ (2 , 2 , 2) curve from a 1 to 0 in a complement domain of η 2 [0 , τ 2 ] in D with force points η 2 ( τ 2 ) , a 1 , b 1 ; and η 2 is likewise. 11 / 37

4 -SLE These η 1 and η 2 both end at 0 and do not intersect with each other at other points. Given ( η 1 , η 2 ), if we further draw chordal SLE κ curves γ 1 and γ 2 in two complement domains of η 1 ∪ η 2 in D from b 1 and b 2 , respectively, to 0, then ( η 1 , η 2 , γ 1 , γ 2 ) form a 4-SLE κ with connection pattern ( D ; a j → 0 , b j → 0 , j = 1 , 2): if we condition on any three of them, the remaining curve is a chordal SLE κ curve. It can be understood as a 2-SLE κ with connection pattern ( D ; a 1 ↔ b 1 , a 2 ↔ b 2 ) conditioned on the event that both curves pass through 0. 12 / 37

Comparing Laws Let P 2 denote the joint law of the radial Loewner driving functions of a 2-SLE κ with connection pattern ( D ; a 1 → b 1 , a 2 → b 2 ). Let P 4 denote the joint law of the radial Loewner driving functions of the curves starting from a 1 and a 2 of a 4-SLE κ with connection pattern ( D ; a j → 0 , b j → 0 , j = 1 , 2). Using Girsanov Theorem and some study of two-time-parameter martingales, we can conclude that P 2 is locally absolutely continuous w.r.t. P 4 , and derive the local Radon-Nikodym derivatives. 13 / 37

Two-parameter Filtration Here is the setup. Let Σ = � 0 <T ≤∞ C ([0 , T ) , R ). For f ∈ Σ, let T f denote its lifetime, which may be finite or infinite. For each t ≥ 0, let F t := σ ( { f ∈ Σ : T f > s, f ( s ) ∈ A } : 0 ≤ s ≤ t, A ∈ B ( R )) . Then we get a filtration ( F t ) t ≥ 0 . We mainly work on the space Σ 2 , and understand that P 2 and P 4 are probability measures on Σ 2 . The first and second coordinates of Σ 2 respectively generate filtrations t ) t ≥ 0 . Let Q = [0 , ∞ ) 2 be the first quadrant with partial ( F 1 t ) t ≥ 0 and ( F 2 order: t = ( t 1 , t 2 ) ≤ s = ( s 1 , s 2 ) if t 1 ≤ s 1 and t 2 ≤ s 2 . Then we get an Q -indexed filtration ( F t ) t ∈ Q by F ( t 1 ,t 2 ) = F 1 t 1 ∨ F 2 t 2 . An ( F t )-stopping time is a function T : Σ 2 → Q such that { T ≤ t } ∈ F t , ∀ t ∈ Q . 14 / 37