Makarov s LIL for SLE

Transcription

1 Makarov s LIL for SLE Nam-Gyu Kang Department of Mathematics, M.I.T. Workshop at IPAM, 2007 Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

2 Outline 1 Introduction and Preliminaries Khintchines LIL Makarovs LIL 2 Results An upper half of LIL for SLE Sketch of proof 3 Further Discussion 2-point correlation function for pre-schwarzian Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

3 Khintchine s LIL Introduction and Preliminaries Khintchine s LIL The law of the iterated logarithm for Brownian motion describes the oscillations of Brownian motion B t near t = 0 and as t. Khintchine s LIL for a Brownian motion B t : lim sup t B t 2t log log t = 1 almost surely. The time inversion of a Brownian motion X t = tb 1/t is a Brownian motion. Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

4 Bloch Functions Introduction and Preliminaries Makarov s LIL An analytic function g on D is called a Bloch function if g B = sup z D (1 z 2 ) g (z) <. This defines a semi-norm. The Bloch functions form a complex Banach space B with the norm g(0) + g B. A connection between Bloch functions and univalent functions: For any conformal map f : D C, log f is a Bloch function and log f B 6. Conversely if g B 1 then g = log f for some conformal map f. Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

5 Makarov s LIL Introduction and Preliminaries Makarov s LIL Khintchine s LIL for a Brownian motion B t : lim sup t B t 2t log log t = 1 almost surely. Makarov s LIL: If g B then, for almost every ζ D, lim sup r 1 T. Lyons synthetic proof of Makarov s LIL. g(rζ) g B. log r log log log 1 r Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

6 Hausdorff Measure Introduction and Preliminaries Makarov s LIL L-functions and Measure Functions A real valued function ϕ on an interval (0, r 0) is called an L-function if ϕ is defined by a finite algebraic combination of exponential functions and logarithmic functions. A measure function is a positive, increasing and continuous function. Hausdorff Measure For a measure function ϕ, define Λ δ ϕ(e) = inf X ϕ(rj) : E [ B(z j, r j), r j δ. Then the limit Λ ϕ(e) = lim δ 0 Λ δ ϕ(e) exists in [0, ], and Λ ϕ(e) is called the ϕ-hausdorff measure of E. Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

7 Introduction and Preliminaries Makarov s LIL Harmonic Measure and Hausdorff Measure Setting ϕ : an L-measure function satisfying lim r 0 ϕ(r)/r =. f : a conformal mapping from D onto a s.c. Ω. ω : the harmonic measure for some point w 0 Ω. ω Λ ϕ ω Λ ϕ lim inf r 1 lim inf r 1 (1 r 2 ) f (rζ) ϕ 1 (1 r) (1 r 2 ) f (rζ) ϕ 1 (1 r) > 0 for a.e. ζ D. = 0 for a.e. ζ D. Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

8 Makarov s LIL Introduction and Preliminaries Makarov s LIL Theorem (Makarov) dim supp ω = 1. For ϕ C (r) = re C log 1 r log log log 1 r (C > 0), ΛϕC is the proper measure for the size of supp ω. Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

9 Jones Theorem Introduction and Preliminaries Makarov s LIL Theorem (Jones) If g B 1 and if there exists β and M < such that for all z 0 D, sup (1 z 2 ) g (z) β, {z:ρ(z,z 0)<M} then there exists c = c(β, M) > 0 such that almost every ζ D, lim sup r 1 Re g(rζ) c. log r log log log 1 r Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

10 Student Version of MATLAB Student Version of MATLAB Introduction and Preliminaries Schramm-Loewner Evolution Makarov s LIL H f t 2 SLE κ map g t (z): t g t (z) = g t (z), g 0 (z) = z. κb t SLE κ backward flow f t = g t : t f t (z) = 2 f t (z) + κb t, f 0 (z) = z. f t (z) has the same distribution as z g 1 t (z + κb t ) κb t. Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

11 Main Results Results An upper half of LIL for SLE Conjecture: Let D = [ 1/2, 1/2] (0, 1). For a fixed time t and a sufficiently small ɛ > 0, on the event E = {inf z D Im f t (z) ɛ}, lim sup y 0 Re log f t (x + iy) log 1 y log log log 1 y for almost all x [ 1/2, 1/2], almost surely. Theorem: An upper bound. For given x [ 1/2, 1/2] \ {0}, = c(κ) := 1 κ/8 + 2/κ, E Re log f T 0 (x + iy) 2 = 1 2 c(κ)2 log 1 y + o(log 1 y ), where T u = T u (z) := inf{t 0 : Im f t (z) e u }. Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

12 Corollary Results An upper half of LIL for SLE Suppose ω is a harmonic measure restricted to {f t (x) : x [ 1/2, 1/2]}. For a sufficiently small ɛ > 0, almost surely, on the event E, Theorem: ω Λ ϕc for c > c(κ) = 1 κ/8 + 2/κ. Conjecture: ω Λ ϕc for c c(κ) = 1 κ/8 + 2/κ. Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

13 Duplantier s Duality Results An upper half of LIL for SLE From the dimension estimate for the trace and outer boundary of the hull, B. Duplantier conjectured that SLE κ (κ = 16/κ) should describe the boundary of the hull of SLE κ when κ > 4. Note that c(κ 1 ) = c(κ) =. κ/8 + 2/κ Figure: The boundary of the hull of SLE 6 describes SLE 8/3. Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

14 E Re log f T 0 (x + iy) 2 Results Sketch of proof E Re log f T 0 (x + iy) 2 = 1 2 c(κ)2 log 1 y + o(log 1 y ), c(κ) = 1 κ/8 + 2/κ. E log f T 0 (x + iy) 2 = c(κ) 2 log 1 y + o(log 1 y ). Itô calculus on log f t (z) and log(f t (z) + κb t ): d log f t (z) = 0 2 db t κ/4 + 4/κ f t (z) + 4 κb t 4 + κ d log(f t(z) + κb t ). [ T0 db t E 0 f t (z) + ] [ T 0 2 dt ] = E κb t 0 f t (z) + κb t 2 [ T0 db t E f t (z) + ] [ 2 1 ] = E t(z) κb t 2 log Im f T0 = log 1 y. Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

15 E Re log f T 0 (x + iy) 2 Results Sketch of proof E Re log f T 0 (x + iy) 2 = 1 2 c(κ)2 log 1 y + o(log 1 y ), c(κ) = 1 κ/8 + 2/κ. d log f t (z) = db t 2c(κ) Re f t (z) + +. κb t dm t := Re db t f t (z) + κb t. W 2 u d M Tu u = Wu 2 du. 1 2 T u = T u(z) := inf{t 0 : Im f t(z) e u }. f Tu (z) + κb Tu = X u + iy u, W u := X u/y u. dw u = 2W udu + p κ/2 p 1 + W 2 u de Bu. W 2 u 1 + Wu 2 du = κ 4(κ + 4) du 1 2κ 2(κ + 4) d W log(1+w2 u)+ u db u. 2(κ + 4) 1 + W 2 u Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

16 Results Distribution of Re log f T 0 (x + iy) Sketch of proof (S. Rohde, O. Schramm) y p E [ (1 + X 2 0 )b f T 0(z) (z) p] = (1 + x 2 /y 2 ) b y q. z = x + iy H with y < 1. For each b R, p := 2b + κb(1 b)/2, q := 4b + κb(1 2b)/2. E f T 0 (z) p Cy p( p/µ p/µ ). [ f T (K) P 0 (z) ] µ log(λy) > λ C erfc( ), µ = κ/ /κ. y 2 log λ (K) f t is a.s. h-hölder (κ 4) if h < h(κ) = 1 1 µ µ 2 µ. (Choose p = (2 + 1 µ ) + (µ + 1) 1 for Chebyshev.) µ µ (I. Binder) f (1/h(κ)) = 0 for the multifractal spectrum f of SLE κ. (D. Beliaev) The average integral means spectrum for the SLE. (J. Lind) Hölder regularity for the SLE trace. Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

17 Results Distribution of Re log f T 0 (x + iy) Sketch of proof (S. Rohde, O. Schramm) y p E [ (1 + X 2 0 )b f T 0(z) (z) p] = (1 + x 2 /y 2 ) b y q. z = x + iy H with y < 1. For each b R, p := 2b + κb(1 b)/2, q := 4b + κb(1 2b)/2. [ f T P 0 (z) ] µ log(λy) > λ C erfc( ), µ = κ/ /κ. y 2 log λ [ ] P n = P log f T 0 (x + iy n ) > α log 1yn log log log 1yn. For log 1 = r n (r > 1), log λy n = α ( log(n log r) 1 + O ( log(n log r) ) ) y n log λ r n. P n C(n log r) µ 2 α2, and Borel-Cantelli. Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

18 Results Distribution of Re log f T 0 (x + iy) Sketch of proof (S. Rohde, O. Schramm) y p E [ (1 + X 2 0 )b f T 0(z) (z) p] = (1 + x 2 /y 2 ) b y q. z = x + iy H with y < 1. For each b R, p := 2b + κb(1 b)/2, q := 4b + κb(1 2b)/2. E f T 0 (z) p Cy p( p/µ p/µ ). [ ] P n = P log f T 0 (x + iy n ) > α log 1yn log log log 1yn. For log 1 y n = r n (r > 1), choose p = µα log(n log r)r n/2 for Chebyshev. P C(n log r) µ 2 α2, and Borel-Cantelli. Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

19 Further Discussion 2-point correlation function for pre-schwarzian The Linear Dependence Correlation coefficients S n = log f t (x + ia n ) (A > 1), S n = X X n. f t (z n ) X n C Re y n f t (z n ). yf t (z) L t = L t (z) := f t (z) + 4 ( yf t (z) 4 + κ f t (z) + y ) κb t z 2 t yf s (z) L t = κ/4 + 4/κ (f s (z) + κb s ) db s. 2 0 The complex martingale L t (z) at t = has exponential decay of correlations: EL (z 1 )L (z 2 ) 1 = 2( κ/4 + d H (z 1, z 2 ) 4/κ) 2 cosh 2, 2 where cosh d H (z 1, z 2 ) = 1 + z1 z2 2 2y 1y 2 = 1 + z1 z2 2 2y 1y 2. Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

20 Further Discussion 2-point correlation function for pre-schwarzian The Linear Dependence Correlation coefficients L t (z) = 2 t yf s (z) κ/4 + 4/κ (f s (z) + κb s ) db s. 2 t 0 yf s (z) N t (z) := 0 (f s (z) + κb s ) db s. 2 [ yvf t (z)f t (w) ] Cov(N (z), N (w)) = E 0 (f t (z) + κb t ) 2 (f t (w) + κb t ) dt. 2 A chordal version of Goluzin s identities: d ft(z) ft(w) log = d f t (z)f t (w) dt z w z w dt (f t(z) f. t(w)) 2 = 2 z w (f t(z) + U t)(f t(w) + U = 2f t (z)f t (w) t) (f t(z) + U t) 2 (f t(w) + U. t) 2 [ 1 d yvf t (z)f t (w) ] Cov(N (z), N (w)) = E 2 dt (f t (z) f t (w)) dt = 1 yv 2 2 (z w) 2. 0 Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

21 Mixing Coefficients Further Discussion 2-point correlation function for pre-schwarzian Philosophically, LIL holds for any process for which the Borel-Cantelli lemma, CLT with a reasonably good remainder, and a certain maximal inequality are valid. Several strongly mixing coefficients were introduced to measure the dependence between the σ-fields and were used in the context of limit theorems for dependent random variables. For a certain class of weakly dependent random variables, their partial sums can be approximated by Brownian motion with probability one. Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

22 Further Discussion 2-point correlation function for pre-schwarzian Mixing Coefficients For any two σ-fields A and B F, define α(a, B) := ρ(a, B) := sup P(A B) P(A)P(B) ; A A, B B sup f L 2 (A), g L 2 (B) E[fg] E[f ]E[g] f 2 g 2. Suppose {X n } n= is a sequence of random variables. For m n, define the σ-field F n m := σ(x k, m k n). Define α(n) := sup j Z α(f j, F j+n); ρ(n) := sup j Z ρ(f j, F j+n). {X n } n= is said to be strongly mixing (or α-mixing) if α(n) 0 as n, ρ-mixing if ρ(n) 0 as n. Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

23 Further Discussion Mixing Conditions and LIL 2-point correlation function for pre-schwarzian Theorem (Yoshihara) Suppose that {X n} n=1 is a (not necessarily P strictly stationary) sequence of random variables with E[X n] = 0 and E[Xn 2] <. Let Sn = n k=1 X k, σn 2 = E[S2 n ], and σ2 m,n = E[(Sn Sm)2 ]. Suppose that {X n} n=1 is almost surely uniformly bounded and σ n+1 σ n σn 2 σn 2 1 as n, 0 < lim inf lim sup n n n n <, σ2 m,n = (σ2 n σ2 m )(1 + o(1)) as n m. Moreover, suppose that {X n } n=1 satisfies a strong mixing condition with α(n) = O(n 1 ɛ ), for some ɛ > 0. Then almost surely lim sup n S n 2σ 2 n log log σ 2 n = 1. Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24

24 More Computations Further Discussion 2-point correlation function for pre-schwarzian L t (z) = 2 t yf s (z) κ/4 + 4/κ (f s (z) + κb s ) db s. 2 t yf s (z) N t (z) := 0 (f s (z) + κb s ) db s. 2 For two points z = x + iy and w = u + iv H, Cov(N (z), N (w)) = 1 yv 2 (z w). 2 Cov(N (z), 2 N (w)) 2 = 1 yv 2. 2 (z w) 2 0 For z 0 = x 0 + iy 0, z 1 = x 1 + iy 1, w 0 = u 0 + iv 0, and w 1 = u 1 + iv 1 H, Cov(N (z 0 )N (z 1 ), N (w 0 )N (w 1 )) = 1 4 y 0 v 0 (z 0 w 0 ) 2 y 1 v 1 (z 1 w 1 ) 2 + y 0v 1 (z 0 w 1 ) 2 y 1 v 0 (z 1 w 0 ) 2. Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, / 24