Bridge Decomposition of Restriction Measures Tom Alberts joint work with Hugo Duminil (ENS) with lots of help from Wendelin Werner University of

Transcription

1 Bridge Decomposition of Restriction Measures Tom Alberts joint work with Hugo Duminil (ENS) with lots of help from Wendelin Werner University of Toronto

2 Self-Avoiding Walk Bridges

3 Self-Avoiding Walk Bridges

4 Self-Avoiding Walk Bridges

5 Self-Avoiding Walk Bridges

6 Self-Avoiding Walk Bridges

7 Self-Avoiding Walk Bridges

8 Self-Avoiding Walk Bridges

9 Self-Avoiding Walk Bridges

10 Self-Avoiding Walk Bridges

11 SAW Irreducible Bridges

12 SAW Irreducible Bridges

13 SAW Irreducible Bridges

14 SAW Irreducible Bridges

15 Bridge Decomposition of SAW

16 Bridge Decomposition of SAW

17 Bridge Decomposition of SAW

18 Bridge Decomposition of SAW

19 Bridge Decomposition of SAW # length-n SAW in H β n [Kes63, Kes64] proves that # length-n SAW bridges in H β n and irreducible bridges ω β ω = 1

20 Bridge Decomposition of SAW [MS93], [LSW04] use Kesten s relation to show that the half-plane SAW measure (the weak limit of the uniform measure on n-step half-plane SAWs) can be obtained by concatenating an i.i.d sequence of irreducible bridges. Means that the SAW refreshes itself after each irreducible bridge.

21 SLE(8/3) as the SAW Scaling Limit

22 Does SLE(8/3) Have a Bridge Decomposition?

23 Bridge Points and Heights for SLE(8/3) What are the analogues of bridge points and lines? Definition: L > 0 is a bridge height for the curve γ if γ intersects the horizontal line y = L only once. The corresponding point of intersection is called a bridge point.

24 Bridge Points and Heights for SLE(8/3)

25 Bridge Points and Heights for SLE(8/3) Question: Does an SLE path have bridge points and lines?

26 Bridge Points and Heights for SLE(8/3) Question: Does an SLE path have bridge points and lines? Answer: Calculate their Hausdorff dimension!

27 Restriction Property of SLE(8/3)

28 Restriction Property of SLE(8/3) A

29 Restriction Property of SLE(8/3) A

30 Key restriction formula: Restriction Property of SLE(8/3) P (γ stays in H\A) = φ A(0) 5/8 where φ A is a conformal map from H\A to H such that φ A (z) z as z. A 0 φ A 0

31 Existence of Bridge Points and Lines for SLE(8/3) 0

32 Existence of Bridge Points and Lines for SLE(8/3) ǫ ǫ 0

33 Existence of Bridge Points and Lines for SLE(8/3) ǫ ǫ 0

34 Existence of Bridge Points and Lines for SLE(8/3) ǫ ǫ φ A (0) π 8y 2 cosh 2 ( π 2 x y) ǫ2 0

35 Existence of Bridge Points and Lines for SLE(8/3) φ A(0) 5/8 F(x, y)ǫ 5/4 dim H (bridge points) = 2 5/4 = 3/4

36 Existence of Bridge Points and Lines for SLE(8/3) φ A(0) 5/8 F(x, y)ǫ 5/4 dim H (bridge points) = 2 5/4 = 3/4

37 Existence of Bridge Points and Lines for SLE(8/3) φ A(0) 5/8 F(x, y)ǫ 5/4 dim H (bridge points) = 2 5/4 = 3/4

38 Existence of Bridge Points and Lines for SLE(8/3) φ A(0) 5/8 F(x, y)ǫ 5/4 dim H (bridge points) = 2 5/4 = 3/4

39 Existence of Bridge Points and Lines for SLE(8/3) φ A(0) 5/8 F(x, y)ǫ 5/4 dim H (bridge points) = 2 5/4 = 3/4 dim H (bridge heights) = 3/4

40 Bridge Decomposition of SLE(8/3)

41 Bridge Decomposition of SLE(8/3)

42 Bridge Decomposition of SLE(8/3)

43 Bridge Decomposition of SLE(8/3)

44 Bridge Decomposition of SLE(8/3)

45 Bridge Decomposition of SLE(8/3)

46 Bridge Decomposition of SLE(8/3)

47 Bridge Decomposition of SLE(8/3)

48 Refreshing Property of SLE(8/3) at Bridge Heights

49 Refreshing Property of SLE(8/3) at Bridge Heights

50 Refreshing Property of SLE(8/3) at Bridge Heights

51 Refreshing Property of SLE(8/3) at Bridge Heights

52 Refreshing Property of SLE(8/3) at Bridge Heights Refreshing is with respect to an enlarged filtration: G t := σ (γ[0,t], bridge heights of γ in γ[0,t]) If you know you are currently at a bridge height, then the law of the future curve is just SLE(8/3) in the half-plane above the bridge line.

53 Refreshing Property of SLE(8/3) at Bridge Heights Important Consequence: Can do excursion theory of bridges. Run at the appropriate speed, say L(t), γ(l(t)) is a Poisson Point Process with respect to G t. Implies the existence of a measure on continuous irreducible bridges. SLE(8/3) curve can be constructed by taking an i.i.d. sample from this measure and gluing the pieces together. Similar work already done for Brownian Excursions and certain SLE(κ, ρ) at cutpoints, see [Vir03], [Dub06].

54 Refreshing Property of SLE(8/3) at Bridge Heights

55 Refreshing Property of SLE(8/3) at Bridge Heights Important Consequence: Can do excursion theory of bridges. Run at the appropriate speed, say L(t), γ(l(t)) is a Poisson Point Process with respect to G t. Implies the existence of a measure on continuous irreducible bridges. SLE(8/3) curve can be constructed by taking an i.i.d. sample from this measure and gluing the pieces together. Similar work already done for Brownian Excursions and certain SLE(κ, ρ) at cutpoints, see [Vir03], [Dub06].

56 Are Lines Special for SLE(8/3)?

57 Are Lines Special for SLE(8/3)? Using reversibility, we can immediately see that the answer is no.

58 Decomposition by Circles z 1/z

59 Decomposition by Circles

60 Decomposition by General Shapes

61 Decomposition by General Shapes

62 Decomposition by General Shapes

63 Decomposition by General Shapes

64 Decomposition by General Shapes

65 Is SLE(8/3) Special for the Decomposition?

66 General Restriction Measures There exists a one-parameter family of measures P α on hulls that satisfies the restriction property for α 5/8. α = 5/8: SLE(8/3) α = 1: Brownian Excursions Only α = 5/8 is supported on simple curves. All others are supported on hulls. Key restriction formula: P α (hull stays in H\A) = φ A(0) α where φ A is a conformal map from H\A to H such that φ A (z) z as z.

67 Bridge Decomposition for Restriction Measures Using the same calculation as before, restriction measures have bridge points and lines for α < 1 with dim H (bridge points) = dim H (bridge lines) = 2 2α Refreshing property still holds at bridge heights. Can decompose the restriction hulls at their bridge heights to get a Poisson Point Process.

68 Open Questions Do bridge heights and lines exist for other SLE(κ)? Can one give a more in depth description of the twodimensional bridge point process? Can one do some excursion theory? And is there a continuous analogue of Kesten s relation? Is the Poisson Point Process and the decomposition more natural if the curve is in the natural time parameterization?

69 Problem of Time Parameterization ˆγ(s) := γ(t + s) γ(t) should be have the law of SLE(8/3) in the half plane above the bridge line, when γ is the curve in the correct time parameterization. That is, the right parameterization should have some sort of additivity property.

70 Problem of Time Parameterization The usual time parameterization of SLE will not have this property. Recall it is defined so that the half-plane capacity of γ[0,t] is 2t. A promising candidate for the right time parameterization has been identified by Lawler and Sheffield [LS08]. A related problem has already been solved by Alberts and Sheffield [AS08a].

71 Time Parameterization of SLE(8/3)

72 Time Parameterization of SLE(8/3)

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144 Doob-Meyer Decomposition of the Measure +

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217 Doob-Meyer Decomposition of the Measure Total mass of increasing measure at time n is n. Try to replicate the same idea in the continuum to construct the natural time parameterization for SLE.

218 Natural Time Parameterization for SLE Need to estimate the probability that an SLE curve goes through a point z H. 1 4 θ t(z) dist(z, γ[0,t]) 4θ t (z) where θ t (z) is the conformal distance from z to γ[0,t]. Using a basic stochastic calculus argument, can show that P (θ (z) r) G(z)r 1 κ 8, r 0 G(re iθ ) = r κ 8 1 sin κ 8 +8 κ 2 θ

219 Natural Time Parameterization for SLE Given the curve γ[0,t], have P (θ (z) r γ[0,t]) G ( g t (z) κb t ) (r g t (z) ) 1 κ 8 z r g t γ([0, t]) g t (z)r g t (z) 0 κbt

220 Natural Time Parameterization for SLE Thus for each z H, M t (z) := lim r 0 P (θ (z) r γ[0,t]) r 1 κ 8 is a local martingale. = g t(z) 1 κ 8 G ( g t (z) κb t ) It is important to note that M t (z) is not an honest martingale. Can show that M (z) = {, z γ[0, ) 0, z γ[0, )

221 Natural Time Parameterization for SLE The supermartingale part is replaced by M t (z)d 2 z, H since dim H γ[0, ) < 2 and Lebesgue measure doesn t see it. Does the supermartingale have a Doob-Meyer decomposition as M t (z)d 2 z = N t A t H where N t is a martingale and A t is a non-decreasing process?

222 Natural Time Parameterization for SLE In the natural time parameterization should have that the mass of the increasing measure at time t is t. Need to show that A t is continuous and strictly increasing. If it is then find σ(t) such that A σ(t) = t Then γ(t) := γ(σ(t)) is the curve in the natural time parameterization. Open Question: Does γ have the right additivity property for the bridge decomposition? Or should one be using the bigger filtration G t to create the correct time parameterization for the bridge decomposition?

223 Constructing a Boundary Measure for SLE For 4 < κ < 8, the set γ[0, ) R is fractal with dim H γ[0, ) R = 2 8/κ =: d with probability one, see [AS08b], [SZ07]. Using the same idea as for the natural time parameterization, [AS08a] constructs a measure on γ[0, ) R.

224 Constructing a Boundary Measure for SLE The supermartingale R M t (x) dx is shown to have a non-trivial Doob-Meyer decomposition, i.e. its increasing part is not identically zero. This is accomplished using a correlation bound E [M τ (x)m τ (y)] cx β x y β that comes from a two-point martingale in [SZ07]. correlation bound can be used to show that M t (x) dx is of class D. R The

225 References [AS08a] Tom Alberts and Scott Sheffield. The covariant measure of SLE on the boundary. arxiv: v1 [math.pr], [AS08b] Tom Alberts and Scott Sheffield. Hausdorff dimension of the sle curve intersected with the real line. Electron. Jour. Probab., 13: (electronic), [Dub06] Julien Dubédat. Excursion decompositions for SLE and Watts crossing formula. Probab. Theory Related Fields, 134(3): , [Kes63] Harry Kesten. On the number of self-avoiding walks. J. Mathematical Phys., 4: , [Kes64] Harry Kesten. On the number of self-avoiding walks. II. J. Mathematical Phys., 5: , 1964.

226 [LS08] Gregory F. Lawler and Scott Sheffield. Construction of the natural parameterization for SLE curves. In preparation, [LSW04] Gregory F. Lawler, Oded Schramm, and Wendelin Werner. On the scaling limit of planar self-avoiding walk. In Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2, volume 72 of Proc. Sympos. Pure Math., pages Amer. Math. Soc., Providence, RI, [MS93] Neal Madras and Gordon Slade. The self-avoiding walk. Probability and its Applications. Birkhäuser Boston Inc., Boston, MA, [SZ07] Oded Schramm and Wang Zhou. Boundary proximity of SLE. arxiv: v2 [math.pr], [Vir03] Bálint Virág. Brownian beads. Probab. Theory Related Fields, 127(3): , 2003.

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