Self-avoiding walk ensembles that should converge to SLE

Transcription

1 Tom Kennedy UC Davis, May 9, 2012 p. 1/4 Self-avoiding walk ensembles that should converge to SLE Tom Kennedy University of Arizona, MSRI

2 Tom Kennedy UC Davis, May 9, 2012 p. 2/4 Outline Variety of ensembles of self-avoiding walk. Quick definition of SLE Point to point in simply connected domain Point to boundary arc in simply connected domain Fixed length ensemble Caveat: should - all statements in this talk are conjectures. Most are supported by simulations of the self-avoiding walk. Many of these conjectures appeared in LSW: Lawler, Schramm, Werner, On the scaling limit of planar self-avoiding walk

3 Tom Kennedy UC Davis, May 9, 2012 p. 3/4 Schramm-Loewner Evolution (SLE) One parameter family of random processes in the plane. It satisfies restriction property for κ = 8/3 and so should give the scaling limit of self-avoiding walk if SAW is conformally invariant. It gives a probability measure on simple curves in simply connected domains which is conformally invariant (by definition). Chordal SLE: boundary point to boundary point Radial SLE: boundary point to interior point Dipolar SLE: boundary point to boundary arc By taking a limit of radial SLE in exterior of a small disc from boundary to we also have Full plane SLE: origin to in full plane

4 SLE - 1 flip Tom Kennedy UC Davis, May 9, 2012 p. 4/4

5 SLE - 2 flips Tom Kennedy UC Davis, May 9, 2012 p. 5/4

6 SLE - 3 flips Tom Kennedy UC Davis, May 9, 2012 p. 6/4

7 SLE - 5 flips Tom Kennedy UC Davis, May 9, 2012 p. 7/4

8 SLE -10 flips Tom Kennedy UC Davis, May 9, 2012 p. 8/4

9 Tom Kennedy UC Davis, May 9, 2012 p. 9/4 SLE - definition Keep the angle fixed, let length of slit 0. Normalization Map always sends origin to tip of slit. Chordal: map fixes, has derivative 1 at. Gives curve from 0 to. Radial: map fixes i. Gives curve from 0 to i. Dipolar: map fixes 1 and 1. Gives curve from 0 to R \ [ 1, 1].

10 Tom Kennedy UC Davis, May 9, 2012 p. 10/4 Self-avoiding walk - chordal ensemble Let δ > 0 (the lattice spacing). We work on the lattice δz 2. A self-avoiding walk ω is a finite length nearest walk which does not visit any site more than once. c N is number with N steps starting at 0 in full plane. Lattice connectivity constant : µ = lim N c1/n N Chordal ensemble: Let D be simply connected domain. Fix two boundary points a,b. Look at all SAW s in D between a and b. Define probability of walk ω to be proportional to µ ω. ω is number of steps in ω. Normalize to get a probability measure. Let δ 0 to get a probability measure on curves in D from a to b.

11 Tom Kennedy UC Davis, May 9, 2012 p. 11/4 SAW chordal ensemble Prediction: (LSW) Scaling limit of chordal ensemble is chordal SLE 8/3. No good algorithm to simulate this ensemble, except... If D is half plane H = {z : Im(z) 0} and boundary points are 0 and, then definition of ensemble is different. Take all SAW in H with N steps with uniform probability measure. Let N, then δ 0. We should get chordal SLE in H from 0 to. Monte Carlo simulations show good agreement.

12 SAW - scaling limit Tom Kennedy UC Davis, May 9, 2012 p. 12/4

13 Tom Kennedy UC Davis, May 9, 2012 p. 13/4 SAW radial ensemble Radial ensemble: Again, let D be a simply connected domain, but now a is on the boundary and b is an interior point. Prediction: (LSW) Scaling limit of radial ensemble is radial SLE 8/3. Full plane ensemble: Forget D. Take all N steps SAW s starting at 0 with uniform probabililty measure, let N and then let δ 0. Prediction: (LSW) Scaling limit of full plane ensemble is full plane SLE 8/3. No Monte Carlo tests of these two ways.

14 Tom Kennedy UC Davis, May 9, 2012 p. 14/4 SAW - point to boundary arc Point to arc ensembles Let D be a simply connected domain. Let a be an interior point. We take all SAW s that start at a and stay in D until they end somewhere on the boundary of D. Usual weight of µ ω. Condition on the endpoint and you get radial ensemble of SAW. Just need to know the distribution of the endpoint along the boundary. Caution: Scaling limit of boundary point to boundary arc is not dipolar SLE.

15 Tom Kennedy UC Davis, May 9, 2012 p. 15/4 Conformal covariance of SAW Look at SAW starting at interior point, ending on boundary. One red segment is image of other under the conformal map. Probabilities walk ends in the two red arcs are not equal.

16 Tom Kennedy UC Davis, May 9, 2012 p. 16/4 Conformal covariance of SAW This hitting distribution is conformally covariant, not invariant. Assume boundary is smooth so that distribution of hitting point has a density ρ a,d (z) with respect to Lebesgue measure along the curve. Let ρ a,d (z) be density for domain D. f conformal map on D. Then LSW conjectured that (up to lattice effects) ρ a,d (z) = c f (z) b ρ f(a),f(d) (f(z)), b = 5/8 So if u a,d (z) is density for harmonic measure, then ρ a,d (z) = c u a,d (z) b

17 Where does this come from? Tom Kennedy UC Davis, May 9, 2012 p. 17/4

18 Tom Kennedy UC Davis, May 9, 2012 p. 18/4 Point to boundary Prediction (LSW): Let D be simply connected domain, a an interior point. Consider all SAW s in D which start at a and end somewhere on the boundary of D. Then scaling limit is (up to lattice effects) D P z a SLE ρ a,d (z) d z where P z a SLE is radial SLE in D from z to a. Prediction (LSW): Let D be simply connected domain, a a boundary pt, C a boundary arc not containing a. Consider all SAW s in D which start at a and end somewhere in C. Then scaling limit is (up to lattice effects) C P z a SLE ρ a,d (z) d z where P z a SLE is chordal SLE in D from z to a. No good way to simulate these SAW ensembles (not quite true).

19 Tom Kennedy UC Davis, May 9, 2012 p. 19/4 Simulations of point to boundary Infinite SAW in the half plane. Condition on it crossing curve only once.

20 Tom Kennedy UC Davis, May 9, 2012 p. 20/4 Interior point to boundary Density of the crossing point should be ρ int (θ)ρ ext (θ) = [sin(θ)] 5/8 [sin(θ)] 5/8 Run the simulation and find small but statistically significant discrepancies.

21 Tom Kennedy UC Davis, May 9, 2012 p. 21/4 Lattice effects SAW in full plane Condition to hit a circle exactly once.

22 Tom Kennedy UC Davis, May 9, 2012 p. 22/4 Lattice effects R=0.1 R=0.2 R=0.3 R= Deviation from uniform cdf Polar angle

23 Tom Kennedy UC Davis, May 9, 2012 p. 23/4 Lattice effects Square Triangular Hexagonal Deviation from uniform cdf Polar angle

24 Tom Kennedy UC Davis, May 9, 2012 p. 24/4 Lattice effects Conjecture (K, Lawler): There is function l(θ) (depends on the type of lattice) such that ρ a,d (z) = c u a,d (z) b l(θ(z)) where θ(z) is the angle of the tangent to the boundary at z. l(θ) only depends on the angle, not on the domain. It is a local effect. It does depend on def of ending on boundary or crossing only once. C w

25 Tom Kennedy UC Davis, May 9, 2012 p. 25/4 Lattice effects Number of N steps SAW on one side of line l N (θ)µ N N γ 1 ρ

26 Tom Kennedy UC Davis, May 9, 2012 p. 26/4 Lattice effect function Lattice effect function l(theta) - square lattice theta

27 Tom Kennedy UC Davis, May 9, 2012 p. 27/4 Full plane hitting circle Full plane SAW hitting circle: difference of simulation and conjecture (100K steps, 31M samples) Without l(theta) correction 0 With l(theta) correction theta

28 Tom Kennedy UC Davis, May 9, 2012 p. 28/4 Half plane hitting semi-circle Half plane hitting semicircle: difference of simulation and conjecture (100K steps, 73M samples) Without l(theta) correction 0 With l(theta) correction theta

29 Tom Kennedy UC Davis, May 9, 2012 p. 29/4 Fixed length ensemble Take all SAW starting at 0 in full plane with N steps. With δ = 1, E[ ω(n) 2 ] grows like N 2ν with ν = 3/4. Take δ = N ν. As N we should get a probability measure on simple curves in full plane starting at 0 and ending at a random point whose distance from 0 is O(1). It is not full plane SLE stopped at some fixed natural length. Can do the same in the half plane H = {z : Im(z) 0} with SAW starting at 0. Call this the half plane fixed length ensemble - FLE. If γ is a curve in FLE, let φ γ be (random) Möbius transformation of H which takes endpoint of γ to i. Now φ γ (γ) is random curve in H from 0 to i, as is radial SLE. Prediction (K): γ(end) is endpoint of γ. A is event for curves from 0 to i. E FLE [1(φ γ (γ) A) γ(end) p ] E FLE [ γ(end) p ] = P 0 i SLE(A) where P 0 i SLE is radial SLE in H from 0 to i. p = (ρ γ)/ν = 61/48.

30 Tom Kennedy UC Davis, May 9, 2012 p. 30/4 Simulation test X S Y R

31 Tom Kennedy UC Davis, May 9, 2012 p. 31/4 Simulation test X N=200K N=500K N=1000K

32 Tom Kennedy UC Davis, May 9, 2012 p. 32/4 Simulation test Y N=200K N=500K N=1000K

33 Tom Kennedy UC Davis, May 9, 2012 p. 33/4 Derivation Look at all finite length walks in upper half plane from 0 on δz 2. Z = ω:ω(n) A µ ω A = {z : r 1 z r 2 }. Z = z:z A ω:0 z µ ω Sum on ω will converge to radial SLE in H from 0 to z. Mapping z to i transforms this to SLE from 0 to i.

34 Tom Kennedy UC Davis, May 9, 2012 p. 34/4 Derivation Z = µ ω N=0 ω:ω(n) A, ω =N Number of half plane N step SAW s: b N cµ N N γ 1 ρ. Z = N=0 N=0 µ N b N 1 b N N γ 1 ρ 1 b N ω:ω(n) A, ω =N ω:ω(n) A, ω =N N is very large, so N ν δω is approximately a sample γ from fixed length ensemble. ω(n) A N ν δ 1 γ A Interchange sums, then sum on N gives factor of γ(end) p.

35 Tom Kennedy UC Davis, May 9, 2012 p. 35/4 Fixed length ensemble itself Can you express fixed length ensemble in terms of SLE? Prediction (???) You get the fixed length ensemble by following: Sample a full plane SLE γ from 0 to 1. Let l(γ) be its natural length. Rotate it by a uniformly random angle. Scale it by l(γ) ν so it has natural length 1. Weight it by l(γ) 2ν(b 1), b = 5/48

36 Tom Kennedy UC Davis, May 9, 2012 p. 36/4 Endpoint of fixed length ensemble Look at distance of endpoint of fixed length SAW from origin. Let ρ(r) be its density. Conjectures from 70 s, 80 s : ρ(r) exp( r 1/ν ), r ρ(r) r θ, r 0 with θ = 1 1/ν 2b = 35/24. Previous slide yields ρ(r) = σ(r 1/ν )r 1 1/ν 2b where σ(l) is density of length of SLE curve in full plane from 0 to 1. Hueristic argument gives behavior of σ(l) as l and gives same value of θ. Can you derive σ(l) exp( 1/l) as l 0?

37 Tom Kennedy UC Davis, May 9, 2012 p. 37/4 Dilation ensemble Let D be simply connected domain containing 0 such that rays from 0 intersect boundary only once. Want the scaling limit of the radial ensemble of SAW. Take all SAW s of length N. Give them equal probability. Let λ(ω) > 0 be such that ω/λ(ω) ends on D. ω/λ(ω) may or may not lie in D. Condition on the event that it does. So ω/λ(ω) is a curve in D from 0 to D. Radial SLE?

38 Tom Kennedy UC Davis, May 9, 2012 p. 38/4 Dilation Prediction (K): Let X be a RV for simple curves in D from 0 to D. ESLE 0 D [X] = lim N E N [λ(ω) p W( ω λ(ω) ) X( ω λ(γ) ) ω/λ(ω) D] E N [λ(ω) p W( ω λ(ω) ) ω/λ(ω) D] p = ρ γ = 61/48 ν (radial case), p = 2ρ γ = 3/4 ν (chordal case). E N is uniform probability measure on N-step SAW s. W(ω/λ(ω)) only depends on ω(n) and contains lattice effect and... Simulations have been done to test the LSW conjecture for the distribution of the endpoint along the boundary.

39 Tom Kennedy UC Davis, May 9, 2012 p. 39/4 Dilation ensemble - Equilateral triangle Polar angle Polar angle

40 Tom Kennedy UC Davis, May 9, 2012 p. 40/4 Lattice effect function Dilation ensemble Natural ensemble Cut-curve ensemble

41 Tom Kennedy UC Davis, May 9, 2012 p. 41/4 Conclusions More ensembles: We treated curves as unparametrized curves. Scaling limit of SAW has a natural parametrization. Should correspond to natural length of SLE. Bridges - SAW in half plane Horizontal lines that intersect it only once Conclusion: We have a wide variety of ways in which the self-avoiding walk should converge to SLE 8/3, simulations that support many of them, but no proofs of any of them.

42 Bridges Tom Kennedy UC Davis, May 9, 2012 p. 42/4