Minicourse II Symplectic Monte Carlo Methods for Random Equilateral Polygons

Transcription

1 Minicourse II Symplectic Monte Carlo Methods for Random Equilateral Polygons Jason Cantarella and Clayton Shonkwiler University of Georgia Georgia Topology Conference July 9, 2013

2 Basic Definitions Symplectic Structure Connections Conclusion Summary of the Constructions Summary of Tom Needham s Talk The fixed edgelength polygon space Pol(n; r) is symplectic. M n 2 (C) V 2 (C n )=µ 1 U(2) (0) //U(2) = proj. µ 1 U(1) n (r) = j S3 rj //U(1) n =Hopf Gr 2 (C n ) µ 1 U(1) n (r) //U(1) n µ 1 SO(3) (0) j S2 r j //SO(3) Pol r (n) Tom Needham Symplectic Structures of Polygon Spaces

3 Basic Definitions Symplectic Structure Summary Connections of Tom Needham s Talk II Conclusion Summary of the Constructions Pol(n; r) s symplectic volume is the standard measure. M n 2 (C) V 2 (C n )=µ 1 U(2) (0) //U(2) = proj. µ 1 U(1) n (r) = j S3 rj //U(1) n =Hopf Gr 2 (C n ) µ 1 U(1) n (r) //U(1) n µ 1 SO(3) (0) j S2 r j //SO(3) Pol r (n) Tom Needham Symplectic Structures of Polygon Spaces

4 Basic Summary Definitions of Tom Needham s Talk III Symplectic Structure Connections Measure on the Grassmannian Conclusion of variable edgelength polygons is a natural combination of measures on fixed edgelength polygons. Summary of the Constructions M n 2 (C) V 2 (C n )=µ 1 U(2) (0) //U(2) = proj. µ 1 U(1) n (r) = j S3 rj //U(1) n =Hopf Gr 2 (C n ) µ 1 U(1) n (r) //U(1) n µ 1 SO(3) (0) j S2 r j //SO(3) Pol r (n)

5 Why do we care? Symplectic Geometry Recap A symplectic manifold M 2n has a special 2-form ω called the symplectic form. The n-th power of this form ω n is the volume form on M 2n. The circle acts by symplectomorphisms on M 2n if the action preserves ω. A circle action generates a vector field X on M 2n. We can contract the vector field X with ω to generate a one-form: ι X ω( v) = ω(x, v) If ι X ω is exact, the map is called Hamiltonian and it is dh for some smooth function H on M 2n. The function H is called the momentum associated to the action, or the moment map.

6 Symplectic Geometry Recap II A torus T k which acts by symplectomorphisms on M so that the action is Hamiltonian induces a moment map µ : M R k where the action preserves the fibers. Theorem (Atiyah, Guillemin Sternberg, 1982) The image of µ is a convex polytope in R k called the moment polytope. Theorem (Duistermaat Heckman, 1982) The pushforward of Liouville measure to the moment polytope is piecewise polynomial. If k = n the manifold is called a toric symplectic manifold and the pushforward measure is the uniform measure on the polytope.

7 A Down-to-Earth Example Let (M, ω) be the 2-sphere with the standard area form. Let T 1 = S 1 act by rotation around the z-axis. Then the moment polytope is the interval [ 1, 1], and S 2 is a toric symplectic manifold. Theorem (Archimedes, Duistermaat Heckman) The pushforward of the standard measure on the sphere to the interval is 2π times Lebesgue measure. π Illustration by Kuperberg.

8 Probability and Toric Symplectic Manifolds If M 2n is a toric symplectic manifold with moment polytope P R n, then the inverse image of each point in P is an n-torus. This yields α : P T n M which parametrizes a full-measure subset of M by action-angle coordinates. Proposition The map α : P T n M is measure-preserving. Therefore, we can sample M with respect to Liouville measure by sampling P and T n independently and uniformly. For example, we can sample S 2 uniformly by choosing z and θ independently and uniformly.

9 An Extended Example: Equilateral Arm Space Toric symplectic manifold

10 An Extended Example: Equilateral Arm Space Toric symplectic manifold equilateral arms Arm(n; 1) = ΠS 2.

11 An Extended Example: Equilateral Arm Space Toric symplectic manifold equilateral arms Arm(n; 1) = ΠS 2. Torus action

12 An Extended Example: Equilateral Arm Space Toric symplectic manifold equilateral arms Arm(n; 1) = ΠS 2. Torus action spin each edge around z axis

13 An Extended Example: Equilateral Arm Space Toric symplectic manifold equilateral arms Arm(n; 1) = ΠS 2. Torus action spin each edge around z axis Moment map µ

14 An Extended Example: Equilateral Arm Space Toric symplectic manifold equilateral arms Arm(n; 1) = ΠS 2. Torus action spin each edge around z axis Moment map µ µ(e 1,..., e n ) = (z 1,..., z n ), z-coordinates of e i.

15 An Extended Example: Equilateral Arm Space Toric symplectic manifold equilateral arms Arm(n; 1) = ΠS 2. Torus action spin each edge around z axis Moment map µ µ(e 1,..., e n ) = (z 1,..., z n ), z-coordinates of e i. Moment polytope P

16 An Extended Example: Equilateral Arm Space Toric symplectic manifold equilateral arms Arm(n; 1) = ΠS 2. Torus action spin each edge around z axis Moment map µ µ(e 1,..., e n ) = (z 1,..., z n ), z-coordinates of e i. Moment polytope P hypercube [ 1, 1] n.

17 An Extended Example: Equilateral Arm Space Toric symplectic manifold equilateral arms Arm(n; 1) = ΠS 2. Torus action spin each edge around z axis Moment map µ µ(e 1,..., e n ) = (z 1,..., z n ), z-coordinates of e i. Moment polytope P hypercube [ 1, 1] n. Action-angle coordinates α

18 An Extended Example: Equilateral Arm Space Toric symplectic manifold equilateral arms Arm(n; 1) = ΠS 2. Torus action spin each edge around z axis Moment map µ µ(e 1,..., e n ) = (z 1,..., z n ), z-coordinates of e i. Moment polytope P hypercube [ 1, 1] n. Action-angle coordinates α α(z i, θ i ) = ( 1 zi 2 cos θ i, 1 zi 2 sin θ i, z i ).

19 An Extended Example: Equilateral Arm Space Toric symplectic manifold equilateral arms Arm(n; 1) = ΠS 2. Torus action spin each edge around z axis Moment map µ µ(e 1,..., e n ) = (z 1,..., z n ), z-coordinates of e i. Moment polytope P hypercube [ 1, 1] n. Action-angle coordinates α α(z i, θ i ) = ( 1 zi 2 cos θ i, 1 zi 2 sin θ i, z i ). Proposition (with Shonkwiler) The pushforward of z i (the projection of e i on the z-axis) from Arm(n; 1) to R is the pushforward of z i from the hypercube [ 1, 1] n to R.

20 Lemma (with Shonkwiler) The pushforward of uniform measure on the hypercube [ 1, 1] n to [ n, n] by the linear function x i is given by f n (x) = 1 2 n n SA(x, [ 1, 1]n ), where SA(x, [ 1, 1] n ) is the volume of the slice of the hypercube by the plane x i = x. Proposition (with Shonkwiler) The pdf of the end-to-end distance l [0, n] over Arm(n; 1) is φ(l) = l SA(l 1, [ 1, 1]n 1 ) SA(l + 1, [ 1, 1] n 1 ) 2 n 1 n 1

21 Proof The pdf of z i is f n (x). Since this is a projection of the spherically symmetric distribution of the end-to-end vector in R 3 to the z-axis, the pdf of end-to-end distance l is given by To differentiate f n (l), observe f n (x) = rn φ(l) = 2lf n(l). r n f n 1 (x y) 1 2r n dy = F n 1(x + r n ) F n 1 (x r n ) 2r n, where F n 1 (x) is the integral of f n 1. Differentiate, and substitute in the results of the Lemma.

22 Is that the Rayleigh/Chandrasekhar sinc formula? Let sinc x = sin x/x. Pólya (1912) showed that the volume of the central slab of the hypercube [ 1, 1] n 1 given by a 0 x i a 0 is given by Vol(a 0 ) = 2n a 0 π 0 sinc a 0 y sinc n 1 y dy. Our SA(x, [ 1, 1] n 1 ) is the n 1 dimensional volume of a face of this slab; since it is this face (and its symmetric copy) which sweep out n-dimensional volume as a 0 increases, we can deduce that n 1 SA(x, [ 1, 1] n 1 ) = Vol (x), 2 so SA(x, [ 1, 1] n 1 ) = 2n 1 n 1 π 0 cos xy sinc n 1 y dy.

23 The sinc formula explained This implies that SA(l 1, [ 1, 1] n 1 ) SA(l + 1, [ 1, 1] n 1 ) = Multiplying by = 2n 1 n 1 π 0 2 sin y sin ly sinc n 1 y dy = 2n n 1 π l 2 n 1 gives the pdf for l as n 1 Theorem (Rayleigh, 1919) φ(l) = 2l π 0 y sin ly sinc n y dy, 0 y sin ly sinc n y dy.

24 Part II: Toric Symplectic Structure on Closed Polygons Theorem (Kapovich Millson 1996, Hitchin 1987, Howard-Manon-Millson 2011) If we let Pol(n; r) be the space of closed space polygons with edgelengths (r 1,..., r n ) = r (mod SO(3)), then given an abstract triangulation T of the n-gon: symplectic structure Pol(n; r) is the 2n 6 dimensional symplectic reduction of n i=1 S2 (r i ) by the (Hamiltonian) diagonal SO(3) action. torus action fold polygon around n 3 chords in T moment map µ lengths of n 3 chords in T moment polytope P determined by triangle inequalities action-angle coordinates α build triangles and embed with given dihedrals

25 The Fan Triangulation We call this the fan triangulation: The corresponding moment polytope is given by the inequalities: 0 d 1 r 1 + r 2 r i+2 d i + d i+1 d i d i+1 r i+2 0 d n 3 r n + r n 1

26 Moment Polytopes The moment polytope corresponding to a fan triangulation is determined by the fan triangulation inequalities 0 d 1 r 1 + r 2 r i+2 d i + d i+1 d i d i+1 r i+2 0 d n 3 r n + r n 1 Moment polytope for Pol(5; 1) Moment polytope for Pol(6; 1)

27 Expected Value of Chord Lengths Proposition (with Shonkwiler) The expected length of a chord skipping k edges in an n-edge equilateral polygon is the k 1st coordinate of the center of mass of the moment polytope for Pol(n; 1). n k = , , ,035 97, , , ,059 91, ,499 78, , ,337 91, ,608 78, , ,337 91, ,985 78,400 1, ,059 91, ,608 78, ,121 91, ,499 78,400 97,456 78,400

28 Part III: Markov Chain on Toric Symplectic Manifold We can sample any toric symplectic manifold (such as Pol(n; r)) using a Markov chain in action-angle coordinates: TORIC-SYMPLECTIC-MCMC( p, θ, β) prob = UNIFORM-RANDOM-VARIATE(0, 1) if prob < β then Get point in P using hit-and-run. v = RANDOM-RN-DIRECTION(n) (t 0, t 1 ) = FIND-INTERSECTION-ENDPOINTS(P, p, v) t = UNIFORM-RANDOM-VARIATE(t 0, t 1 ) p = p + t v else Get point in T n uniformly. for ind = 1 to n do θ ind = UNIFORM-RANDOM-VARIATE(0, 2π) return ( p, θ) We can then convert a point (p, θ) P T n to a point in M 2n using the map α.

29 Convergence of TSMCMC(β) Proposition (with Shonkwiler) TORIC-SYMPLECTIC-MCMC(β) converges geometrically to Liouville measure on M 2n.

30 Convergence of TSMCMC(β) Proposition (with Shonkwiler) TORIC-SYMPLECTIC-MCMC(β) converges geometrically to Liouville measure on M 2n. Proposition (with Shonkwiler) Suppose that M 2n is a toric symplectic manifold with moment polytope P and action-angle coordinates α: P T n M 2n. Further, let P m ( p, θ, ) be the m-step transition probability of the Markov chain given by TORIC-SYMPLECTIC-MCMC(β) and let µ be the measure defined by symplectic volume on M 2n. There are constants R < and ρ < 1 so that for any ( p, θ) int(p) T n, α P m ( p, θ, ) µ < Rρ m. TV

31 Error Analysis for Integration with TSMCMC(β) Suppose f is a function M 2n R. If a run of the TSMCMC(β) algorithm produces (( p 0, θ 0 ), ( p 1, θ 1 ),...), let µ m (f ) := 1 m m f (α( p k, θ k )) k=1 be the sample average of the values of f over the run. Proposition (with Shonkwiler) If f is square-integrable, there exists a real number σ(f ) so that m( µm (f ) E(f )) w N (0, σ(f ) 2 ), the Gaussian distribution with mean 0 and standard deviation σ(f ) 2. Here the superscript w denotes weak convergence and E(f ) is the expectation of f.

32 Estimating the error variance σ(f ) We can use Geyer s technique to construct a statistically consistent estimator for the variance σ(f ) using the sample variance and covariance over a run. Let the lag-k covariance of a run γ k (f ) be given by γ k (f ) = 1 m k [f (α( p i, θ m i )) µ m (f )][f (α( p i+k, θ i+k )) µ m (f )] i=1 where µ m (f ) is the sample mean over the run, and let Γ k (f ) = γ 2k (f ) + γ 2k+1 (f ) Theorem (Geyer, 1992) If the points in a run were sampled from the stationary distribution of the Markov chain, the sequence of Γ k (f ) would be positive, decreasing, and convex.

33 The Initial Positive Sequence Estimator Definition (Geyer, 1992) Given a function f and a length-m run of the TSMCMC algorithm as above, let N be the largest integer so that Γ 1 (f ),..., Γ N (f ) are all strictly positive. Then the initial positive sequence estimator for σ(f ) is σ m (f ) := σ m,n (f ) = γ 0 (f ) + 2 γ 1 (f ) + 2 N Γ k (f ). k=1 Given a run, after we compute µ m (f ) and σ m (f ) a 95%-confidence interval for the true expectation of f is then given by E(f ) µ m (f ) ± 1.96 σ m (f )/ m

34 Example Computations Total Curvature for Equilateral 64-gons 1,000,000 steps, about 1 minute runtime TSMCMC(0.5) Perm Sample Mean IPS Error Estimate

35 Focusing on mostly permutation runs Proposition (with Shonkwiler) Expected total curvature of an equilateral 64-gon is Dihedral Moment Polytope Perm Mean IPS Error Actual

36 Confined Polygons Definition A polygon p Pol(n; r) is in rooted spherical confinement of radius r if each diagonal length d i r. Such a polygon is contained in a sphere of radius r centered at the first vertex. Illustration by Diao et al.

37 Sampling Confined Polygons Proposition (with Shonkwiler) Polygons in Pol(n; r) in rooted spherical confinement in a sphere of radius r are a toric symplectic manifold with moment polytope 0 d 1 r 1 + r 2 r i+2 d i + d i+1 d i d i+1 r i+2 0 d n 3 r n + r n 1 and the additional linear inequalities d i r. These polytopes are simply subpolytopes of the fan triangulation polytopes. Many other confinement models are possible!

38 Unconfined 100-gons

39 Unconfined 100-gons

40 Unconfined 100-gons

41 Unconfined 100-gons

42 50-confined 100-gons

43 50-confined 100-gons

44 50-confined 100-gons

45 50-confined 100-gons

46 20-confined 100-gons

47 20-confined 100-gons

48 20-confined 100-gons

49 20-confined 100-gons

50 10-confined 100-gons

51 10-confined 100-gons

52 10-confined 100-gons

53 10-confined 100-gons

54 5-confined 100-gons

55 5-confined 100-gons

56 5-confined 100-gons

57 5-confined 100-gons

58 2-confined 100-gons

59 2-confined 100-gons

60 2-confined 100-gons

61 2-confined 100-gons

62 1.1-confined 100-gons

63 1.1-confined 100-gons

64 1.1-confined 100-gons

65 1.1-confined 100-gons

66 Thank you for listening! Thank you!

67 References Probability Theory of Random Polygons from the Quaternionic Viewpoint Jason Cantarella, Tetsuo Deguchi, and Clayton Shonkwiler arxiv: To appear in Communications on Pure and Applied Mathematics. The Expected Total Curvature of Random Polygons Jason Cantarella, Alexander Y. Grosberg, Robert Kusner, and Clayton Shonkwiler arxiv: Symplectic Methods for Random Equilateral Polygons and the Moment Polytope Sampling Algorithm Jason Cantarella and Clayton Shonkwiler In preparation.