## The Annals of Applied Probability

### The symplectic geometry of closed equilateral random walks in 3-space

#### Abstract

A closed equilateral random walk in 3-space is a selection of unit length vectors giving the steps of the walk conditioned on the assumption that the sum of the vectors is zero. The sample space of such walks with $n$ edges is the $(2n-3)$-dimensional Riemannian manifold of equilateral closed polygons in $\mathbb{R}^{3}$. We study closed random walks using the symplectic geometry of the $(2n-6)$-dimensional quotient of the manifold of polygons by the action of the rotation group $\operatorname{SO}(3)$.

The basic objects of study are the moment maps on equilateral random polygon space given by the lengths of any $(n-3)$-tuple of nonintersecting diagonals. The Atiyah–Guillemin–Sternberg theorem shows that the image of such a moment map is a convex polytope in $(n-3)$-dimensional space, while the Duistermaat–Heckman theorem shows that the pushforward measure on this polytope is Lebesgue measure on $\mathbb{R}^{n-3}$. Together, these theorems allow us to define a measure-preserving set of “action-angle” coordinates on the space of closed equilateral polygons. The new coordinate system allows us to make explicit computations of exact expectations for total curvature and for some chord lengths of closed (and confined) equilateral random walks, to give statistical criteria for sampling algorithms on the space of polygons and to prove that the probability that a randomly chosen equilateral hexagon is unknotted is at least $\frac{1}{2}$.

We then use our methods to construct a new Markov chain sampling algorithm for equilateral closed polygons, with a simple modification to sample (rooted) confined equilateral closed polygons. We prove rigorously that our algorithm converges geometrically to the standard measure on the space of closed random walks, give a theory of error estimators for Markov chain Monte Carlo integration using our method and analyze the performance of our method. Our methods also apply to open random walks in certain types of confinement, and in general to walks with arbitrary (fixed) edgelengths as well as equilateral walks.

#### Article information

Source
Ann. Appl. Probab., Volume 26, Number 1 (2016), 549-596.

Dates
Revised: January 2015
First available in Project Euclid: 5 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1452003247

Digital Object Identifier
doi:10.1214/15-AAP1100

Mathematical Reviews number (MathSciNet)
MR3449326

Zentralblatt MATH identifier
06554528

Subjects
Primary: 53D30: Symplectic structures of moduli spaces
Secondary: 60G50: Sums of independent random variables; random walks

#### Citation

Cantarella, Jason; Shonkwiler, Clayton. The symplectic geometry of closed equilateral random walks in 3-space. Ann. Appl. Probab. 26 (2016), no. 1, 549--596. doi:10.1214/15-AAP1100. https://projecteuclid.org/euclid.aoap/1452003247

#### References

• [1] Alvarado, S., Calvo, J. A. and Millett, K. C. (2011). The generation of random equilateral polygons. J. Stat. Phys. 143 102–138.
• [2] Andersen, H. C. and Diaconis, P. (2007). Hit and run as a unifying device. J. Soc. Fr. Stat. & Rev. Stat. Appl. 148 5–28.
• [3] Atiyah, M. F. (1982). Convexity and commuting Hamiltonians. Bull. Lond. Math. Soc. 14 1–15.
• [4] Avis, D. and Fukuda, K. (1992). A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Discrete Comput. Geom. 8 295–313.
• [5] Barakat, R. (1973). Isotropic random flights. J. Phys. A 6 796–804.
• [6] Benham, C. J. and Mielke, S. P. (2005). DNA mechanics. Annu. Rev. Biomed. Eng. 7 21–53.
• [7] Bernardi, O., Duplantier, B. and Nadeau, P. (2010). A bijection between well-labelled positive paths and matchings. Sém. Lothar. Combin. 63 Art. B63e, 13.
• [8] Blum, J. R. and Pathak, P. K. (1972). A note on the zero-one law. Ann. Math. Statist. 43 1008–1009.
• [9] Boneh, A. and Golan, A. (1979). Constraints redundancy and feasible region boundedness by random feasible point generator (RGPG). In Third European Congress on Operations Research—EURO III. Association of European Operational Research Societies, Leeds, UK.
• [10] Borwein, D. and Borwein, J. M. (2001). Some remarkable properties of sinc and related integrals. Ramanujan J. 5 73–89.
• [11] Brion, M. (1991). Cohomologie équivariante des points semi-stables. J. Reine Angew. Math. 421 125–140.
• [12] Buonocore, A., Pirozzi, E. and Caputo, L. (2009). A note on the sum of uniform random variables. Statist. Probab. Lett. 79 2092–2097.
• [13] Bustamante, C., Bryant, Z. and Smith, S. B. (2003). Ten years of tension: Single-molecule DNA mechanics. Nature 421 423–426.
• [14] Calvo, J. A. (2001). The embedding space of hexagonal knots. Topology Appl. 112 137–174.
• [15] Cannas da Silva, A. (2001). Lectures on Symplectic Geometry. Lecture Notes in Math. 1764. Springer, Berlin.
• [16] Cantarella, J., Deguchi, T. and Shonkwiler, C. (2014). Probability theory of random polygons from the quaternionic viewpoint. Comm. Pure Appl. Math. 67 1658–1699.
• [17] Cantarella, J., Grosberg, A. Y., Kusner, R. B. and Shonkwiler, C. (2015). The expected total curvature of random polygons. Amer. J. Math. 137 411–438.
• [18] Caravenna, F. (2005). A local limit theorem for random walks conditioned to stay positive. Probab. Theory Related Fields 133 508–530.
• [19] Chan, K. S. and Geyer, C. J. (1994). Discussion: Markov chains for exploring posterior distributions. Ann. Statist. 22 1747–1758.
• [20] Chazelle, B. (1993). An optimal convex hull algorithm in any fixed dimension. Discrete Comput. Geom. 10 377–409.
• [21] Cogburn, R. (1972). The central limit theorem for Markov processes. In Proc. Sixth Berkeley Symp. Math. Statist. Probab. 2 485–512. Univ. California Press, Berkeley, CA.
• [22] Diao, Y., Ernst, C., Montemayor, A., Rawdon, E. J. and Ziegler, U. (2014). The knot spectrum of confined random equilateral polygons. Molecular Based Mathematical Biology 2 19–33.
• [23] Diao, Y., Ernst, C., Montemayor, A. and Ziegler, U. (2011). Generating equilateral random polygons in confinement. J. Phys. A 44 405202, 16.
• [24] Diao, Y., Ernst, C., Montemayor, A. and Ziegler, U. (2012). Generating equilateral random polygons in confinement II. J. Phys. A 45 275203, 15.
• [25] Diao, Y., Ernst, C., Montemayor, A. and Ziegler, U. (2012). Generating equilateral random polygons in confinement III. J. Phys. A 45 465003, 16.
• [26] Duistermaat, J. J. and Heckman, G. J. (1982). On the variation in the cohomology of the symplectic form of the reduced phase space. Invent. Math. 69 259–268.
• [27] Ernst, C. and Ziegler, U. Personal communication.
• [28] Gawrilow, E. and Joswig, M. (2000). polymake: A framework for analyzing convex polytopes. In Polytopes—Combinatorics and Computation (Oberwolfach, 1997). DMV Sem. 29 43–73. Birkhäuser, Basel.
• [29] Geyer, C. J. (1992). Practical Markov chain Monte Carlo. Statist. Sci. 7 473–483.
• [30] Grosberg, A. Y. (2008). Total curvature and total torsion of a freely jointed circular polymer with $n\gg 1$ segments. Macromolecules 41 4524–4527.
• [31] Guillemin, V. and Sternberg, S. (1982). Convexity properties of the moment mapping. Invent. Math. 67 491–513.
• [32] Hausmann, J.-C. and Knutson, A. (1997). Polygon spaces and Grassmannians. Enseign. Math. (2) 43 173–198.
• [33] Hausmann, J.-C. and Knutson, A. (1998). The cohomology ring of polygon spaces. Ann. Inst. Fourier (Grenoble) 48 281–321.
• [34] Hitchin, N. J., Karlhede, A., Lindström, U. and Roček, M. (1987). Hyper-Kähler metrics and supersymmetry. Comm. Math. Phys. 108 535–589.
• [35] Howard, B., Manon, C. and Millson, J. (2011). The toric geometry of triangulated polygons in Euclidean space. Canad. J. Math. 63 878–937.
• [36] Hughes, B. D. (1995). Random Walks and Random Environments. Vol. 1: Random Walks. Clarendon, New York.
• [37] Kamiyama, Y. and Tezuka, M. (1999). Symplectic volume of the moduli space of spatial polygons. J. Math. Kyoto Univ. 39 557–575.
• [38] Kapovich, M. and Millson, J. J. (1996). The symplectic geometry of polygons in Euclidean space. J. Differential Geom. 44 479–513.
• [39] Khoi, V. T. (2005). On the symplectic volume of the moduli space of spherical and Euclidean polygons. Kodai Math. J. 28 199–208.
• [40] Kirwan, F. (1992). The cohomology rings of moduli spaces of bundles over Riemann surfaces. J. Amer. Math. Soc. 5 853–906.
• [41] Klenin, K. V., Vologodskii, A. V., Anshelevich, V. V., Dykhne, A. M. and Frank-Kamenetskii, M. D. (1988). Effect of excluded volume on topological properties of circular DNA. Journal of Biomolecular Structure and Dynamics 5 1173–1185.
• [42] Łatuszyński, K., Roberts, G. O. and Rosenthal, J. S. (2013). Adaptive Gibbs samplers and related MCMC methods. Ann. Appl. Probab. 23 66–98.
• [43] Lord, R. D. (1954). The use of the Hankel transform in statistics. I. General theory and examples. Biometrika 41 44–55.
• [44] Lovász, L. (1999). Hit-and-run mixes fast. Math. Program. 86 443–461.
• [45] Lovász, L. and Vempala, S. (2006). Hit-and-run from a corner. SIAM J. Comput. 35 985–1005 (electronic).
• [46] Lua, R. C., Moore, N. T. and Grosberg, A. Yu. (2005). Under-knotted and over-knotted polymers. II. Compact self-avoiding loops. In Physical and Numerical Models in Knot Theory (J. A. Calvo, K. C. Millett, E. J. Rawdon and A. Stasiak, eds.). Ser. Knots Everything 36 385–398. World Scientific, Singapore.
• [47] Mandini, A. (2014). The Duistermaat–Heckman formula and the cohomology of moduli spaces of polygons. J. Symplectic Geom. 12 171–213.
• [48] Mardia, K. V. and Jupp, P. E. (2000). Directional Statistics. Wiley, Chichester.
• [49] Marichal, J.-L. and Mossinghoff, M. J. (2008). Slices, slabs, and sections of the unit hypercube. Online J. Anal. Comb. 3 Art. 1, 11.
• [50] Marsden, J. and Weinstein, A. (1974). Reduction of symplectic manifolds with symmetry. Rep. Mathematical Phys. 5 121–130.
• [51] Meyer, K. R. (1973). Symmetries and integrals in mechanics. In Dynamical Systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) 259–272. Academic Press, New York.
• [52] Millett, K. C. (1994). Knotting of regular polygons in $3$-space. J. Knot Theory Ramifications 3 263–278.
• [53] Moore, N. T. and Grosberg, A. Y. (2005). Limits of analogy between self-avoidance and topology-driven swelling of polymer loops. Phys. Rev. E (3) 72 061803.
• [54] Moore, N. T., Lua, R. C. and Grosberg, A. Y. (2004). Topologically driven swelling of a polymer loop. Proc. Natl. Acad. Sci. USA 101 13431–13435.
• [55] Moore, N. T., Lua, R. C. and Grosberg, A. Yu. (2005). Under-knotted and over-knotted polymers. I. Unrestricted loops. In Physical and Numerical Models in Knot Theory (J. A. Calvo, K. C. Millett, E. J. Rawdon and A. Stasiak, eds.). Ser. Knots Everything 36 363–384. World Scientific, Singapore.
• [56] Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W., eds. (2010). NIST Handbook of Mathematical Functions. U.S. Dept. Commerce, National Institute of Standards and Technology, Washington, DC.
• [57] Orlandini, E. and Whittington, S. G. (2007). Statistical topology of closed curves: Some applications in polymer physics. Rev. Modern Phys. 79 611–642.
• [58] Pennec, X. (2006). Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements. J. Math. Imaging Vision 25 127–154.
• [59] Pólya, G. (1912). On a few questions in probability theory and some definite integrals related to them. Ph.D. thesis, Eötvös Lorànd Univ., Budapest.
• [60] Polya, G. (1913). Berechnung eines bestimmten Integrals. Math. Ann. 74 204–212.
• [61] Rayleigh, L. (1919). On the problem of random vibrations, and of random flights in one, two, or three dimensions. Philosophical Magazine Series 5 37 321–347.
• [62] Roberts, G. O. and Rosenthal, J. S. (1997). Geometric ergodicity and hybrid Markov chains. Electron. Commun. Probab. 2 13–25 (electronic).
• [63] Sendler, W. (1975). A note on the proof of the zero-one law of J. R. Blum and P. K. Pathak: “A note on the zero-one law” (Ann. Math. Statist. 43 (1972) 1008–1009). Ann. Probab. 3 1055–1058.
• [64] Smith, R. L. (1980). Monte Carlo procedures for generating random feasible solutions to mathematical programs. In A Bulletin of the ORSA/TIMS Joint National Meeting. Univ. Pittsburgh, Pittsburgh, PA.
• [65] Smith, R. L. (1984). Efficient Monte Carlo procedures for generating points uniformly distributed over bounded regions. Oper. Res. 32 1296–1308.
• [66] Soteros, C. Personal communication.
• [67] Stanley, R. P. (1999). Enumerative Combinatorics. Vol. 2. Cambridge Studies in Advanced Mathematics 62. Cambridge Univ. Press, Cambridge.
• [68] Strick, T. R., Croquette, V. and Bensimon, D. (2000). Single-molecule analysis of DNA uncoiling by a type II topoisomerase. Nature 404 901–904.
• [69] Takakura, T. (2001). Intersection theory on symplectic quotients of products of spheres. Internat. J. Math. 12 97–111.
• [70] Tierney, L. (1994). Markov chains for exploring posterior distributions. Ann. Statist. 22 1701–1762.
• [71] Treloar, L. R. G. (1946). The statistical length of long-chain molecules. Trans. Faraday Soc. 42 77–82.
• [72] Varela, R., Hinson, K., Arsuaga, J. and Diao, Y. (2009). A fast ergodic algorithm for generating ensembles of equilateral random polygons. J. Phys. A 42 095204, 14.
• [73] Vologodskii, A. V., Anshelevich, V. V., Lukashin, A. V. and Frank-Kamenetskii, M. D. (1979). Statistical mechanics of supercoils and the torsional stiffness of the DNA double helix. Nature 280 294–298.
• [74] Wästlund, J. (2012). A random walk with uniformly distributed steps. MathOverflow. Available at http://mathoverflow.net/questions/94262 (version: 2012-04-17).
• [75] Wuite, G. J., Smith, S. B., Young, M., Keller, D. and Bustamante, C. (2000). Single-molecule studies of the effect of template tension on T7 DNA polymerase activity. Nature 404 103–106.
• [76] Zirbel, L. and Millett, K. C. (2012). Characteristics of shape and knotting in ideal rings. J. Phys. A 45 225001.