Generating equilateral random polygons in confinement III

Diao, Yuanan; Ernst, Claus; Montemayor, Anthony; Ziegler, Uta

doi:10.1088/1751-8113/45/46/465003

Cited by 15 publications

(27 citation statements)

References 18 publications

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“…The generation methods of equilateral confined random polygons introduced in [3,4,5] all rely on the use of explicit probability density functions that guide the generation of the polygons step-by-step. However these probability density functions are quite complicated.…”

Section: Introductionmentioning

confidence: 99%

Curvature of random walks and random polygons in confinement

Diao

Ernst

Montemayor

et al. 2013

J. Phys. A: Math. Theor.

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The purpose of this paper is to study the curvature of equilateral random walks and polygons that are confined in a sphere. Curvature is one of several basic geometric properties that can be used to describe random walks and polygons. We show that confinement affects curvature quite strongly, and in the limit case where the confinement diameter equals the edge length the unconfined expected curvature value doubles from π/2 to π. To study curvature a simple model of an equilateral random walk in spherical confinement in dimensions two and three is introduced. For this simple model we derive explicit integral expressions for the expected value of the total curvature in both dimensions. These expressions are functions that depend only on the radius R of the confinement sphere. We then show that the values obtained by numeric integration of these expressions agrees with numerical average curvature estimates obtained from simulations of random walks. Finally, we compare the confinement effect on curvature of random walks with random polygons.

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Section: Introductionmentioning

confidence: 99%

Curvature of random walks and random polygons in confinement

Diao

Ernst

Montemayor

et al. 2013

J. Phys. A: Math. Theor.

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“…The values of − cos θ which lie within these categories are: marking the angle which would step to the boundary. The probability density function (PDF) proposed by [43] from which − cos θ is sampled is uniform for angles further than a unit from the boundary, rises linearly within a unit of the boundary, and is zero outside of the boundary. Explicitly, this is PDF(− cos θ) =      for normalisation.…”

Section: Appendix a Generating Ideal Chains In Spheres Tubes And Slitsmentioning

confidence: 99%

Knotting and weak knotting in confined, open random walks using virtual knots

Alexander

Taylor

Dennis

2020

J. Phys. A: Math. Theor.

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We probe the character of knotting in open, confined polymers, assigning knot types to open curves by identifying their projections as virtual knots. In this sense, virtual knots are transitional, lying in between classical knot types, which are useful to classify the ambiguous nature of knotting in open curves.Modelling confined polymers using both lattice walks and ideal chains, we find an ensemble of random, tangled open curves whose knotting is not dominated by any single knot type, a behaviour we call weakly knotted. We compare cubically confined lattice walks and spherically confined ideal chains, finding the weak knotting probability in both families is quite similar and growing with length, despite the overall knotting probability being quite different. In contrast, the probability of weak knotting in unconfined walks is small at all lengths investigated. For spherically confined ideal chains, weak knotting is strongly correlated with the degree of confinement but is almost entirely independent of length. For ideal chains confined to tubes and slits, weak knotting is correlated with an adjusted degree of confinement, again with length having negligible effect.

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“…This p.d.f. is key to determining the Green's function for closed polygons, which in turn is fundamental to the Moore-Grosberg [53] and Diao-Ernst-Montemayor-Ziegler [23][24][25] sampling algorithms and to expected total curvature calculations [17,30]. For mathematicians, we note that this p.d.f.…”

Section: 3mentioning

confidence: 99%

“…They fall into two main categories: Markov chain algorithms such as polygonal folds [52] or crankshaft moves [41,73] (cf. [1] for a discussion of these methods) and direct sampling methods such as the "triangle method" [54] or the "generalized hedgehog" method [72] and the methods of Moore and Grosberg [53] and Diao, Ernst, Montemayor and Ziegler [23][24][25] which are both based on the "sinc integral formula" (7).…”

Section: Markov Chain Monte Carlo For Closed and Confined Random Walksmentioning

confidence: 99%

“…• Now that we can sample confined polygons quickly, with solid error bars on our calculations, what frontiers does this open in the numerical study of confined polymers? We take our cues from the pioneering work of Diao, Ernst, Montemayor and Ziegler [22][23][24][25], but are eager to explore this new experimental domain. For instance, sampling tightly confined n-gons might be a useful form of "enriched sampling" in the hunt for complicated knots of low equilateral stick number, since very entangled polygons are likely to be geometrically compact as well.…”

Section: 5mentioning

confidence: 99%

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

Cantarella¹,

Shonkwiler²

2016

Ann. Appl. Probab.

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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.Comment: Published at http://dx.doi.org/10.1214/15-AAP1100 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Generating equilateral random polygons in confinement III

Abstract: In this paper we continue our earlier studies (Diao et al 2011

Cited by 15 publications

References 18 publications

Curvature of random walks and random polygons in confinement

Curvature of random walks and random polygons in confinement

Knotting and weak knotting in confined, open random walks using virtual knots

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

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