Open and closed random walks with fixed edgelengths in $\mathbb{R}^d$

@article{Cantarella2018OpenAC,
  title={Open and closed random walks with fixed edgelengths in \$\mathbb\{R\}^d\$},
  author={Jason Cantarella and Kyle Leland Chapman and Philipp Reiter and Clayton Shonkwiler},
  journal={arXiv: Statistical Mechanics},
  year={2018}
}
In this paper, we consider fixed edgelength $n$-step random walks in $\mathbb{R}^d$. We give an explicit construction for the closest closed equilateral random walk to almost any open equilateral random walk based on the geometric median, providing a natural map from open polygons to closed polygons of the same edgelength. Using this, we first prove that a natural reconfiguration distance to closure converges in distribution to a Nakagami$(\frac{d}{2},\frac{d}{d-1})$ random variable as $n… 
1 Citations

Figures from this paper

New Stick Number Bounds from Random Sampling of Confined Polygons

A Monte Carlo approach is adopted, producing very large ensembles of random polygons in tight confinement to look for new examples of knots constructed from few segments, yielding either the exact stick number or an improved upper bound for more than 40% of the knots with 10 or fewer crossings.

References

SHOWING 1-10 OF 20 REFERENCES

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

It is proved rigorously that the algorithm converges geometrically to the standard measure on the space of closed random walks, a theory of error estimators for Markov chain Monte Carlo integration using the method is given, and the performance of the algorithm is analyzed.

The Toric Geometry of Triangulated Polygons in Euclidean Space

Abstract Speyer and Sturmfels associated Gröbner toric degenerations $\text{G}{{\text{r}}_{2}}{{({{\mathbb{C}}^{n}})}^{\mathcal{T}}}$ of $\text{G}{{\text{r}}_{2}}{{({{\mathbb{C}}^{n}})}^{{}}}$ with

The Knotting of Equilateral Polygons in R3

  • Y. Diao
  • Mathematics, Computer Science
  • 1995
If EPn is an equilateral random polygon of n steps, then it is proved that n is large enough, where ∊ is some positive constant, and the knotting probability of a Gaussian random polyagon goes to 1 as the length of the polygon goes to infinity.

Geometric folding algorithms - linkages, origami, polyhedra

Aimed primarily at advanced undergraduate and graduate students in mathematics or computer science, this lavishly illustrated book will fascinate a broad audience, from high school students to researchers.

The Shapes of Random Walks

A theoretical description of the shape of a random object is presented that is analytically simple in application but quantitatively accurate, and can be extended to yield an approximate, but extremely accurate, expression for the probability distribution function directly.

The symplectic geometry of polygons in Euclidean space

We study the symplectic geometry of moduli spaces M r of polygons with xed side lengths in Euclidean space. We show that M r has a natural structure of a complex analytic space and is

User-Friendly Tail Bounds for Sums of Random Matrices

  • J. Tropp
  • Mathematics
    Found. Comput. Math.
  • 2012
This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices and provides noncommutative generalizations of the classical bounds associated with the names Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid.

A note on Fermat's problem

  • H. Kuhn
  • Mathematics
    Math. Program.
  • 1973
This note calls attention to the work of Weiszfeld in 1937, who may have been the first to propose an iterative algorithm for the General Fermat Problem.

On the symplectic volume of the moduli space of Spherical and Euclidean polygons

In this paper, we study the symplectic volume of the moduli space of polygons by using Witten’s formula. We propose to use this volume as a measure for the flexibility of a polygon with fixed