# 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…

## One Citation

### New Stick Number Bounds from Random Sampling of Confined Polygons

- Computer ScienceExperimental Mathematics
- 2021

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.

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