Clayton Shonkwiler PRO
Mathematician and artist
Direct Sampling of Confined Polygons in Linear Time
/jmm25
this talk!
Results on Curves and Surfaces Inspired by Experiments
January 10, 2025
Key Point
Symplectic geometry and some baby combinatorics give a surprisingly fast algorithm for sampling random polygons in tight confinement.
Random Knot Model: Equilateral Polygons
\(\operatorname{Pol}(n)\) is the space of equilateral \(n\)-gons in \(\mathbb{R}^3\); it consists of (congruence classes of) piecewise-linear maps \(S^1 \to \mathbb{R}^3\) with \(n\) unit-length segments.
Equilateral Polygons
\(\operatorname{Pol}(n)\) can be constructed as a symplectic reduction (see Kapovich–Millson and Hausmann–Knutson):
\operatorname{Pol}(n)=(S^2)^n/\!/_{\vec{0}}SO(3)
Continuous symmetry \(\Rightarrow\) conserved quantity
\(n-3\) commuting symmetries
Rotations around \(n-3\) chords \(d_i\) by \(n-3\) angles \(\theta_i\) commute.
More precisely, \(\operatorname{Pol}(n)\) is (almost) toric, and the \(d_i\) and \(\theta_i\) are action-angle coordinates.
Chord distributions
Theorem [with Cantarella]
The joint distribution of \(d_1,\ldots , d_{n-3}\) and \(\theta_1, \ldots , \theta_{n-3}\) are all uniform on their domains.
Therefore, sampling \(\operatorname{Pol}(n)\) is equivalent to sampling random points in the convex polytope of \(d_i\)’s and random angles \(\theta_i\).
A polytope
The \((n-3)\)-dimensional moment polytope \(\mathcal{P}_n \subset \mathbb{R}^{n-3}\) is defined by the triangle inequalities
0 \leq d_i \leq 2
1 \leq d_i + d_{i-1}
|d_i - d_{i-1}| \leq 1
0 \leq d_{n-3} \leq 2
Confined Polygons
If we want to sample polygons in rooted, spherical confinement of radius \(R\), then we simply add the constraints \(d_i \leq R\) for all \(i\).
Sampling
Algorithm [w/ Cantarella]
A uniformly ergodic Markov chain for sampling random equilateral \(n\)-gons in rooted spherical confinement (implemented in plCurve).
Algorithm [Diao–Ernst–Montemayor–Ziegler]
An algorithm for directly sampling successive marginals of Lebesgue measure on \(\mathcal{P}_n(R)\).
“The data shows that the runtime grows slower than an exponential function, but faster than a power function.”
Tightest Confinement
When \(R=1\), the inequalities reduce to
\(0\leq d_i \leq 1 \qquad 1 \leq d_i + d_{i+1}\)
Define \(\phi: (d_1, d_2, \dots ) \mapsto (d_1, 1-d_2, d_3, 1-d_4, \dots )\), which sends \(\mathcal{P}_n(1)\) to
which define a polytope \(\mathcal{P}_n(1) \subset [0,1]^{n-3}\).
\(\mathcal{O_{n-3}} := \{(s_1, \dots , s_{n-3}) \in [0,1]^{n-3} : s_1 \geq s_2 \leq s_3 \geq s_4 \leq \dots\}\),
The order polytope of the zig-zag (fence) poset.
Each possible total ordering \(s_{\sigma(i_1)} \leq \dots \leq s_{\sigma(i_{n-3})}\) corresponds to a linear extension of the zig-zag poset or, equivalently, an alternating permutation.
Each alternating permutation \(\sigma\) determines an orthoscheme \(\Delta_\sigma\) in a unimodular triangulation of \(\mathcal{O}_{n-3}\).
\(\Leftrightarrow d_1 \geq 1-d_2 \leq d_3 \geq 1-d_3 \leq \dots\)
Tightest Confinement
When \(R=1\), the inequalities reduce to
\(0\leq d_i \leq 1 \qquad 1 \leq d_i + d_{i+1}\)
which define a polytope \(\mathcal{P}_n(1) \subset [0,1]^{n-3}\).
Each alternating permutation determines an orthoscheme \(\Delta_\sigma\) in a triangulation of \(\mathcal{O}_{n-3}\).
Alternating Permutations and Euler Numbers
\((1)\)
\((21)\)
\((213),(312)\)
\((2143),(3142),(3241),(4132),(4231)\)
\((21435),(21534),(31425),(31524),\dots\)
\((214365),(215364),(216354),\dots\)
\(n\)
\(E_n\)
\(1\)
\(1\)
\(2\)
\(1\)
\(2\)
\(3\)
\(4\)
\(5\)
\(5\)
\(16\)
\(6\)
\(61\)
The number of alternating permutations on \(n\) letters is denoted \(E_n\) and called an Euler number.
\(\text{Vol}(\mathcal{P}_n(1)) = \frac{E_{n-3}}{(n-3)!} \sim 2 \left(\frac{2}{\pi}\right)^{n-2}\)
Rejection sampling won’t work!
Marchal’s Magic Algorithm
CPOP
Theorem [w/ Theis]
CPOP generates random polygons in \(\operatorname{Pol}(n;1)\) with average time complexity \(\Theta(n)\).
CPOP
Expected Total Curvature
For unconfined polygons:
Theorem [Grosberg]
\(\mathbb{E}_{\operatorname{Pol}(n)}(\kappa) = \frac{\pi}{2}n + \frac{3\pi}{8} + O\left(\frac{1}{n}\right)\)
Conjecture [w/ Theis]
\(\mathbb{E}_{\operatorname{Pol}(n;1)}(\kappa) = 2.14625 n - 0.46742 + o(1).\)
Theorem [w/ Theis]
This is the asymptotic limit, and we have an explicit (terrible!) formula.
Questions
- What about other \(R\)?
- How does expected total curvature relate to \(R\)? (cf. Diao–Ernst–Rawdon–Ziegler)
- What can we learn about knotting phenomena of tightly confined random polygons?
Thank you!
Funding: NSF
References
Direct sampling of confined polygons in linear time
Clayton Shonkwiler and Kandin Theis
Preprint, 2025. arXiv: 2501.04885 [math.GT]
The symplectic geometry of closed equilateral random walks in 3-space
Jason Cantarella and Clayton Shonkwiler
Annals of Applied Probability 26 (2016), no. 1, 549–596
Direct Sampling of Confined Polygons in Linear Time
By Clayton Shonkwiler
