Finding Good Coordinates for Sampling The Importance of Geometry

Clayton Shonkwiler

Colorado State University

shonkwiler.org

/cu24

this talk!

September 13, 2024

Random points on a line

Random points on a circle

Polar/angular coordinates!

Random points in a square

Rectangular coordinates!

Random points in a disk

Random points in a disk

Rectangular coordinates, sorta

Probability that a random point in an \(n\)-dimensional cube lies inside the inscribed sphere

\mathbb{P}_n = \frac{\pi^{n/2}}{2^n \Gamma(n/2+1)}

Random points in a disk

Random points in a disk

Polar coordinates?

Random points in a disk

Polar coordinates?

Polar coordinates, non-uniformly

Random points on a sphere

Random points on a sphere

Spherical coordinates?

Random points on a circle (or sphere)

2D standard Gaussian points

Normalized 2D standard Gaussian points

Multivariate Gaussian

Distance distribution for 2D Gaussian

Distance distribution for 5D Gaussian

Distance distribution for 20D Gaussian

Random points on a sphere

Cylindrical coordinates?

Archimedes’ Theorem:

The red and blue areas are equal!

Random points on a sphere

Cylindrical coordinates!

Lewis Carroll’s Pillow Problem #58

Earlier versions

W. S. B. Woolhouse, Educational Times 18 (1865), p. 189

J. J. Sylvester, Educational Times 18 (1865), p. 68

W. S. B. Woolhouse, The Lady's and Gentleman's Diary 158 (1861), p. 76

Random Points on an Infinite Plane

Probability \(p\)

\(p\)

\(p\)

\(p\)

\(p\)

Uh oh!

A probabilist’s answer

Proposition [Portnoy]

If the vertices of the triangle are chosen from the standard Gaussian on \(\mathbb{R}^2\), then 

\(\mathbb{P}(\text{obtuse}) = \frac{3}{4}\)

Choose three vertices uniformly in the disk:

\(\mathbb{P}(\text{obtuse})=\frac{9}{8}-\frac{4}{\pi^2}\approx 0.7197\)

Restricted domain?

Choose three vertices uniformly in the square:

\(\mathbb{P}(\text{obtuse})=\frac{97}{150}-\frac{\pi}{40}\approx 0.7252\)

Side Lengths?!

Sidelengths \((a,b,c)\) uniquely determine a triangle (SSS).

Obtuseness is scale-invariant, so pick a perimeter \(P\) and we have \(a+b+c=P\).

\(b+c<a\)

\(a+b<c\)

\(a+c<b\)

\(\mathbb{P}(\text{obtuse})=9-12\ln 2 \approx 0.68\)

\(b^2+c^2=a^2\)

\(a^2+b^2=c^2\)

\(a^2+c^2=b^2\)

Carroll’s answer

Suppose \(AB\) is the longest side. Then

\(\mathbb{P}(\text{obtuse})=\frac{\pi/8}{\pi/3-\sqrt{3}/4} \approx 0.64\)

But if \(AB\) is the second longest side, 

\(\mathbb{P}(\text{obtuse}) = \frac{\pi/2}{\pi/3+\sqrt{3}/2} \approx 0.82\)

reentrant

J.J. Sylvester, Educational Times, April 1864

Sylvester’s four point problem

Sylvester and Cayley’s solution

J.J. Sylvester, Phil. Trans. R. Soc. London  154 (1864), p. 654, footnote 64(b)

W.S.B. Woolhouse, Mathematical Questions with Their Solutions VII (1867), p. 81

A. De Morgan’s solution

A. De Morgan, Trans. Cambridge Phil. Soc.  XI (1871), pp. 147–148

W.S.B. Woolhouse, Mathematical Questions with Their Solutions  VI (1866), p. 52

C.M. Ingleby, Mathematical Questions with Their Solutions V (1865), p. 82

C.M. Ingleby’s solution

G. C. De Morgan’s solution

G.C. De Morgan, Mathematical Questions with Their Solutions V (1865), p. 109

W.S.B. Woolhouse’s solution

W.S.B. Woolhouse, Mathematical Questions with Their Solutions VI (1866), p. 52

W.S.B. Woolhouse’s solutions

W.S.B. Woolhouse, Mathematical Questions with Their Solutions  VIII (1868), p. 105

Quadrilaterals in convex bodies

\(\mathbb{P}(\text{reflex})=\frac{1}{3}\)

\(\mathbb{P}(\text{reflex})=\frac{35}{12\pi^2}\approx 0.296\)

Theorem [Blaschke, 1917]

\(\frac{35}{12\pi^2}\leq\mathbb{P}(\text{reflex})\leq\frac{1}{3}\)

J.M. Wilson’s solution

J.M. Wilson, Mathematical Questions with Their Solutions  V (1866), p. 81

image/svg+xml

We need measure theory!

W.A. Whitworth, Mathematical Questions with Their Solutions  VIII (1868), p. 36

Report on J.J. Sylvester’s presentation of his paper “On a Special Class of Questions on the Theory of Probabilities” to the British Association for the Advancement of Science, 1865

We need geometry!

Protonated P2VP

Polymers in solution

Random flight model

Ring Polymers

Knotted DNA

Random polygon model

Symmetries \(\Rightarrow\) Coordinates

Action-angle coordinates

Action Coordinates

Polytope determined by triangle inequalities

Action-Angle Coordinates

Angle Coordinates

Action-Angle Coordinates are Good

Theorem [with Cantarella]

Sampling action-angle coordinates uniformly is equivalent to sampling equilateral polygons uniformly.

Theorem [with Cantarella and Schumacher]

We can generate random equilateral \(n\)-gons in expected time \(\Theta(n^2)\).

What is the Probability of Knotting?

Frisch–Wasserman–Delbrück Conjecture

In any reasonable model of random knots parametrized by a “size,” the probability of knotting goes to 1 as the size goes to infinity.

Random polygon

Random diagram

Random petal knot

What is the Probability of Knotting?

Frisch–Wasserman–Delbrück Conjecture

In any reasonable model of random knots parametrized by a “size,” the probability of knotting goes to 1 as the size goes to infinity.

Random polygon

Theorem [Diao]

For random equilateral \(n\)-gons, the probability of knotting is

\(\geq 1-e^{-n^{\epsilon}}\) for some \(\epsilon > 0\).

\mathbb{P}(\text{unknot}) \approx e^{-n/251}\left[1-0.51\sqrt{n}-\frac{1.2}{n}\right]

What About More Complicated Topological Polymers?

Synthetic chemists can now produce simple topological polymers in usable quantities.

\(\theta\)-curves in solution at the Tezuka lab

What are good coordinates for random embeddings of graphs?

Thank You!

shonkwiler.org/cu24

References

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

New stick number bounds from random sampling of confined polygons

Thomas D. Eddy and Clayton Shonkwiler

Experimental Mathematics 31 (2022), no. 4, 1373–1395

Random triangle and polygons in the plane

Jason Cantarella, Tom Needham, Clayton Shonkwiler, and Gavin Stewart

The American Mathematical Monthly 126 (2019), no. 2, 113–134

A faster direct sampling algorithm for equilateral closed polygons and the probability of knotting

Jason Cantarella, Henrik Schumacher, and Clayton Shonkwiler

Journal of Physics A: Mathematical and Theoretical 57 (2024), no. 28, 28205

Finding Good Coordinates for Sampling: The Importance of Geometry

By Clayton Shonkwiler

Finding Good Coordinates for Sampling: The Importance of Geometry

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