### Clayton Shonkwiler PRO

Mathematician and artist

A ring polymer in solution takes on an ensemble of random shapes, with topology (knot type!) as the unique conserved quantity.

Knotted DNA

Wassermann et al.

*Science* 229, 171–174

Knot complexity in DNA from P4 tailless mutants

Arsuaga et al., *PNAS* 99 (2002), 5373–5377

**Is this surprising?**

A polymer in solution takes on an ensemble of random shapes, with topology as the unique conserved quantity.

*Modern polymer physics is based on the analogy between a polymer chain and a random walk.*

– Alexander Grosberg

Protonated P2VP

Roiter/Minko

Clarkson University

Plasmid DNA

Alonso–Sarduy, Dietler Lab

EPF Lausanne

Generate \(n\) independent uniform random points on \(S^2\) and treat them as an ordered list of edge vectors.

Alvarado, Calvo, Millett, *J. Stat. Phys.* 143 (2011), 102–138

*Random Polygon* \(\Leftrightarrow\) *point in some (nice!) conformation space*

Knowledge of the (differential, symplectic, algebraic) geometry of these conformation spaces leads to both

**theorems and fast numerical algorithms** for studying and comparing polygons in \(\mathbb{R}^3\).

The edges of an \(n\)-gon (up to translation) are unit vectors, so the space of \(n\)-gons is the submanifold of \((S^2)^n\) implicitly defined by \(e_1 + \ldots + e_n = 0\), which is codimension 3.

The natural measure is Hausdorff measure; equivalently, the volume form induced by the Riemannian submanifold metric.

A random closed \(n\)-edge polygon is a \(k\)-edge random flight and an \((n-k)\)-edge random flight, conditioned on having the same end-to-end distance.

**Classical Fact:** The density of the end-to-end vector of an \(n\)-step random walk is

\phi_n(\ell) = \frac{2\ell}{\pi} \int_0^\infty x \sin \ell x \,\mathrm{sinc}^n x \,\mathrm{d}x

= \frac{2^{-n-1}}{\pi
\ell(n-2)!} \sum_{k=0}^{n-1} (-1)^k \binom{n-1}{k} \left([-2k+\ell +n-2]_+^
{n-2}\right.

\left.-
[-2 k+\ell +n]_+^{n-2}\right)

**Proof:** Fourier transform (since \(\mathrm{sinc}\) is the transform of the boxcar function).

This is piecewise-polynomial in \(\ell\) of degree \(n-3\)

**Proposition (with Cantarella)** The pdf of the chord connecting \(v_1\) with \(v_{k+1}\) in an \(n\)-gon is

\frac{4 \pi \ell^2}{C_n} \phi_k(\ell) \phi_{n-k}(\ell)

where \(C_n = 2^{n-5}\pi^{n-4} \int_{-\infty}^{\infty} x^2 \,\mathrm{sinc}^n x \,\mathrm{d}x\).

**Fact:** This is piecewise-polynomial in \(\ell\) of degree \(n-4\).

Why are the expectations rational?

Why degree \(n-4\)?

*Where are these crazy polynomials coming from?*

Continuous symmetry \(\Rightarrow\) conserved quantity

**Duistermaat–Heckman Theorem (stated informally)**

*On a \(2m\)-dimensional symplectic manifold, \(d\) commuting Hamiltonian symmetries (a Hamiltonian \(T^d=(S^1)^d\)-action) induce \(d\) conserved quantities (momenta).*

*The joint distribution of the momenta on \(\mathbb{R}^d\) is continuous, piecewise polynomial, degree \(\leq m-d\).*

A symplectic manifold \((M,\omega)\) is an even-dimensional manifold \(M\) together with a closed, non-degenerate 2-form \(\omega\).

**Example:** \((S^2,d\theta\wedge dz)\)

**Example:** \((\mathbb{R}^2,dx \wedge dy) = (\mathbb{C},\frac{i}{2}dz \wedge d\bar{z})\)

**Example:** \((T^*\mathbb{R}^n, \sum_{i=1}^n dq_i \wedge dp_i)\)

**Example:** \((S^2,\omega)\), where \(\omega_p(u,v) = (u \times v) \cdot p\)

**Example:** \((\mathbb{R}^2,\omega)\) where \(\omega(u,v) = \langle i u, v \rangle \)

This is the right setting for Hamiltonian mechanics.

**Example:** \((\mathbb{C}^n, \frac{i}{2} \sum dz_k \wedge d\overline{z}_k)\)

**Theorem (Archimedes, Duistermaat–Heckman)**

*Let \(f: S^2 \to \mathbb{R}\) be given by \(f(x,y,z) = z\). Pushing forward the uniform measure on \(S^2\) to the image \([-1,1]\) gives Lebesgue measure.*

**Proposition** *If we rotate both factors of \(S^2 \times S^2\) simultaneously around the \(z\)-axis, the conserved quantity is \(z_1 + z_2\), and the pushforward measure on \([-2,2]\) has density given by the triangle function.*

**Duistermaat–Heckman Theorem (stated informally)** *On a \(2m\)-dimensional symplectic manifold, \(d\) commuting Hamiltonian symmetries (a Hamiltonian \(T^d\)-action) induce \(d\) conserved quantities (momenta).*

*The joint distribution of the momenta is continuous, piecewise polynomial, degree \(\leq m-d\).*

\(n\)-gons up to rotation are \(2m=(2n-6)\)-dimensional, so chord length is piecewise polynomial of degree \(\leq\)

\(m-1=(n-3)-1=n-4\)

**We actually have more symmetries!**

Rotations around \(n-3\) chords \(d_i\) by \(n-3\) angles \(\theta_i\) commute.

**Theorem (with Cantarella, ’16)**

The joint distribution of \(d_1,\ldots , d_{n-3}\) and \(\theta_1, \ldots , \theta_{n-3}\) are all uniform on their domains.

**Proof:** Check that D–H applies (this is the hard part, since the torus action is not defined everywhere).

Then count: \(m=n-3\) and we have \(d=n-3\) symmetries, so the pdf of the \(d_i\) is piecewise polynomial of degree \(\leq\)

\(m-d = (n-3)-(n-3)=0\).

Since the pdf is continuous and the domain is connected, it must be constant.

**Definition:** If \((M,\omega)\) is \(2m\)-dimensional and has \(m\) commuting symmetries, we call \(M\) *toric*.

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

There exists an almost-everywhere defined map \(\alpha: \mathcal{P}_n \times (S^1)^{n-3} \to \{n\text{-gons}\}\) which is measure-preserving.

This is only sensible as a map to polygons modulo translation and rotation.

We have reduced the problem of sampling a random polygon to the problem of sampling a (specific) convex polytope.

**Theorem (Smith, 1984)**

For any convex polytope \(\mathcal{P}\), the hit-and-run Markov chain is uniformly ergodic with respect to Lebesgue measure on \(\mathcal{P}\).

**Definition (TSMCMC\((\beta)\))**

If \(x_k = (p_k,\theta_k) \in \mathcal{P_n} \times T^{n-3}\), define \(x_{k+1}\) by:

- With probability \(\beta\), update \(p_k\) by a hit-and-run step on \(\mathcal{P}_n\)
- Else, replace \(\theta_k\) by a new point sampled uniformly from \(T^{n-3}\)

At each step, construct the corresponding polygon \(\alpha(p_n, \theta_n)\).

**Theorem (with Cantarella, 2016)**

TSMCMC(\(\beta\)) is uniformly ergodic with respect to the standard probability measure on random polygons modulo translation and rotation.

**Theorem (with Cantarella, 2016)**

If \((M,\omega)\) is toric symplectic with moment polytope \(\mathcal{P} \subset \mathbb{R}^m\), then TSMCMC(\(\beta\)) on \(\mathcal{P} \times T^m\) is uniformly ergodic with respect to the symplectic volume measure on \(M\).

For example, polygons in (rooted) spherical confinement.

For unconfined polygons, we can do better: no Markov chains required!

Introduce fake chordlengths \(d_0=1=d_{n-2}\) and make the linear change of variables

\(s_i = d_i - d_{i-1} \text{ for } 1 \leq i \leq n-2\).

Then \(\sum s_i = d_{n-2} - d_0 = 0\), so \(s_{n-2}\) is determined by \(s_1, \ldots , s_{n-3}\)

and the 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

become

\(-1 \leq s_i \leq 1, -1 \leq \sum_{i=1}^{n-3} s_i \leq 1\)

\(\sum_{j=1}^i s_j + \sum_{j=1}^{i+1}s_j \geq -1\)

Let \(\mathcal{C}_n \subset \mathbb{R}^{n-3}\) be determined by

\(-1 \leq s_i \leq 1, -1 \leq \sum_{i=1}^{n-3} s_i \leq 1\)

\(\sum_{j=1}^i s_j + \sum_{j=1}^{i+1}s_j \geq -1\)

\(\mathcal{C}_5\)

\(\mathcal{C}_6\)

**Proposition (with Cantarella, Duplantier, Uehara, 2016)**

The probability that a point in the hypercube lies in \(\mathcal{C}_n\) is asymptotic to

\(6 \sqrt{\frac{6}{\pi}}\frac{1}{n^{3/2}}\)

**Theorem (Khoi, Takakura, Mandini, 2000s)**

The volume of \(\mathcal{C}_n\) is exactly

\text{Vol} (\mathcal{C}_n) = \frac{1}{2 (n-3)!}\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor} (-1)^{k+1} \binom{n}{k}(n-2 k)^{n-3}

**Observation (Edwards, 1922)**

\text{Vol}(\mathcal{C}_n) = \frac{2^{n-1}}{2\pi} \int_{-\infty}^\infty \frac{\sin^n x}{x^{n-2}}dx

[T]here is in these days far too great a tendency on the part of teachers to push on their pupils so fast to the Higher Branches of Analysis or to Physical Mathematics that many have neither time nor opportunity for the cultivation of real personal proficiency, or for the acquirement of that individual manipulative skill which is essential to any real confidence of the student in his own power to conduct unaided investigation.

= \frac{2^{n-1}}{2\pi} \int_{-\infty}^\infty \mathrm{sinc}^n( x) x^2 dx

= \frac{2^{n-1}}{2\pi}\int_{-\infty}^\infty \mathrm{sinc}^n\left(\frac{y}{\sqrt{n}}\right) \frac{y^{2}dy}{n^{3/2}}

x=\frac{y}{\sqrt{n}}

\mathrm{sinc}\left(\frac{y}{\sqrt{n}}\right)=1-\frac{y^2}{6n}+o\left(\frac{1}{n}\right)

\lim_{n \to \infty} \left(1-\frac{a}{n}+o\left(\frac{1}{n}\right)\right)^n=e^{-a}

\sim \frac{2^{n-1}}{2\pi}\frac{1}{n^{3/2}} \int_{-\infty}^\infty e^{-\frac{y^2}{6}} y^{2} dy

= 3 \, \sqrt{\frac{3}{\pi }}
\, 2^{n-\frac{3}{2}} \, \frac{1}{n^{3/2}}

**Action-Angle Method**

**Theorem (with Cantarella, Duplantier, Uehara, 2016)**

The action-angle method directly samples polygon space in expected time \(\Theta(n^{5/2})\).

- Generate \((s_1,\ldots , s_{n-3})\) uniformly on \([-1,1]^{n-3}\)
- Test whether \((s_1,\ldots , s_{n-3})\in \mathcal{C}_n\)
- Change to \((d_1, \ldots , d_{n-3})\) coordinates
- Generate dihedral angles from \(T^{n-3}\)
- Build corresponding polygon

\(O(n)\) time

acceptance probability \(\sim \frac{1}{n^{3/2}}\)

```
RandomDiagonals[n_] :=
Accumulate[
Join[{1}, RandomVariate[UniformDistribution[{-1, 1}], n]]];
InMomentPolytopeQ[d_] :=
And[Last[d] >= 0, Last[d] <= 2,
And @@ (Total[#] >= 1 & /@ Partition[d, 2, 1])];
DiagonalSample[n_] := Module[{d},
For[d = RandomDiagonals[n], ! InMomentPolytopeQ[d], ,
d = RandomDiagonals[n]];
d[[2 ;;]]
];
```

Not the first direct sampler (Grosberg–Moore), but simple and fast.

Frequency plot of the HOMFLY polynomials produced by sampling 10 million random 60-gons (there were a total of 6371 distinct HOMFLYs).

cf. Baiesi–Orlandini–Stella

If we round every point \((d_1),\ldots , d_{n-3})\) in the moment polytope to the nearest point with half-integer coordinates \(\frac{1}{2}(x_1, \ldots , x_{n-3})\) …

… and subdivide the regions of attraction by

ordering and sign of \(d_i - \frac{1}{2} x_i\), we get a collection of identical, orthogonal simplices of equal area (this works for any \(n\)):

**Proposition*:** The number of half-integer points in \(\mathcal{P}_n\) is the Motzkin number

M_{n-3} = \sum_{k=0}^{\lfloor \frac{n-2}{2}\rfloor} \binom{n-2}{2k} C_k

\sim \sqrt{3} \frac{1 + \frac{1}{16(n-3)}}{(2n-3)\sqrt{(n-1)\pi}} 3^{n-2}

Therefore, there are exponentially many simplices in this decomposition. ☹️

**Theorem (K. Chapman, in progress):** With \(O(n^3)\) preprocessing of the transition probabilities, a direct sampling algorithm for equilateral \(n\)-gons in \(O(n^2 (\log n)^2)\) time.

The simplices are indexed by certain permutations.

One can recursively construct Lehmer codes of valid permutations using a Markov chain.

What’s the corresponding story for planar \(n\)-gons? Or \(n\)-gons in \(\mathbb{R}^d\)?

The space of planar \(n\)-gons up to translation and rotation is \((n-3)\)-dimensional, so there is no chance of a symplectic structure.

**Proposition:** The space of \(n\)-gons in \(\mathbb{R}^d\) is the quotient of (almost all) of the space \((S^{d-1})^n\) by the (diagonal) action of the Möbius group.

**Current Project:** Use this structure to provide robust coordinates for polygons in all dimensions.

**Question:** Can this be adapted to give an algorithm for sampling the natural probability measure on \(n\)-gons?

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

J. Cantarella & C. Shonkwiler

*Annals of Applied Probability* **26** (2016), no. 1, 549–596

A fast direct sampling algorithm for equilateral closed polygons

J. Cantarella, B. Duplantier, C. Shonkwiler, & E. Uehara

*Journal of Physics A* **49** (2016), no. 27, 275202

Sampling knots using orthoschemes

K. Chapman

In preparation

Funding: Simons Foundation

By Clayton Shonkwiler

In statistical physics, the basic (and highly idealized) model of a ring polymer like bacterial DNA is a closed random flight in 3-space with equal-length steps, often called an equilateral random polygon. While random flights without the closure condition are easy to simulate and analyze, the fact that the steps in a random polygon are not independent has made it challenging to develop practical yet provably correct sampling and numerical integration techniques for polygons. In this talk I will describe a geometric approach to the study of random polygons which overcomes these challenges. The symplectic geometry of the space of polygon conformations can be exploited to produce both Markov chain and direct sampling algorithms; in fact, this approach can be generalized to produce a sampling theory for arbitrary toric symplectic manifolds. This is joint work with Jason Cantarella, Bertrand Duplantier, and Erica Uehara.

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