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 Cambrian explosion of topological polymers

Tezuka Lab, Tokyo

**How do we know which is smaller?**

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 in \(\mathbb{R}^3\) according to your favorite probability distribution and treat them as an ordered list of edge vectors.

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

*Topologically constrained random walk* \(\Leftrightarrow\) *point in some (nice!) configuration space*

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

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

Kapovich–Millson, Hausmann–Knutson, Howard–Manon–Millson

Let \(P \in G_2(\mathbb{C}^n)\) and let \((\vec{u},\vec{v})\) be an orthonormal basis for \(P\).

\(q_\ell := u_\ell + v_\ell \mathbf{j} \in \mathbb{H}\)

\(\vec{e}_\ell := \overline{q}_\ell \mathbf{i} q_\ell\) is purely imaginary.

**Proposition.** \(\vec{e}_1, \dots , \vec{e}_n\) form the edges of a closed polygon in \(\mathbb{R}^3\) of perimeter 2.

\(\sum \vec{e}_\ell = \sum (\overline{u}_\ell - v_\ell \mathbf{j})\mathbf{i}(u_\ell + \mathbf{i}v_\ell) \mathbf{i}\)

\(\sum |\vec{e}_\ell| = \sum u_\ell \overline{u}_\ell + v_\ell \overline{v}_\ell = \|\vec{u}\|^2 + \|\vec{v}\|^2 = 2\)

\(=\sum \mathbf{i}(\overline{u}_\ell u_\ell - \overline{v}_\ell v_\ell+2 \overline{u}_\ell v_\ell \mathbf{j})\)

\( =\left(\|\vec{u}\|^2 - \|\vec{v}\|^2 + 2\langle \vec{u},\vec{v}\rangle\mathbf{j} \right)\)

\(=0\)

This is really a story about *framed* polygons; the map \(q \mapsto [\overline{q}\mathbf{i}q \,|\, \overline{q}\mathbf{j}q \,|\, \overline{q}\mathbf{k}q]\) is just the standard double covering of \(SO(3)\) by \(SU(2) \simeq S^3\).

**Definition [w/ Cantarella & Deguchi]**

The *symmetric measure* on \(n\)-gons of perimeter 2 up to translation and rotation is the pushforward of Haar measure on \(G_2(\mathbb{C}^n)\).

Therefore, \(U(n)\) acts transitively on \(n\)-gons and preserves the symmetric measure.

Sampling is easy: generate 2 random Gaussians in \(\mathbb{C}^n\) and apply Gram–Schmidt. This is \(O(n)\) complexity!

**Theorem [w/ Cantarella, Grosberg, Kusner]**

The expected total curvature of a random space \(n\)-gon is exactly

\(\frac{\pi}{2}n + \frac{\pi}{4} \frac{2n}{2n-3}\).

**Corollary**

At least \(\frac{1}{3}\) of hexagons and \(\frac{1}{11}\) of heptagons are knotted.

**Theorem [w/ Cantarella & Deguchi, also Zirbel–Millett]**

For any probability distribution on \(n\)-gons which is invariant under permuting edges, the expected squared radius of gyration (mean squared distance to the center of mass) is

E[\text{Gyradius}] = \frac{n+1}{12} E\left[|\vec{e}_i|^2\right].

$E[\text{Gyradius}] = \frac{n+1}{12} E\left[|\vec{e}_i|^2\right].$

**Corollary.** For the symmetric measure, \(E[\text{Gyradius}]=\frac{1}{2n}\).

**Corollary.** For equilateral polygons with unit edges, \(E[\text{Gyradius}] = \frac{n+1}{12}\).

**Corollary.** For standard Gaussian polygons, \(E[\text{Gyradius}]=\frac{n^2-1}{4n}\).

There’s a similar story for planar polygons using complex numbers rather than quaternions and \(z\mapsto z^2\) rather than \(q \mapsto \overline{q} \mathbf{i}q\)

\(\mathbb{P}(\text{obtuse})=\frac{1}{4\pi}\text{Area} = \frac{24}{4\pi} \int_R d\theta dz\)

**Theorem [w/ Cantarella, Needham, Stewart]**

The probability that a random triangle is obtuse is

\(\frac{3}{2}-\frac{3\ln 2}{\pi}\approx0.838\)

Symmetric triangles

Isosceles

triangles

Degenerate

triangles

**Theorem [with Bowden, Haynes, Shukert]**

The least symmetric triangle has side length ratio

1 : 3-\frac{1}{\sqrt{2}} : 3

$1 : 3-\frac{1}{\sqrt{2}} : 3$

convex

reflex/reentrant

self-intersecting

**Modern Reformulation:** What is the probability that all vertices of a random quadrilateral lie on its convex hull?

**Theorem [w/ Cantarella, Needham, Stewart]**

Convex, reflex, and self-intersecting quadrilaterals are all equiprobable.

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

**Definition.** A *symplectic manifold* is a smooth manifold \(M\) together with a closed, non-degenerate 2-form \({\omega \in \Omega^2(M)}\).

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

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

\(\omega^{\wedge n} = \omega \wedge \dots \wedge \omega\) is a volume form on \(M\), and induces a measure

m(U) := \int_U \omega^{\wedge n}

$m(U) := \int_U \omega^{\wedge n}$

called *Liouville measure* on \(M\).

Let \(G\) be a Lie group, and let \(\mathfrak{g}\) be its Lie algebra. If \(G\) acts on \((M,\omega)\), then each \(V \in \mathfrak{g}\) determines a vector field \(X_V\) on \(M\) by

X_V(p) = \left.\frac{d}{dt}\right|_{t=0}\exp(t V) \cdot p

$X_V(p) = \left.\frac{d}{dt}\right|_{t=0}\exp(t V) \cdot p$

\(SO(3)\) acts on \(S^2\) by rotations.

\(X_{V_{(a,b,c)}}((x,y,z)) = (bz-cy)\frac{\partial}{\partial x} + (cx - az) \frac{\partial}{\partial y} + (ay - bx) \frac{\partial}{\partial z}\)

For \(V_{(a,b,c)} = \begin{bmatrix} 0 & -c & b \\ c & 0 & -a \\ -b & a & 0 \end{bmatrix} \in \mathfrak{so}(3)\),

\(= (a,b,c) \times (x,y,z)\)

**Definition.** An action of \(G\) on \((M,\omega)\) is *Hamiltonian* if each one-parameter subgroup action is Hamiltonian. Equivalently, there exists a map

\mu: M \to \mathfrak{g}^*

$\mu: M \to \mathfrak{g}^*$

so that \(\omega_p(X_V, X) = D_p \mu(X)(V)\) for each \(p \in M\), \(X \in T_pM\), and \(V \in \mathfrak{g}\).

\(X_{V_{(a,b,c)}}(x,y,z) = (a,b,c) \times (x,y,z)\)

\((\iota_{X_{V_{(a,b,c)}}}\omega)_{(x,y,z)} = a dx + b dy + c dz \)

\(\mu(x,y,z)(V_{(a,b,c)}) = (x,y,z)\cdot(a,b,c)\)

i.e., \(\mu(x,y,z)=(x,y,z)\).

The diagonal \(SO(3)\) action on \(S^2 \times \dots \times S^2\) is Hamiltonian, with moment map

\(\mu:(\vec{e}_1, \dots , \vec{e}_n) \mapsto \vec{e}_1 + \dots + \vec{e}_n\)

Therefore, the space of equilateral polygons is \(\mu^{-1}(\vec{0})\), and the space of equilateral polygons modulo rotations is symplectic:

\(\operatorname{Pol}(n) := \mu^{-1}(\vec{0})/SO(3) = (S^2)^n /\!/\!_{\vec{0}} SO(3)\)

In fact, the same holds for polygons with any fixed edgelengths

**Theorem [Khoi]**

The space of equilateral polygons is larger than any other fixed edgelength polygon space.

**Classical Fact:** The density of the end-to-end vector of an \(n\)-step random walk in \(\mathbb{R}^3\) is

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

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

= \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.

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

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

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

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

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

**Proposition [w/ 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)

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

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 \(\leq m-d\).*

**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.*

**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 [w/ Cantarella]**

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.

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

$0 \leq d_i \leq 2$

1 \leq d_i + d_{i-1}

$1 \leq d_i + d_{i-1}$

|d_i - d_{i-1}| \leq 1

$|d_i - d_{i-1}| \leq 1$

0 \leq d_{n-3} \leq 2

$0 \leq d_{n-3} \leq 2$

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.

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

$0 \leq d_i \leq 2$

1 \leq d_i + d_{i-1}

$1 \leq d_i + d_{i-1}$

|d_i - d_{i-1}| \leq 1

$|d_i - d_{i-1}| \leq 1$

0 \leq d_{n-3} \leq 2

$0 \leq d_{n-3} \leq 2$

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 [w/ Cantarella, Duplantier, Uehara]**

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

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

**Action-Angle Method**

**Theorem [w/ Cantarella, Duplantier, Uehara]**

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

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

**Theorem [w/ Cantarella, Deguchi, Uehara]**

In a *Gaussian* random embedding of a graph \(G\), the vector of \(x\)-coordinates of the vertices is distributed as

\mathcal{N}(0,L^+)

$\mathcal{N}(0,L^+)$

where \(L\) is the graph Laplacian of \(G\) and \(L^+\) is its Moore–Penrose pseudoinverse.

**Theorem [w/ Cantarella, Deguchi, Uehara]**

In a *Gaussian* random embedding of a graph \(G\), the vector of \(x\)-coordinates of the vertices is distributed as

\mathcal{N}(0,L^+)

$\mathcal{N}(0,L^+)$

where \(L\) is the graph Laplacian of \(G\) and \(L^+\) is its Moore–Penrose pseudoinverse.

**Theorem [w/ Cantarella, Deguchi, Uehara]**

If \(\lambda_i\) are the eigenvalues of \(L\), the expected radius of gyration of a Gaussian random embedding of \(G\) in \(\mathbb{R}^d\) is

\frac{d}{|V|} \sum \frac{1}{\lambda_i} = \frac{d}{|V|} \operatorname{tr} L^+

$\frac{d}{|V|} \sum \frac{1}{\lambda_i} = \frac{d}{|V|} \operatorname{tr} L^+$

This is the *Kirchhoff index* of the graph.

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

Funding: Simons Foundation

Probability theory of random polygons from the quaternionic viewpoint

J. Cantarella, T. Deguchi, & C. Shonkwiler

*Communications on Pure and Applied Mathematics* **67** (2014), no. 10, 1658–1699

The expected total curvature of random polygons

J. Cantarella, A.Y. Grosberg, R. Kusner, & C. Shonkwiler

*American Journal of Mathematics* **137** (2015), no. 2, 411–438

Spherical geometry and the least symmetric triangle

L. Bowden, A. Haynes, C. Shonkwiler, & A. Shukert

*Geometriae Dedicata* (2018), https://doi.org/10.1007/s10711-018-0327-4

Random triangles and polygons in the plane

J. Cantarella, T. Needham, C. Shonkwiler, & G. Stewart

*The American Mathematical Monthly*, to appear, arXiv:1702.01027 [math.MG]