or

/ne18

This talk!

*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

1QMG – Acetohydroxyacid isomeroreductase

Suppose \(e_1,\ldots , e_n\) are the edges of a random walk in \(\mathbb{R}^d\).

\mathbb{P}\left(\left\|\frac{1}{n}\sum_i e_i \right\| < r\right) \geq 1-2d e^{-\frac{nr^2}{2d}}

$\mathbb{P}\left(\left\|\frac{1}{n}\sum_i e_i \right\| < r\right) \geq 1-2d e^{-\frac{nr^2}{2d}}$

By Chernoff’s inequality,

end-to-end distance: 16.99

distance to closed: 5.64

end-to-end distance: 17.76

distance to closed: 0.68

**Conjecture.** For \(n \to \infty\), distance to the closest polygon to a random walk in \(\mathbb{R}^d\) follows a Nakagami\(\left(\frac{d}{2},\frac{d}{d-1}\right)\) distribution.

**Definition**

A *geometric median* (or *Fermat-Weber point*) of a collection \(X=\{x_1,\ldots , x_n\}\) of points in \(\mathbb{R}^d\) is any point closest to the \(x_i\):

\(\text{gm}(X)=\text{argmin}_y \sum \|x_i-y\|\)

**Definition**

A point cloud has a *nice* geometric median if:

- \(\text{gm}(X)\) is unique (\(\Leftarrow X\) is not linear)
- \(\text{gm}(X)\) is not one of the \(x_i\)

**Definition**

If the edge cloud \(X\) of an equilateral arm has a nice geometric median, the *geometric median closure* \(\text{gmc}(X)\) recenters the edge cloud at the geometric median.

\(\text{gmc}(X)_i = \frac{x_i-\text{gm}(X)}{\|x_i - \text{gm}(X)\|}\)

**Proposition (with Cantarella and Reiter)**

If it exists, the geometric median closure of an arm is a closed polygon.

**Proof**

\(\text{gm}(X)\) minimizes the average distance function

\(\mathrm{Ad}_X(y) = \frac{1}{n}\sum_i \|x_i - y\|\),

which is convex everywhere and smooth away from the \(x_i\), and

\(\nabla \mathrm{Ad}_X(y) = \frac{1}{n}\sum_i \frac{x_i-y}{\|x_i-y\|}\).

**Definition**

An arm or polygon \(X\) is given by \(n\) edge vectors \(x_i \in \mathbb{R}^d\), or a single point in \(\mathbb{R}^{dn}\). The *distance* between \(X\) and \(Y\) is the Euclidean distance between these points in \(\mathbb{R}^{dn}\).

**Theorem (with Cantarella and Reiter)**

If \(X\) is an equilateral arm in \(\mathbb{R}^d\) with a geometric median closure, then \(\text{gmc}(X)\) is the closest equilateral polygon to \(X\).

**Proof**

Depends on the neat fact that if \(\|x_i\|=\|y_i\|\), then

\(\langle X, Y -X \rangle \leq 0\).

Suppose \(X=(x_1,\ldots , x_n)\) consists of the edges of an \(n\)-step random walk in \(\mathbb{R}^d\). Let \(\mu=\|\mathrm{gm}(X)\|\).

**Lemma.** \(d(X,\mathrm{Pol}(n,d))<\mu\sqrt{2}\sqrt{n}\)

In fact, \(d(X,\mathrm{Pol}(n,d)) \sim \mu\sqrt{\frac{d-1}{d}}\sqrt{n}\).

**Lemma.** If \(d_\mathrm{max-angular}(X,Y):=\max_i \angle(x_i,y_i)\), then

d_\mathrm{max-angular}(X,\mathrm{Pol}(n,d)) < \frac{\pi}{2}\mu

$d_\mathrm{max-angular}(X,\mathrm{Pol}(n,d)) < \frac{\pi}{2}\mu$

**Theorem* (with Cantarella & Reiter)**

If \(X\) consists of the edges of a random walk in \(\mathbb{R}^d\) and \(\mu=\|\mathrm{gm}(X)\|\), then for any \(r<\frac{3}{7}\),

\mathbb{P}(\mu < r) \geq 1-2de^{-n\frac{r^2}{20d}}

$\mathbb{P}(\mu < r) \geq 1-2de^{-n\frac{r^2}{20d}}$

**Corollary**

For any \(\alpha < \frac{3}{7} \sqrt{\frac{n(d-1)}{d}}\),

\mathbb{P}(d(X,\mathrm{Pol}(n,d)<\alpha)\geq 1-2d e^{-\frac{\alpha^2}{20(d-1)}}

$\mathbb{P}(d(X,\mathrm{Pol}(n,d)<\alpha)\geq 1-2d e^{-\frac{\alpha^2}{20(d-1)}}$

For \(n\) large enough

\mathbb{P}(d(X,\mathrm{Pol}(n,3))<16)\geq 0.99

$\mathbb{P}(d(X,\mathrm{Pol}(n,3))<16)\geq 0.99$

Recall that \(\mathrm{gm}(X)\) is the unique minimizer of the convex function \(\mathrm{Ad}_X(y)\).

- The minimum eigenvalue of the Hessian of \(\mathrm{Ad}_X\) is very likely to be bounded below near the origin.
- \(\|\nabla \mathrm{Ad}_X(0)\|\) is very likely to be small.
- Since \(\mathrm{Ad}_X\) is strictly convex, \(\nabla \mathrm{Ad}_X(y)=0\) for some \(y\) near the origin...but this \(y\) is exactly the point \(\mathrm{gm}(X)\).

Closing a random walk is very unlikely to mess up the local structure of the walk.

Random walks are surprisingly close to closed polygons, for any \(n\) and in any dimension.

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

Concentration of measure for equilateral polygons in \(\mathbb{R}^d\)

J. Cantarella, P. Reiter, & C. Shonkwiler

In preparation

Funding: Simons Foundation