or

### Loop Closure is Surprisingly Non-Destructive

Clayton Shonkwiler

http://shonkwiler.org

04.21.18

/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

### Is it almost closed?

1QMG – Acetohydroxyacid isomeroreductase

### Most random walks are almost closed(?)

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,

### Distance to polygons is tricky

end-to-end distance: 16.99

distance to closed: 5.64

end-to-end distance: 17.76

distance to closed: 0.68

### Distance to a closed polygon

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.

### The geometric median

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 $$x_i$$

### Geometric median closure

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.

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

### The geometric median closure is closed

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

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

### The geometric median closure is optimal

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

$$\langle X, Y -X \rangle \leq 0$$.

### Some neat bounds

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

### Main Theorem

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

### $$d=3$$

For $$n$$ large enough

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

### Flow of the proof

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

1. The minimum eigenvalue of the Hessian of $$\mathrm{Ad}_X$$ is very likely to be bounded below near the origin.
2. $$\|\nabla \mathrm{Ad}_X(0)\|$$ is very likely to be small.
3. 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 $$\mathrm{gm}(X)$$.

### Moral of the story

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.

# Thank you!

### References

J. Cantarella & C. Shonkwiler

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

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

#### Random Walks are Almost Closed

By Clayton Shonkwiler

# Random Walks are Almost Closed

Loop closure is surprisingly non-destructive

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