Random Walks are Almost Closed

or

Loop Closure is Surprisingly Non-Destructive

Clayton Shonkwiler

Colorado State University

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

Random walks

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}}
P(1niei<r)12denr22d\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 of a triangle

Geometric median of a quadrilateral

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.

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

Closing a 17-edge arm

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
dmaxangular(X,Pol(n,d))<π2μd_\mathrm{max-angular}(X,\mathrm{Pol}(n,d)) < \frac{\pi}{2}\mu

Loop closure doesn’t change much!

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}}
P(μ<r)12denr220d\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)}}
P(d(X,Pol(n,d)<α)12deα220(d1)\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
P(d(X,Pol(n,3))<16)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

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

J. Phys. A Highlight of 2016

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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