Most random walks are almost closed(?)

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

Fixing edgelengths is trickier

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 equilateral polygon

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, Chapman, 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 \(\mathbb{R}^{dn}\).

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

Loop closure doesn’t change much!

Small knots in proteins persist

3L05A

...whereas large knots often don’t

2HOCA

Closure mostly preserves small knots

Knotted core size vs. Knotted closure probability

Knotted core size likelihood

~70% of closures knotted

~15% of closures knotted

~15% of proteins

~70% of proteins

For all proteins in KnotProt as of July 2, 2018

Main Theorem

Theorem (with Cantarella, Chapman, and Reiter)

If \(X\) consists of the edges of a random walk in \(\mathbb{R}^3\) and \(\mu=\|\mathrm{gm}(X)\|\), then for any \(r<\frac{5}{1000}\),

Corollary

For any \(\alpha < \frac{5}{1000} \sqrt{\frac{n}{2}}\),

Similar results hold in any dimension.

Provable vs. actual distance to closure

Provable: For \(n>\)1,280,000, \(\mathbb{P}(d(X,\mathrm{Pol}(n,3))<4)\geq 0.999\).

Actual: For \(n\geq10\), \(\mathbb{P}(d(X,\mathrm{Pol}(n,3))<3)\geq 0.999\).

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. By Taylor’s theorem, the radial directional derivative must be positive outside some small ball, so the point where \(\nabla \mathrm{Ad}(y) = 0\)—namely, \(\mathrm{gm}(X)\)—must be inside this ball, and hence close to the origin.