Most random walks are almost closed(?)

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

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

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

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!

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 \(r<\frac{3}{7}\),

Corollary

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

\(d=3\)

For \(n\) large enough

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 \(\mathrm{gm}(X)\).