On a random entanglement problem

TL;DR

This paper models the entanglement of a 2D reflecting Brownian motion in a divided region with multiple windows, using a Markov chain on the fundamental groupoid, and establishes limit theorems for entanglement quantification.

Contribution

It introduces a novel approach to quantify entanglement via a Markov chain on the fundamental groupoid and derives limit theorems with explicit constants.

Findings

Law of large numbers for entanglement measure

Central limit theorem with explicit constants

Quantitative description of entanglement behavior

Abstract

We study a model for the entanglement of a two-dimensional reflecting Brownian motion in a bounded region divided into two halves by a wall with three or more small windows. We map the Brownian motion into a Markov Chain on the fundamental groupoid of the region. We quantify entanglement of the path with the length of the appropriate element in this groupoid. Our main results are a law of large numbers and a central limit theorem for this quantity. The constants appearing in the limit theorems are expressed in terms of a coupled system of quadratic equations.