# Loop-erased walks and random matrices

## TL;DR

This paper explores the connections between loop-erased paths, non-intersecting Brownian motions, and random matrix ensembles, extending classical reflection principles to planar processes and revealing new links to Cauchy-type and circular orthogonal ensembles.

## Contribution

It generalizes Fomin's reflection principle to planar processes and affine settings, establishing new links between loop-erased paths, Brownian motions, and random matrix ensembles.

## Key findings

- Derived new examples of Cauchy-type ensembles from planar Brownian motions.
- Extended Fomin's identity to the affine setting with applications to the circular orthogonal ensemble.
- Provided a novel interpretation of the circular orthogonal ensemble via Brownian motions in an annulus.

## Abstract

It is well known that there are close connections between non-intersecting processes in one dimension and random matrices, based on the reflection principle. There is a generalisation of the reflection principle for more general (e.g. planar) processes, due to S. Fomin, in which the non-intersection condition is replaced by a condition involving loop-erased paths. In the context of independent Brownian motions in suitable planar domains, this also has close connections to random matrices. An example of this was first observed by Sato and Katori (Phys. Rev. E, 83, 2011). We present further examples which give rise to various Cauchy-type ensembles. We also extend Fomin's identity to the affine setting and show that in this case, by considering independent Brownian motions in an annulus, one obtains a novel interpretation of the circular orthogonal ensemble.