Loop-erased walks and random matrices

Jonas Arista, Neil O'Connell

Published 2019-01-23Version 1

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. We show that in the context of independent Brownian motions in suitable planar domains, this also has close connections to random matrices. 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.