Loop-erased walks and random matrices

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.

References in corpus (9)

• Euler integrals for commuting SLEs
• Gap probabilities in the finite and scaled Cauchy random matrix ensembles
• Nonintersecting Brownian motions on the unit circle
• Loop-erased random walk and Poisson kernel on planar graphs
• Multiple SLE type scaling limits: from local to global
• Nonintersecting lattice paths on the cylinder
• The scaling limit of Fomin's identity for two paths in the plane
• Determinantal Correlations of Brownian Paths in the Plane with Nonintersection Condition on their Loop-Erased Parts
• Loop-Erasure of Plane Brownian Motion