Interacting diffusions on positive definite matrices

Abstract

We consider systems of Brownian particles in the space of positive definite matrices, which evolve independently apart from some simple interactions. We give examples of such processes which have an integrable structure. These are related to $K$-Bessel functions of matrix argument and multivariate generalisations of these functions. The latter are eigenfunctions of a particular quantisation of the non-Abelian Toda chain.

arXiv commentv3: substantial revision, includes new section on complex case

References in corpus (9)

• Directed polymers and the quantum Toda lattice
• Tropical Combinatorics and Whittaker functions
• Geometric RSK correspondence, Whittaker functions and symmetrized random polymers
• Exponential functionals of Brownian motion and class-one Whittaker functions
• Determinantal structures in the O'Connell-Yor directed random polymer model
• Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble
• BKP hierarchy and Pfaffian point process
• Geometric RSK and the Toda lattice
• Stochastic Bäcklund transformations

Cited by in corpus (4)

• Matrix Whittaker processes
• Matsumoto-Yor and Dufresne type theorems for a random walk on positive definite matrices
• Yang-Baxter maps and independence preserving property
• On two-dimensional extensions of Bougerol's identity in law