A Finitization of the Bead Process

TL;DR

This paper introduces a finitized version of the bead process, linking it to random matrix eigenvalues, and analyzes its correlation structure and scaling limits, revealing new insights into particle configurations and their asymptotic behavior.

Contribution

It defines a finitized bead process with a determinantal structure and explicit correlation kernel, connecting it to random matrix theory and scaling limits.

Findings

The finitized bead process is determinantal with an explicit correlation kernel.

Global scaling limit determines the shape of particle support.

Bulk scaling limit recovers the bead process kernel of Boutillier.

Abstract

The bead process is the particle system defined on parallel lines, with underlying measure giving constant weight to all configurations in which particles on neighbouring lines interlace, and zero weight otherwise. Motivated by the statistical mechanical model of the tiling of an $abc$-hexagon by three species of rhombi, a finitized version of the bead process is defined. The corresponding joint distribution can be realized as an eigenvalue probability density function for a sequence of random matrices. The finitized bead process is determinantal, and we give the correlation kernel in terms of Jacobi polynomials. Two scaling limits are considered: a global limit in which the spacing between lines goes to zero, and a certain bulk scaling limit. In the global limit the shape of the support of the particles is determined, while in the bulk scaling limit the bead process kernel of Boutillier is reclaimed, after approriate identification of the anisotropy parameter therein.