Eigenvalue detachment, BBP transition and constrained Brownian motion

TL;DR

This paper explores the eigenvalue detachment transition and BBP-like phase transition in statistical systems, linking fluctuation scaling near convex boundaries to models like JT gravity and constrained Brownian motion.

Contribution

It provides a numerical analysis of the BBP transition in various ensembles and connects these phenomena to gravitational models and constrained path trajectories.

Findings

Identification of the critical permeability value for the transition.

Demonstration of the transition from Gaussian to Tracy-Widom fluctuations.

Connection of fluctuation regimes to JT gravity with a radial cutoff.

Abstract

We discuss the eigenvalue detachment transition in terms of scaling of fluctuations in ensembles of paths located near convex boundaries of various physical nature. We consider numerically the BBP-like (Baik-Ben Arous-P\'ech\'e) transition from the Gaussian to the Tracy-Widom scaling of fluctuations in several statistical systems for both canonical and microcanonical ensembles and identify the corresponding control parameter in each case. In particular, for fixed path length (microcanonical) ensemble of paths located in the vicinity of a partially permeable semicircle, the transition occurs at the critical value of a permeability. The Tracy-Widom regime and the BBP-like transition for fluctuations are discussed in terms of the Jakiw-Teitelbom (JT) gravity with a radial cutoff which, in turn, has an interpretation as a ensemble of fixed length world-line geometrically constrained trajectories of a charged particle in an effective transversal magnetic field.