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We discuss the eigenvalue detachment transition in terms of scaling offluctuations in ensembles of paths located near convex boundaries of variousphysical nature. We consider numerically the BBP-like (Baik-BenArous-P\'ech\'e) transition from the Gaussian to the Tracy-Widom scaling offluctuations in several statistical systems for both canonical andmicrocanonical ensembles and identify the corresponding control parameter ineach case. In particular, for fixed path length (microcanonical) ensemble ofpaths located in the vicinity of a partially permeable semicircle, thetransition occurs at the critical value of a permeability. The Tracy-Widomregime and the BBP-like transition for fluctuations are discussed in terms ofthe Jakiw-Teitelbom (JT) gravity with a radial cutoff which, in turn, has aninterpretation as a ensemble of fixed length world-line geometricallyconstrained trajectories of a charged particle in an effective transversalmagnetic field.

Eigenvalue detachment, BBP transition and constrained Brownian motion