Identify the Brownian last-passage law beyond the isotropic regime

Identify the law of the Brownian last-passage functional F_Σ beyond the isotropic covariance regime, where a Gaussian maximal-eigenvalue representation need not exist.

Background

The paper studies the Brownian last-passage functional F_Σ arising as the scaling limit of longest weakly increasing subsequences in finite-state Markov random words. Under the projected-isotropy condition QΣQ=cQ, the authors identify F_Σ with the largest eigenvalue of an appropriately centered Gaussian Hermitian matrix and establish rigidity results for such representations.

Section 5 provides explicit three-state and four-state Markov-chain counterexamples showing that outside the isotropic regime, F_Σ may fail to have the law of the largest eigenvalue of any centered jointly Gaussian Hermitian matrix of the corresponding dimension. The unresolved problem is therefore to determine the distribution of F_Σ in the general non-isotropic setting, even when no Gaussian random-matrix representation is available.