Recovering Parameters from Edge Fluctuations: Beta-Ensembles and Critically-Spiked Models
Pierre Yves Gaudreau Lamarre
TL;DR
The paper provides a new inverse-spectral result for stochastic Airy operators, showing that a single edge-spectrum realization Λ^θ from beta-ensembles or Critically-Spiked models determines either the inverse temperature β or the Robin-Dirichlet boundary count r_0 through a deterministic map 𝒯 defined by a small-time trace limit. The authors develop and apply multivariate SAO semigroup theory to obtain precise trace asymptotics: E Tr[e^{-tH_{θ,η}/2}] = (r/(√{2π}κ)) t^{-3/2} + (1/4)(2r_0 - r + rσ^2/κ + r(r-1)ν^2/κ) + o(1) as t→0^+, with a covariance bound, enabling almost-sure recovery of r_0+1/β via a law of large numbers over a vanishing sequence of t. This leads to concrete corollaries for temperature recovery in beta-ensembles and for detecting near-threshold spikes in spiked Wishart/Gaussian models from edge data alone, and provides insights into rigidity and cancellations of subcritical coordinates within SAO spectra. The work links inverse spectral problems to edge fluctuations and rigidity phenomena in random matrix theory, introducing explicit, computable mechanisms (via the trace-expansion) to distinguish SAO spectra across dimensions and boundary conditions. It advances the understanding of how limited edge information encodes global operator parameters with potential applications to statistical inference in high-dimensional random systems.
Abstract
Let $Λ=\{Λ_0,Λ_1,Λ_2,\ldots\}$ be the point process that describes the edge scaling limit of either (i) "regular" beta-ensembles with inverse temperature $β>0$, or (ii) the top eigenvalues of Wishart or Gaussian invariant random matrices perturbed by $r_0\geq1$ critical spikes. In other words, $Λ$ is the eigenvalue point process of one of the scalar or multivariate stochastic Airy operators. We prove that a single observation of $Λ$ suffices to recover (almost surely) either (i) $β$ in the case of beta-ensembles, or (ii) $r_0$ in the case of critically-spiked models. Our proof relies on the recently-developed semigroup theory for the multivariate stochastic Airy operators. Going beyond these parameter-recovery applications, our results also (iii) refine our understanding of the rigidity properties of $Λ$, and (iv) shed new light on the equality (in distribution) of stochastic Airy spectra with different dimensions and the same Robin boundary conditions.