🤖 AI Summary
This study investigates the influence of coupling strength on spectral edge phase transitions in Wigner matrices with row–column shared random factor correlation structures. By leveraging the Karhunen–Loève expansion, random matrix theory, and spectral analysis of compact Volterra integral operators, the authors demonstrate that the spectrum of the associated correlation matrix consists of a vanishing bulk component and a sequence of outlier eigenvalues. The key contribution lies in the discovery of multiple Baik–Ben Arous–Péché (BBP) phase transitions forming discrete critical levels, where the transition points are precisely determined by the singular values of the underlying compact Volterra integral operator. Theoretical predictions align with numerical simulations to within 1% error for the first twenty singular values, accurately capturing the hierarchical emergence of eigenvalues beyond the semicircle law edge.
📝 Abstract
We study a Wigner-type random matrix in which the off-diagonal correlation between entries is generated by a random factor shared among all entries in a given row and column, with the coupling strength held fixed as the matrix size grows. Although the bulk spectral moments remain those of the pure semicircle law, we show that the underlying correlation matrix decomposes into a vanishing bulk together with a countable family of outlier eigenvalues that, at fixed rank $k$, converge to the singular values of a compact Volterra (cumulative-sum) integral operator -- obtained in closed form via the classical Karhunen--Loève expansion of Brownian motion and confirmed numerically to better than one percent across the top twenty such values. Each singular value drives an independent Baik--Ben Arous--Péché (BBP) transition as the coupling strength increases, producing an evenly spaced, discrete hierarchy of critical points -- rather than a single transition -- at each of which one further eigenvalue detaches from the semicircle edge, in close agreement with direct diagonalization. We show that this mechanism generalizes to a broader family of correlation structures, with the critical hierarchy in every case set by the spectrum of an associated compact integral operator.