A Cascade of Volterra-Operator BBP Transitions in a Correlated Wigner Matrix

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--Lo\`eve 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--P\'{e}ch\'{e} (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.