Global spectral density for products with heterogeneous rectangularity parameters

Determine the global limiting spectral density for the squared singular values of the product Y_M = X_M X_{M-1} ... X_1 of independent rectangular complex Ginibre matrices in the asymptotic regime where N = min_j N_j tends to infinity, Delta_{M,N} = \sum_{j=0}^M 1/(N + \nu_j) tends to 0, and N/N_l tends to y_l in (0,1], when the rectangularity ratios y_l are distinct (i.e., not all equal).

Background

The paper analyzes singular value statistics for products of independent rectangular complex Ginibre matrices Y_M = X_M \cdots X_1 with sizes N_j \times N_{j-1}. Under the regime Delta_{M,N} -> 0 and a common limiting rectangularity y_l = y for all l, it derives an explicit parametric form for the global limiting spectral density and proves bulk sine-kernel universality for the log-transformed singular values.

When the rectangularity ratios y_l are not all equal, the Stieltjes transform G(z) still satisfies an algebraic relation, but an explicit determination of the limiting density is not obtained here. The authors state that extending their global analysis from the homogeneous case y_l = y to the heterogeneous case with distinct y_l remains unresolved.