Methods to control multiple crossings in general random matrix models

Develop methods that meaningfully capture and bound the contribution of multiple crossings in combinatorial expansions for arbitrarily structured random matrices, extending beyond single-crossing control available via current interpolation-based techniques.

Background

The interpolation method isolates single-crossing contributions, enabling strong convergence results for general Gaussian and discrete models. However, unlike classical GUE techniques (e.g., genus expansions), there is no general method to handle the simultaneous presence of many crossings in non-GUE contexts.

Such tools would potentially sharpen bounds at spectral edges and could lead to universality-type results for broader classes of models.