This paper proposes a framework to generalize eigenvalue statistics to higher-order members of the Boussinesq hierarchy, extending the connection between random matrix theory and integrable systems. It combines algebro-geometric integration, Riemann-Hilbert analysis, and τ-function representations to derive spectral statistics for the first four hierarchy members and proposes experimental validation through numerical simulations.
Key findings
Establishes the k-th Boussinesq hierarchy member governs eigenvalue statistics at critical points where density vanishes as |x|(2k+1)/3.
Develops algebro-geometric framework based on spectral curves of type (3,3N+1) to construct finite-gap solutions and their eigenvalue statistics.
Derives explicit formulas for correlation kernels associated with the first four hierarchy members using τ-function representations and Riemann-Hilbert methods.
Formulates conjectures about the universality of these statistics across different matrix ensembles, supported by heuristic arguments and numerical evidence.
Proposes numerical experiments using block-tridiagonal matrix ensembles to validate theoretical predictions.
Limitations & open questions
The framework is yet to be experimentally validated for higher-order members beyond the first four.