Higher-order cumulant difference-equation conjecture for induced entropy

Prove that, for every integer l>2, the l-th cumulant κ_l(T) of the induced von Neumann entropy T=∑_{i=1}^m x_i ln x_i in the unconstrained Hilbert–Schmidt (Laguerre unitary) ensemble satisfies the l-th order matrix-dimension difference equation Δ^lκ_l(T)=l!((l−1)ψ_{l−2}(n)+nψ_{l−1}(n)), where Δf_m=f_{m+1}−f_m and n is the environment dimension.

Background

The paper develops Toda-lattice and related integrable-systems methods for deriving linear difference equations in the matrix dimension m for average spectral moments and average entropies of several random-state ensembles. It proposes extending the tau-function construction to multiple deformations in order to generate joint cumulants of arbitrary order.

For the induced entropy T=∑_{i=1}^m x_i ln x_i associated with the unconstrained Hilbert–Schmidt ensemble, the authors note that a recursive scheme based on higher Toda flows does not appear sufficient to establish the conjectured general pattern. The conjecture is inferred from explicit formulas for the first six cumulants and would imply that each κ_l(T) is a polynomial in m of degree l, because its l-th finite difference is independent of m. The authors further state that resolving it likely requires a more intricate tau function that relates cumulants of different orders while avoiding the increasing number of initial conditions needed by recursive calculations.