Poisson limit of higher-order spacings under large superposition

Prove that for any fixed order k ≥ 1 and Dyson index β ∈ {1,2,4} corresponding to a circular ensemble, the modified index β′ defined by P^{(k)}(s, β, m) = P(s, β′) tends to 0 as the number m of superposed independent spectra tends to infinity, implying convergence of the k-th order spacing distribution P^{(k)}(s, β, m) to the Poisson law P(s) = e^{−s}.

Background

From extensive simulations across COE, CUE, and CSE, the authors observe that for fixed k, increasing the number m of superposed spectra reduces β′ and drives the distributions toward Poisson behavior. They therefore formulate a general conjecture that β′ → 0 as m → ∞, i.e., that superposition destroys correlations in the k-th order spacing statistics.

They note special-case analytical support: for k = 1 and β = 2, a Poisson limit has been addressed in prior work for circular ensembles, and for Gaussian ensembles an analogous result for spacing ratios at k = 1 and m → ∞ is known. The present conjecture extends this limiting behavior to higher-order spacings for circular ensembles.