Poisson‑limit conjecture for higher‑order spacings under increasing superposition in circular ensembles

Prove that for fixed spacing order k and for circular ensembles with Dyson index β∈{1,2,4}, the effective modified Dyson index β′ obtained by fitting P^{(k)}(s,β,m) to P(s,β′) tends to 0 as the number of superposed spectra m→∞, thereby implying convergence of the k‑th order spacing distribution to the Poisson nearest‑neighbor spacing distribution.

Background

In simultaneous comparisons across COE, CUE, and CSE, the authors observe that for fixed k the fitted β′ decreases as the number of superposed blocks m increases, suggesting that correlations weaken with growing superposition.

They formulate a conjecture that the limiting distribution becomes Poisson (β′→0) as m→∞ for any fixed k and β, extending known analytical results for special cases (e.g., k=1 for CUE and for Gaussian ensembles) to general k in the circular ensembles.