Smooth disorder and Majorana splitting statistics in the Kitaev chain: an Oke scaling relation and an Apoleia length

The energy splitting δE between the two Majorana end modes of a disordered Kitaev chain is known to follow a log-normal distribution when the disorder is uncorrelated from site to site. Real devices, however, contain smooth, spatially correlated disorder. We ask how the correlation of the disorder changes the wire-to-wire statistics of the splitting. At the point Δ = t we derive that the variance of the log-splitting obeys an Oke scaling relation, Var(ln δE) = N J² K, where N is the number of sites, J the jostle strength of a single site and K the integrated correlation length of the disorder: smooth disorder leaves the typical splitting unchanged but multiplies its spread by K. Exact diagonalization of the full model confirms the relation for flat, Gaussian and exponentially correlated (AR(1)) disorder, at Δ = t and Δ = 0.6t, and identifies two regimes where it fails. Combined with the d₂ constants of statistical quality control, the relation predicts the ratio between the largest and smallest splitting in a batch of n wires. We further find that, at fixed single-site disorder, the fraction of chains whose Majorana pair fails an isolation criterion is largest when the disorder correlation spans an intermediate number of sites, which we call the Apoleia length ℓ_A, and we state this as the Apoleia hypothesis.