Power sums of normalised beta weights and exact arcsine averages of hypercubic lattice invariants

Summary and scope

Power sums of normalised positive weights measure concentration in random partitions and include the inverse participation ratio. Let w1, ..., wd be independent Beta(a, a) variables, let p_mu be w_mu divided by the total weight, and let R_n be the sum over mu of p_mu^n. Combining a Mellin representation of the random denominator with the beta Laplace transform reduces the expectation E[R_n] to an explicit one-dimensional integral, valid for every a > 0, d >= 1 and integer n >= 2. The hypergeometric differential equation supplies a second expression for E[R_2], while an expansion about the mean total weight gives the first finite-d correction. At a = 1/2 the weights follow the arcsine law and acquire a direct lattice interpretation: E[R_n] is the flat Brillouin-zone average of khat^[2n] over (khat^[2])^n, for the standard nearest-neighbour lattice momentum khat_mu = 2 sin(k_mu/2). In d = 2 the averages at n = 2 and n = 3 are exactly 1 - 1/pi and 1 - 3/(2pi). Numerical values for d = 2 to 6 provide reproducible benchmarks, and a supplementary script reproduces the analytic formulae, the tabulated values, and independent sampling and finite-d checks. The results connect normalised random weights with hypercubic lattice geometry. They concern flat, dimensionless averages rather than propagator-weighted Watson integrals or fixed-shell hypercubic extrapolations.

Keywords

How to cite

Elliman, D. (2026). Power sums of normalised beta weights and exact arcsine averages of hypercubic lattice invariants. Neuro-Symbolic Ltd technical report. https://doi.org/10.5281/zenodo.22110150

@techreport{elliman2026arcsineiprmomentsrevised,
  author      = {Elliman, David},
  title       = {Power sums of normalised beta weights and exact arcsine averages of hypercubic lattice invariants},
  institution = {Neuro-Symbolic Ltd},
  year        = {2026},
  doi         = {10.5281/zenodo.22110150},
  url         = {https://neusym.ai/papers/arcsine_ipr_moments_revised/}
}