Short Hyperuniform Random Walks

Random walks of two steps, with fixed sums of lengths of $$1$$1, taken into uniformly random directions in d-dimensional Euclidean spaces (d≥2) are investigated to construct continuous step-length distributions which make them hyperuniform. The endpoint positions of hyperuniform walks are spread out in the unit ball as the projections in the walk space of points uniformly distributed on the surface of the unit hypersphere of some k-dimensional Euclidean space (k>d). Unique symmetric continuous step-length distributions exist for given d and k, provided that d<k<2d. The walk becomes uniform on the unit ball when k=d+2. The symmetric densities reduce to simple polynomials for uniform random walks and are mixtures of two pairs of asymmetric beta distributions.