We study how handedness enters the factorial cumulants and polyspectra of stationary, properly isotropic spatial point processes. Rotational symmetry forces parity-odd cumulants to vanish at every order up to the spatial dimension. At the next order, the odd part of the factorial moment is necessarily connected. We show that this threshold is sharp by constructing a relabelling-invariant measure of simplex handedness and using it to define a chiral Poisson-cluster process. The resulting factorial polyspectrum has a nonzero parity-odd component, which changes sign under reflection and cancels when the two mirror laws are mixed equally.
Assume that a stick with unit length is broken totally randomly into n pieces. The kth largest piece (with positive integer k≤n) is called the kth largest uniform spacing induced by n uniformly distributed order statistics on (0,1). Exact large deviations of the kth largest uniform spacing away from its most probable size lnn/n are studied in this paper, together with an application in goodness-of-fit tests.
A random population of trajectories can be viewed as a point process whose points are entire paths. When the spatial domain contains holes or obstacles, a path also carries global route information, such as how many times it winds around an obstacle. We represent this information by line integrals against a finite family of smooth one-forms supplied by Hodge–Morrey theory; on an annulus, the resulting coordinate is the usual winding number. The main result expresses the cumulant generating function of the aggregate coordinate vector directly in terms of the factorial cumulant measures of the path process. Differentiation yields an all-order partition formula for these topological cumulants. The covariance separates single-trajectory variation from non-Poisson dependence between trajectories. The construction is independent of the chosen closed-form representatives and transforms naturally under a change of coordinate basis. For Poisson path populations the law reduces to compound Poisson, and balanced clockwise and anticlockwise populations give a centred Skellam law.
