Functional limit theorems for power series with rapid decay of moving averages of Hermite processes

We aim to obtain a homogenization theorem for a passive tracer interacting with a fractional, possibly non-Gaussian, noise. To do so, we analyze limit theorems for normalized functionals of Hermite–Volterra processes and extend existing results to cover power series with fast decaying coefficients. We obtain either convergence to a Wiener process, in the short-range dependent case, or to a Hermite process, in the long-range dependent case. Furthermore, we prove convergence in the multivariate case with both, short- and long-range dependent components. Applying this theorem, we obtain a homogenization result for a slow/fast system driven by such Hermite noises.

Functional limit theorems for the euler characteristic process in the critical regime

This study presents functional limit theorems for the Euler characteristic of Vietoris–Rips complexes. The points are drawn from a nonhomogeneous Poisson process on $\mathbb{R}^d$ , and the connectivity radius governing the formation of simplices is taken as a function of the time parameter t, which allows us to treat the Euler characteristic as a stochastic process. The setting in which this takes place is that of the critical regime, in which the simplicial complexes are highly connected and have nontrivial topology. We establish two ‘functional-level’ limit theorems, a strong law of large numbers and a central limit theorem, for the appropriately normalized Euler characteristic process.

Precise Deviations for Disk Counting Statistics of Invariant Determinantal Processes

We consider 2-dimensional determinantal processes that are rotationinvariant and study the fluctuations of the number of points in disks. Based on the theory of mod-phi convergence, we obtain Berry–Esseen as well as precise moderate to large deviation estimates for these statistics. These results are consistent with the Coulomb gas heuristic from the physics literature. We also obtain functional limit theorems for the stochastic process $(\# D_r)_{r>0}$ when the radius $r$ of the disk $D_r$ is growing in different regimes. We present several applications to invariant determinantal processes, including the polyanalytic Ginibre ensembles, zeros of the hyperbolic Gaussian analytic function, and other hyperbolic models. As a corollary, we compute the precise asymptotics for the entanglement entropy of (integer) Laughlin states for all Landau levels.