We study critical branching random walks and introduce the branching capacity and related concepts for any finite subset of ℤ^d, d≥ 5 . By introducing these concepts, we can obtain analogues of various classical results for random walks. In this paper, we develop the basic properties of the branching capacity and show that it is closely related to the probability of visiting a fixed finite set by a critical branching random walk starting from a distant point, as well as to the conditional distribution of the hitting point(s).
We study a general class of nonlinear Ginzburg–Landau SPDEs in infinite volume under weak nonlinearity scaling and with non-equilibrium initial data. We derive the KPZ equation as a continuum limit of these equations. This makes rigorous the original derivation of the KPZ equation from physics [31] in the full-space setting, which was a problem posed in [27]. Our analysis is based on a stochastic heat kernel for a linearization of said SPDEs.
We study zero sets of twisted stationary Gaussian random functions on the complex plane, i.e., Gaussian random functions that are stochastically invariant under the action of the Weyl-Heisenberg group. This model includes translation-invariant Gaussian entire functions (GEFs), and also many other non-analytic examples, in which case winding numbers around zeros can be either positive or negative. We investigate zero statistics both when zeros are weighted with their winding numbers (charged zero set) and when they are not (uncharged zero set). We show that the variance of the charged zero statistic always grows linearly with the radius of the observation disk (hyperuniformity). Importantly, this holds for functions with possibly non-zero means and without assuming additional symmetries such as radiality. With respect to uncharged zero statistics, we provide an example for which the variance grows with the area of the observation disk (non-hyperuniformity). This is used to show that, while the zeros of GEFs are hyperuniform, the set of their critical points fails to be so. Our work contributes to recent developments in statistical signal processing, where the time-frequency profile of a non-stationary signal embedded into noise is revealed by performing a statistical test on the zeros of its spectrogram (“silent points”). We show that empirical spectrogram zero counts enjoy moderate deviations from their ensemble averages over large observation windows (something that was previously known only for pure noise). In contrast, we also show that spectrogram maxima (“loud points”) fail to enjoy a similar property. This gives the first formal evidence for the statistical superiority of silent points over the competing feature of loud points, a fact that has been noted by practitioners. In the same vein, our second order asymptotics for spectrogram maxima show that certain heuristic proxy models used in signal processing are inaccurate at large scales.
We consider an infinite system of interacting Brownian motions that preserves a given random point field invariant. Such dynamics are constructed using Dirichlet form theory, which naturally leads to two Dirichlet forms for the random point field: the upper and the lower Dirichlet forms. A fundamental question is the uniqueness of the Dirichlet form: that is, whether these two forms coincide. This uniqueness has often been imposed as a key assumption in the Dirichlet form approach to the stochastic analysis for infinite particle systems. A sufficient condition for the uniqueness of the Dirichlet forms is known when the random point field is tail trivial. However, tail triviality has been established for only a limited class of random point fields. In this paper, we prove the uniqueness of the Dirichlet form without assuming tail triviality. The main contribution of this work is to establish the tail preserving property, which asserts that global properties of the system, such as particle density, are preserved under time evolution. As a consequence, our results also imply the strong uniqueness of solutions to the associated infinite-dimensional stochastic differential equations.
