In recent years, ``stealthy'' particle systems have gained considerable attention in condensed matter physics. These are particle systems for which the diffraction spectrum or structure function (i.e. the Fourier transform of the truncated pair correlation function) vanishes in a neighbourhood of the origin in the wave space. These systems are believed to exhibit the phenomenon of ``cloaking'', i.e. being invisible to probes of certain frequencies. They also exhibit the phenomenon of hyperuniformity, namely suppressed fluctuations of particle counts, a property that has been shown to arise in a wide array of settings in chemistry, physics and biology. We will demonstrate that stealthy particle systems (and their natural extensions to stealthy stochastic processes) exhibit a highly rigid structure; in particular, their entropy per unit volume is degenerate, and any spatial void in such a system cannot exceed a certain size. Time permitting, we will also discuss the intriguing correlation geometry of such systems and its interplay with the analytical properties of their diffraction spectrum. Based on joint works with Joel Lebowitz and Kartick Adhikari.

In this presentation, we discuss comprehensive frequency domain methods for estimating and inferring the second-order structure of spatial point processes. The main element here is on utilizing the discrete Fourier transform (DFT) of the point pattern and its tapered counterpart. Under second-order stationarity, we show that both the DFTs and the tapered DFTs are asymptotically jointly independent Gaussian even when the DFTs share the same limiting frequencies. Based on these results, we establish an α-mixing central limit theorem for a statistic formulated as a quadratic form of the tapered DFT. As applications, we derive the asymptotic distribution of the kernel spectral density estimator and establish a frequency domain inferential method for parametric stationary point processes. For the latter, the resulting model parameter estimator is computationally tractable and yields meaningful interpretations even in the case of model misspecification. We investigate the finite sample performance of our estimator through simulations, considering scenarios of both correctly specified and misspecified models. Joint work with Yongtao Guan @CUHK-Shenzhen.