Consider the Gaussian entire function Fℂ(z)=∑k=0∞ξkzkk!, z∈ℂ, where {ξk} is a sequence of independent standard complex Gaussians. This random Taylor series is distinguished by the invariance of its zero set with respect to the isometries of the plane ℂ. It has been of considerable interest to study the statistical properties of the zero set, particularly in comparison to other planar point processes. We show that the law of the zero set, conditioned on the function Fℂ having no zeros in a disk of radius r and normalized appropriately, converges to an explicit limiting Radon measure on ℂ as r → ∞. A remarkable feature of this limiting measure is the existence of a large “forbidden region” between a singular part supported on the boundary of the (scaled) hole and the equilibrium measure far from the hole. In particular, this answers a question posed by Nazarov and Sodin, and is in stark contrast to the corresponding result of Jancovici, Lebowitz, and Manificat in the random matrix setting: there is no such forbidden region for the Ginibre ensemble. © 2018 Wiley Periodicals, Inc.

Abstract. Consider the random entire functionwhere the φ n are i.i.d. standard complex Gaussian variables. The zero set of this function is distinguished by invariance of its distribution with respect to the isometries of the plane. We study the probability P H (r) that f has no zeroes in the disk {|z| < r} (hole probability). Improving a result of Sodin and Tsirelson, we show that log P H (r) = − e 2 4 · r 4 + o(r 4 ) as r → ∞. The proof does not use distribution invariance of the zeros, and can be extended to other Gaussian Taylor series.If instead of Gaussians we take Rademacher or Steinhaus random variables φ n , we get a very different result. There exists r 0 so that every random function of the form ( * ) with Rademacher or Steinhaus coefficients must vanish in the disk {|z| < r 0 }.

We consider particle systems (also known as point processes) on the line and in the plane, and are particularly interested in "hole" events, when there are no particles in a large disk (or some other domain). We survey the extensive work on hole probabilities and the related large deviation principles (LDP), which has been undertaken mostly in the last two decades. We mainly focus on the recent applications of LDP-inspired techniques to the study of hole probabilities, and the determination of the most likely configurations of particles that have large holes.As an application of this approach, we illustrate how one can confirm some of the predictions of Jancovici, Lebowitz, and Manificat for large fluctuation in the number of points for the (two-dimensional) β-Ginibre ensembles. We also discuss some possible directions for future investigations.

Abstract. We prove that any power of the logarithm of Fourier series with random signs is integrable. This result has applications to the distribution of values of random Taylor series, one of which answers a long-standing question by J.-P. Kahane.

In this work, we relied on electrophysiological methods to characterize the processing stages that are affected by the presence of regularity in a visual search task. EEG was recorded for 72 participants while they completed a visual search task. Depending on the group, the task contained a consistent-mapping condition, a random-mapping condition, or both consistent and random conditions intermixed (mixed group). Contrary to previous findings, the control groups allowed us to demonstrate that the contextual cueing effect that was observed in the mixed group resulted from interference, not facilitation, to the target selection, response selection, and response execution processes (N2-posterior-contralateral, stimulus-locked lateralized readiness potential [LRP], and response-locked LRP components). When the regularity was highly valid (consistent-only group), the presence of regularity drove performance beyond general practice effects, through facilitation in target selection and response selection (N2-posterior-contralateral and stimulus-locked LRP components). Overall, we identified two distinct effects created by the presence of regularity: a global effect of validity that dictates the degree to which all information is taken into account and a local effect of activating the information on every trial. We conclude that, when considering the influence of regularity on behavior, it is vital to assess how the overall reliability of the incoming information is affected.