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Naomi Feldheim

Publications and source records attributed to Naomi Feldheim.

16 recordsLinked to original sources
arXiv2026

Persistence and entropic repulsion of stationary Gaussian fields with spectral singularity at the origin

We compute the exact log-asymptotics of the persistence probability, and determine the entropic repulsion profile conditioned on persistence, for general $d$-dimensional stationary Gaussian fields with spectral singularity at the origin of order $α\in [0,d)$. Under mild regularity conditions these are shown to be universal, depending only on $α$ and $d$, and to have explicit formulations in terms of the capacity and equilibrium potential of the $α$-Riesz kernel. This generalises a result of Bolthausen, Deuschel and Zeitouni on the Gaussian free field to a wide class of Gaussian fields with spectral singularity.

math.PR↗
arXiv2025

Persistence and Ball Exponents for Gaussian Stationary Processes

Consider a real Gaussian stationary process $f_ρ$, indexed on either $\mathbb{R}$ or $\mathbb{Z}$ and admitting a spectral measure $ρ$. We study $θ_ρ^\ell=-\lim\limits_{T\to\infty}\frac{1}{T} \log\mathbb{P}\left(\inf_{t\in[0,T]}f_ρ(t)>\ell\right)$, the persistence exponent of $f_ρ$. We show that, if $ρ$ has a positive density at the origin, then the persistence exponent exists; moreover, if $ρ$ has an absolutely continuous component, then $θ_ρ^\ell>0$ if and only if this spectral density at the origin is finite. We further establish continuity of $θ_ρ^\ell$ in $\ell$, in $ρ$ (under a suitable metric) and, if $ρ$ is compactly supported, also in dense sampling. Analogous continuity properties are shown for $ψ_ρ^\ell=-\lim\limits_{T\to\infty}\frac{1}{T} \log\mathbb{P}\left(\inf_{t\in[0,T]}|f_ρ(t)|\le \ell\right)$, the ball exponent of $f_ρ$, and it is shown to be positive if and only if $ρ$ has an absolutely continuous component.

math.PR↗
arXiv2024

Hyperuniformity and non-hyperuniformity of zeros of Gaussian Weyl-Heisenberg Functions

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.

math.PR↗
arXiv2023

A sharp transition in zero overcrowding and undercrowding probabilities for Stationary Gaussian Processes

We study the probability that a real stationary Gaussian process has at least $ηT$ zeros in $[0,T]$ (overcrowding), or at most this number (undercrowding). We show that if the spectral measure of the process is supported on $\pm[B,A]$, overcrowding probability transitions from exponential decay to Gaussian decay at $η=\tfrac{A}π$, while undercrowding probability undergoes the reverse transition at $η=\tfrac{B}π$.

math.PR↗
arXiv2023

Typical height of the (2+1)-D Solid-on-Solid surface with pinning above a wall in the delocalized phase

We study the typical height of the (2+1)-dimensional solid-on-solid surface with pinning interacting with an impenetrable wall in the delocalization phase. More precisely, let $Λ_N$ be a $N \times N$ box of $\mathbb{Z}^2$, and we consider a nonnegative integer-valued field $(ϕ(x))_{x \in Λ_N}$ with zero boundary conditions (i.e. $ϕ|_{Λ_N^{\complement}}=0 $) associated with the energy functional $$ \mathcal{V} (ϕ)= β\sum_{x \sim y} \vert ϕ(x)-ϕ(y) \vert- \sum_{x} h \mathbf{1}_{\{ ϕ(x)=0\}},$$ where $β>0$ is the inverse temperature and $h\ge 0$ is the pinning parameter. Lacoin has shown that for sufficiently large $β$, there is a phase transition between delocalization and localization at the critical point $$h_w(β)= \log \left( \frac{e^{4 β}}{e^{4 β}-1}\right).$$ In this paper we show that for $β\ge 1$ and $h \in (0, h_w)$, the values of $ϕ$ concentrate at the height $H= \lfloor (4 β)^{-1} \log N \rfloor$ with constant order fluctuations. Moreover, at criticality $h=h_w$, we provide evidence for the conjectured typical height $H_w= \lfloor (6 β)^{-1} \log N \rfloor$.

math.PR↗
arXiv2022

Efficient computation of the zeros of the Bargmann transform under additive white noise

We study the computation of the zero set of the Bargmann transform of a signal contaminated with complex white noise, or, equivalently, the computation of the zeros of its short-time Fourier transform with Gaussian window. We introduce the adaptive minimal grid neighbors algorithm (AMN), a variant of a method that has recently appeared in the signal processing literature, and prove that with high probability it computes the desired zero set. More precisely, given samples of the Bargmann transform of a signal on a finite grid with spacing $δ$, AMN is shown to compute the desired zero set up to a factor of $δ$ in the Wasserstein error metric, with failure probability $O(δ^4 \log^2(1/δ))$. We also provide numerical tests and comparison with other algorithms.

math.NA↗
arXiv2022

An asymptotic formula for the variance of the number of zeroes of a stationary Gaussian process

We study the variance of the number of zeroes of a stationary Gaussian process on a long interval. We give a simple asymptotic description under mild mixing conditions. This allows us to characterise minimal and maximal growth. We show that a small (symmetrised) atom in the spectral measure at a special frequency does not affect the asymptotic growth of the variance, while an atom at any other frequency results in maximal growth. Our results allow us to analyse a large number of interesting examples.

math.PR↗
arXiv2020

Persistence of Gaussian stationary processes: a spectral perspective

We study the persistence probability of a centered stationary Gaussian process on $\mathbb{Z}$ or $\mathbb{R}$, that is, its probability to remain positive for a long time. We describe the delicate interplay between this probability and the behavior of the spectral measure of the process near zero and infinity.

math.PR↗
arXiv2018

Convergence of the Quantile Admission Process with Veto Power

The quantile admission process with veto power is a stochastic processes suggested by Alon, Feldman, Mansour, Oren and Tennenholtz as a model for the evolution of an exclusive social group. The model itself consists of a growing multiset of real numbers, representing the opinions of the members of the club. On each round two new candidates, holding i.i.d. $μ$-distributed opinions, apply for admission to the club. The one whose opinion is minimal is then admitted if the percentage of current members closer in their opinion to his is at least $r$. Otherwise neither of the candidates is admitted. We show that for any $μ$ and $r$, the empirical distribution of opinions in the club converges to a limit distribution. We further analyse this limit, show that it may be non-deterministic and provide conditions under which it is deterministic. The results rely on a recent work of the authors relating tail probabilities of mean and maximum of any pair of unbounded i.i.d. random variables, and on a coupling of the evolution of the empirical $r$-quantile of the club with a random walk in a changing environment.

math.PR↗
arXiv2017

Exponential concentration for zeroes of stationary Gaussian processes

We show that for any centered stationary Gaussian process of integrable covariance, whose spectral measure has compact support, or finite exponential moments (and some additional regularity), the number of zeroes of the process in $[0,T]$ is within $ηT$ of its mean value, up to an exponentially small in $T$ probability.

math.PR↗
arXiv2016

The winding of stationary Gaussian processes

This paper studies the winding of a continuously differentiable Gaussian stationary process $f:\mathbb{R}\to\mathbb{C}$ in the interval $[0,T]$. We give formulae for the mean and the variance of this random variable. The variance is shown to always grow at least linearly with $T$, and conditions for it to be asymptotically linear or quadratic are given. Moreover, we show that if the covariance function together with its second derivative are in $L^2(\mathbb{R})$, then the winding obeys a central limit theorem. These results correspond to similar results for zeroes of real-valued stationary Gaussian functions by Cuzick, Slud and others.

math.PR↗
arXiv2016

Variance of the Number of Zeroes of Shift-Invariant Gaussian Analytic Functions

Following Wiener, we consider the zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We show that the variance of the number of zeroes in a long horizontal rectangle $[0,T]\times [a,b]$ is asymptotically between $cT$ and $CT^2$, with positive constants $c$ and $C$. We also supply with conditions (in terms of the spectral measure) under which the variance asymptotically grows linearly, as a quadratic function of $T$, or has intermediate growth.

math.PR↗
arXiv2015

The two-dimensional small ball inequality and binary nets

In the current paper we present a new proof of the small ball inequality in two dimensions. More importantly, this new argument, based on an approach inspired by lacunary Fourier series, reveals the first formal connection between this inequality and discrepancy theory, namely the construction of two-dimensional binary nets, i.e. finite sets which are perfectly distributed with respect to dyadic rectangles. This relation allows one to generate all possible point distributions of this type. In addition, we outline a potential approach to the higher-dimensional small ball inequality by a dimension reduction argument. In particular this gives yet another proof of the two-dimensional signed (i.e. coefficients $\pm 1$) small ball inequality by reducing it to a simple one-dimensional estimate. However, we show that an analogous estimate fails to hold for arbitrary coefficients.

math.CA↗
arXiv2015

A note on the convex infimum convolution inequality

We characterize the symmetric measures which satisfy the one dimensional convex infimum convolution inequality of Maurey. For these measures the tensorization argument yields the two level Talagrand's concentration inequalities for their products and convex sets in $\mathbb{R}^n$.

math.PR↗
arXiv2011

Zeroes of Gaussian Analytic Functions with Translation-Invariant Distribution

We study zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We prove that the a limiting horizontal mean counting-measure of the zeroes exists almost surely, and that it is non-random if and only if the spectral measure is continuous (or degenerate). In this case, the mean zero-counting measure is computed in terms of the spectral measure. We compare the behavior with Gaussian analytic function with symmetry around the real axis. These results extend a work by Norbert Wiener.

math.PR↗