arXiv subjects

Alon Nishry

Publications and source records attributed to Alon Nishry.

At least 19 recordsLinked to original sources

arXiv2025

Integer-valued polynomials satisfying growth constraints

We consider polynomials which take integer values on the integers (IVPs), and satisfy an additional growth condition on the natural numbers. Elkies and Speyer, answering a question by Dimitrov, showed there is a critical exponential growth threshold, such that there are infinitely many IVPs with growth above the threshold and finitely many IVPs below that threshold (of arbitrary degree). In this paper, we give more refined estimates for the number of IVPs having exponential growth thresholds. In addition, we consider a similar problem, where there is a (not necessarily symmetric) growth condition on the integers. Notably, the critical threshold is determined by the logarithmic capacity of an explicit domain.

math.NT↗

arXiv2025

Large charge fluctuations in the hierarchical Coulomb gas

The two-dimensional one-component plasma (OCP) is a model of electrically charged particles which are embedded in a uniform background of the opposite charge, and interact through a logarithmic potential. More than 30 years ago, Jancovici, Lebowitz and Manificat discovered an asymptotic law for probabilities of large charge fluctuations in the OCP. We prove that this law holds for the hierarchical counterpart of the OCP. The hierarchical model was recently introduced by Chatterjee, and is inspired by Dyson's hierarchical model of the Ising ferromagnet.

math.PR↗

arXiv2024

The image of random analytic functions: coverage of the complex plane via branching processes

We consider the range of random analytic functions with finite radius of convergence. We show that any unbounded random Taylor series with rotationally invariant coefficients has dense image in the plane. We moreover show that if in addition the coefficients are complex Gaussian with sufficiently regular variances, then the image is the whole complex plane. We do this by exploiting an approximate connection between the coverage problem and spatial branching processes. This answers a long-standing open question of J.-P. Kahane, with sufficient regularity.

math.PR↗

arXiv2023

Universality for outliers in weakly confined Coulomb-type systems

This work concerns weakly confined particle systems in the plane, characterized by a large number of outliers away from a droplet where the bulk of the particles accumulate in the many-particle limit. We are interested in the asymptotic behavior of outliers for two classes of point processes: Coulomb gases at determinantal inverse temperature confined by a regular background, and a class of random polynomials. We observe that the limiting outlier process only depends on the shape of the uncharged region containing them, and the global net excess charge. In particular, for a determinantal Coulomb gas confined by a sufficiently regular background measure, the outliers in a simply connected uncharged region converge to the corresponding Bergman point process. For a finitely connected uncharged region $Ω$, a family of limiting outlier processes arises, indexed by the (Pontryagin) dual of the fundamental group of $Ω$. Moreover, the outliers in different uncharged regions are asymptotically independent, even if the regions have common boundary points. The latter result is a manifestation of screening properties of the particle system.

math.PR↗

arXiv2023

Extremal bounds for Dirichlet polynomials with random multiplicative coefficients

For $X(n)$ a Steinhaus random multiplicative function, we study the maximal size of the random Dirichlet polynomial $$ D_N(t) = \frac1{\sqrt{N}} \sum_{n \leq N} X(n) n^{it}, $$ with $t$ in various ranges. In particular, for fixed $C>0$ and any small $\varepsilon>0$ we show that, with high probability, $$ \exp( (\log N)^{1/2-\varepsilon} ) \ll \sup_{|t| \leq N^C} |D_N(t)| \ll \exp( (\log N)^{1/2+\varepsilon}). $$

math.NT↗

arXiv2022

Gaussian complex zeroes are not always normal: limit theorems on the disc

We study the zeroes of a family of random holomorphic functions on the unit disc, distinguished by their invariance with respect to the hyperbolic geometry. Our main finding is a transition in the limiting behaviour of the number of zeroes in a large hyperbolic disc. We find a normal distribution if the covariance decays faster than a certain critical value. In contrast, in the regime of 'long-range dependence' when the covariance decays slowly, the limiting distribution is skewed. For a closely related model we emphasise a link with Gaussian multiplicative chaos.

math.PR↗

arXiv2021

Moments of polynomials with random multiplicative coefficients

For $X(n)$ a Rademacher or Steinhaus random multiplicative function, we consider the random polynomials $$ P_N(θ) = \frac1{\sqrt{N}} \sum_{n\leq N} X(n) e(nθ), $$ and show that the $2k$-th moments on the unit circle $$ \int_0^1 \big| P_N(θ) \big|^{2k}\, dθ$$ tend to Gaussian moments in the sense of mean-square convergence, uniformly for $k \ll (\log N / \log \log N)^{1/3}$, but that in contrast to the case of i.i.d. coefficients, this behavior does not persist for $k$ much larger. We use these estimates to (i) give a proof of an almost sure Salem-Zygmund type central limit theorem for $P_N(θ)$, previously obtained in unpublished work of Harper by different methods, and (ii) show that asymptotically almost surely $$ (\log N)^{1/6 - \varepsilon} \ll \max_θ|P_N(θ)| \ll \exp((\log N)^{1/2+\varepsilon}), $$ for all $\varepsilon > 0$.

math.NT↗

arXiv2020

The forbidden region for random zeros: appearance of quadrature domains

Our main discovery is a surprising interplay between quadrature domains on the one hand, and the zero process of the Gaussian Entire Function (GEF) on the other. Specifically, consider the GEF conditioned on the rare hole event that there are no zeros in a given large Jordan domain. We show that in the natural scaling limit, a quadrature domain enclosing the hole emerges as a forbidden region, where the zero density vanishes. Moreover, we give a description of those holes for which the forbidden region is a disk. The connecting link between random zeros and potential theory is supplied by a constrained extremal problem for the Zeitouni-Zelditch functional. To solve this problem, we recast it in terms of a seemingly novel obstacle problem, where the solution is forced to be harmonic inside the hole.

math.AP↗

arXiv2020

Fluctuations for Zeros of Gaussian Taylor Series

We study fluctuations in the number of zeros of random analytic functions given by a Taylor series whose coefficients are independent complex Gaussians. When the functions are entire, we find sharp bounds for the asymptotic growth rate of the variance of the number of zeros in large disks centered at the origin. To obtain a result that holds under no assumptions on the variance of the Taylor coefficients we employ the Wiman-Valiron theory. We demonstrate the sharpness of our bounds by studying well-behaved covariance kernels, which we call admissible (after Hayman).

math.PR↗

arXiv2020

Gaussian analytic functions of bounded mean oscillation

We consider random analytic functions given by a Taylor series with independent, centered complex Gaussian coefficients. We give a new sufficient condition for such a function to have bounded mean oscillations. Under a mild regularity assumption this condition is optimal. Using a theorem of Holland and Walsh, we give as a corollary a new bound for the norm of a random Gaussian Hankel matrix. Finally, we construct some exceptional Gaussian analytic functions which in particular disprove the conjecture that a random analytic function with bounded mean oscillations always has vanishing mean oscillations.

math.CV↗

arXiv2019

Rigidity for zero sets of Gaussian entire functions

In this note we consider a certain class of Gaussian entire functions, characterized by some asymptotic properties of their covariance kernels, which we call admissible (as defined by Hayman). A notable example is the Gaussian Entire Function, whose zero set is well-known to be invariant with respect to the isometries of the complex plane. We explore the rigidity of the zero set of Gaussian Taylor series, a phenomenon discovered not long ago by Ghosh and Peres for the Gaussian Entire Function. In particular, we find that for a function of infinite order of growth, and having an admissible kernel, the zero set is "fully rigid". This means that if we know the location of the zeros in the complement of any given compact set, then the number and location of the zeros inside that set can be determined uniquely. As far as we are aware, this is the first explicit construction in a natural class of random point processes with full rigidity.s with full rigidity.

math.PR↗

arXiv2018

Gaussian complex zeros on the hole event: the emergence of a forbidden region

We consider the Gaussian Entire Function (GEF) whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the kth coefficient is 1/k!. This random Taylor series is distinguished by the invariance of its zero set with respect to the isometries of the complex plane. We show that the law of the zero set, conditioned on the GEF having no zeros in a disk of radius r, and properly normalized, converges to an explicit limiting Radon measure in the plane, as r goes to infinity. 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.

math.CV↗

arXiv2018

Point processes, hole events, and large deviations: random complex zeros and Coulomb gases

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.

math.PR↗

arXiv2017

Hole probability for zeroes of Gaussian Taylor series with finite radii of convergence

We study a family of random Taylor series $$F(z) = \sum_{n\ge 0} ζ_n a_n z^n$$ with radius of convergence almost surely $1$ and independent identically distributed complex Gaussian coefficients $(ζ_n)$; these Taylor series are distinguished by the invariance of their zero sets with respect to isometries of the unit disk. We find reasonably tight upper and lower bounds on the probability that $F$ does not vanish in the disk $\{|z|\le r\}$ as $r\uparrow 1$. Our bounds take different forms according to whether the non-random coefficients $(a_n)$ grow, decay or remain of the same order. The results apply more generally to a class of Gaussian Taylor series whose coefficients $(a_n)$ display power-law behavior.

math.CV↗

arXiv2016

Asymptotics of The Hole Probability for Zeros of Random Entire Functions

We study the hole probability of Gaussian random entire functions. More specifically, we work with the flat model (the zero set of this function has a distribution which is invariant with respect to the plane isometries). A hole is the event where the function has no zeros in a disc of radius r. We show that the logarithm of the probability of the hole event decays asymptotically like -1/4 * e^2 * r^4 + o(r^4). We also study the behavior of the hole probability with other types of random coefficients.

math.CV↗

arXiv2016

Entire functions of exponential type represented by pseudo-random and random Taylor series

We study the influence of the multipliers $ξ(n)$ on the angular distribution of zeroes of the Taylor series \[ F_ξ(z) = \sum_{n\ge 0} ξ(n) \frac{z^n}{n!}\,. \] We show that the distribution of zeroes of $ F_ξ$ is governed by certain autocorrelations of the sequence $ ξ$. Using this guiding principle, we consider several examples of random and pseudo-random sequences $ξ$ and, in particular, answer some questions posed by Chen and Littlewood in 1967. As a by-product we show that if $ξ$ is a stationary random integer-valued sequence, then either it is periodic, or its spectral measure has no gaps in its support. The same conclusion is true if $ξ$ is a complex-valued stationary ergodic sequence that takes values from a uniformly discrete set.

math.PR↗

arXiv2015

Distribution of zeroes of Rademacher Taylor series

We find the asymptotics of the counting function of zeroes of random entire functions represented by Rademacher Taylor series. We also give the asymptotics of the weighted counting function, which takes into account the arguments of zeroes. These results answer several questions left open after the pioneering work of Littlewood and Offord of 1948. The proofs are based on our recent result on the logarithmic integrability of Rademacher Fourier series.

math.CV↗

arXiv2014

Topics in the Value Distribution of Random Analytic Functions

This thesis is concerned with the behavior of random analytic functions. In particular, we are interested in the value distribution of Taylor series with independent random coefficients. We begin with a study of the properties of Fourier series with random signs. The main result states that the logarithm of such series is integrable (to any power). Using this result, we answer an old question of J.-P. Kahane, concerning the range of random Taylor series in the unit disk. In addition, we prove a law of large numbers for the number of zeros of entire functions given by Taylor series with random signs. Then we examine some 'rare' events related to the zero set of Gaussian entire functions (given by a Taylor series). In particular, we are interested in the 'hole' event, where the function has no zeros inside a large disk, centered at the origin. We give precise logarithmic asymptotics for the probability of this event, as the radius tends to infinite, depending on the variance of the coefficients of the series. It should be mentioned that our results do not assume any 'regularity' conditions.

math.CV↗