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Oren Yakir

Publications and source records attributed to Oren Yakir.

18 recordsLinked to original sources
arXiv2026

Dragon curves in Littlewood roots

A Littlewood polynomial is a polynomial whose coefficients lie in $\{- 1, +1\}$. While the majority of roots of a Littlewood polynomial of large degree are near the unit circle, numerical experiments suggest that when plotting the roots of \emph{all} Littlewood polynomials of a given large degree, striking fractal structures appear away from the unit circle. These fractals resemble the attractor of a certain iterated function system and are known as \emph{dragon curves}. In this note, we provide a rigorous explanation of this phenomenon, along with an analysis of a random variant, saying that such fractal behavior is typical.

math.CA↗
arXiv2026

A simple proof of local universality for roots of Kac polynomials

Let $f_n$ be a random polynomial of degree $n$ with i.i.d. mean-zero and finite variance random coefficients. It is well known that the roots of $f_n$ cluster uniformly around the unit circle as $n$ grows large. We give a simple and self-contained proof of local universality for the correlation functions of the roots at the microscopic scale $1/n$ around a fixed point on the circle. While previous proofs of local universality were focused on studying the logarithmic potential of $f_n$, we instead directly compare the scaled random polynomial to a limiting Gaussian analytic function, and establish convergence of correlations via a soft argument, using only basic complex analysis and an anti-concentration bound of Esseen.

math.PR↗
arXiv2026

Gaussian limits of lattice Higgs models with complete symmetry breaking

Given any compact connected matrix Lie group $G$ and any lattice dimension $d\ge 2$, we construct a massive Gaussian scaling limit for the $G$-valued lattice Yang-Mills-Higgs theory in the "complete breakdown of symmetry" regime. This limit arises as the lattice spacing tends to zero and the (inverse) gauge coupling constant tends to infinity sufficiently fast, causing the theory to "abelianize" and yield a Gaussian limit. This complements a recent work by Chatterjee (arXiv:2401.10507), which obtained a similar scaling limit in the special case $G= SU(2)$.

math.PR↗
arXiv2026

Optimal factor matchings for point processes on non-amenable unimodular graphs

Consider a unit-intensity point process $Π$ on the vertex set $V$ of a transitive non-amenable unimodular graph. We study invariant matchings between $Π$ and $V$ having small typical matching distances. When $Π$ is either a Poisson process or i.i.d. perturbations of the vertex set, we determine the optimal matching distance and show that it can be attained by a factor matching scheme (that is, a deterministic and equivariant function of $Π$).

math.PR↗
arXiv2025

Correlation decay for U(1) lattice Higgs theory: the case of small mass

We study the lattice Yang-Mills-Higgs model with inverse gauge coupling $β>0$ and Higgs length $α>0$, in the ``complete breakdown of symmetry" regime. For lattice dimension $d\ge 2$ and (abelian) gauge group U(1), we prove that for any $m>0$, if $α= mβ$ and $β$ is large enough, the model exhibits exponential decay of correlations. This extends the classical result of Osterwalder and Seiler (1978), who required in addition that $m$ be sufficiently large. Our result also verifies a phase diagram from the physics literature, predicted by Fradkin and Shenker (1979). The proof is based on the Glimm-Jaffe-Spencer cluster expansion around a massive Gaussian field, following the approach of Balaban et al. (1984).

math.PR↗
arXiv2025

Optimal matchings of randomly perturbed lattices

Consider a point process in Euclidean space obtained by perturbing the integer lattice with independent and identically distributed random vectors. Under mild assumptions on the law of the perturbations, we construct a translation-invariant perfect matching between this point process and the lattice, such that the matching distance has the same tail behavior as the hole probability of the point process, which is a natural lower bound.

math.PR↗
arXiv2025

Law of large numbers for the discriminant of random polynomials

Let $f_n$ be a random polynomial of degree $n$, whose coefficients are independent and identically distributed random variables with mean-zero and variance one. Let $Δ(f_n)$ denote the discriminant of $f_n$, that is $Δ(f_n) = A^{2n-2}\prod_{i < j} (α_j - α_i)^2$ where $A$ is the leading coefficient of $f_n$ and $α_1,\ldotsα_n$ are its roots. We prove that with high probability $$|Δ(f_n)| = n^{2n} e^{-{\sf D}_\ast n(1+o(1))}$$ as $n\to \infty$, for some explicit universal constant ${\sf D}_\ast>0$. A key step in the proof is an analytic representation for the logarithm of the discriminant, which captures both the distributional reciprocal symmetry of the random roots and the cancellations this symmetry induces.

math.PR↗
arXiv2025

Limit law for root separation in random polynomials

Let $f_n$ be a random polynomial of degree $n\ge 2$ whose coefficients are independent and identically distributed random variables. We study the separation distances between roots of $f_n$ and prove that the set of these distances, normalized by $n^{-5/4}$, converges in distribution as $n\to \infty$ to a non-homogeneous Poisson point process. As a corollary, we deduce that the minimal separation distance between roots of $f_n$, normalized by $n^{-5/4}$ has a non-trivial limit law. In the course of the proof, we establish a related result which may be of independent interest: a Taylor series with random i.i.d. coefficients almost-surely does not have a double zero anywhere other than the origin.

math.PR↗
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

Fluctuations in the logarithmic energy for zeros of random polynomials on the sphere

Smale's Seventh Problem asks for an efficient algorithm to generate a configuration of $n$ points on the sphere that nearly minimizes the logarithmic energy. As a candidate starting configuration for this problem, Armentano, Beltrán and Shub considered the set of points given by the stereographic projection of the roots of the random elliptic polynomial of degree $n$ and computed the expected logarithmic energy. We study the fluctuations of the logarithmic energy associated to this random configuration and prove a central limit theorem. Our approach shows that all cumulants of the logarithmic energy are asymptotically linear in $n$, and hence the energy is well-concentrated on the scale of $\sqrt{n}$.

math.PR↗
arXiv2023

The random Weierstrass zeta function I. Existence, uniqueness, fluctuations

We describe a construction of random meromorphic functions with prescribed simple poles with unit residues at a given stationary point process. We characterize those stationary processes with finite second moment for which, after subtracting the mean, the random function becomes stationary. These random meromorphic functions can be viewed as random analogues of the Weierstrass zeta function from the theory of elliptic functions, or equivalently as electric fields generated by an infinite random distribution of point charges.

math.PR↗
arXiv2023

The random Weierstrass zeta function II. Fluctuations of the electric flux through rectifiable curves

Consider a random planar point process whose law is invariant under planar isometries. We think of the process as a random distribution of point charges and consider the electric field generated by the charge distribution. In Part I of this work, we found a condition on the spectral side which characterizes when the field itself is invariant with a well-defined second-order structure. Here, we fix a process with an invariant field, and study the fluctuations of the flux through large arcs and curves in the plane. Under suitable conditions on the process and on the curve, denoted $Γ$, we show that the asymptotic variance of the flux through $R\,Γ$ grows like $R$ times the signed length of $Γ$. As a corollary, we find that the charge fluctuations in a dilated Jordan domain is asymptotic with the perimeter, provided only that the boundary is rectifiable. The proof is based on the asymptotic analysis of a closely related quantity (the complex electric action of the field along a curve). A decisive role in the analysis is played by a signed version of the classical Ahlfors regularity condition.

math.PR↗
arXiv2022

Fluctuations of linear statistics for Gaussian perturbations of the lattice $\mathbb{Z}^d$

We study the point process $W$ in $\mathbb{R}^d$ obtained by adding an independent Gaussian vector to each point in $\mathbb{Z}^d$. Our main concern is the asymptotic size of fluctuations of the linear statistics in the large volume limit, defined as \[ N(h,R) = \sum_{w\in W} h\left(\frac{w}{R}\right), \] where $h\in \left(L^1\cap L^2\right)(\mathbb{R}^d)$ is a test function and $R\to \infty$. We will also consider the stationary counter-part of the process $W$, obtained by adding to all perturbations a random vector which is uniformly distributed on $[0,1]^d$ and is independent of all the Gaussians. We focus on two main examples of interest, when the test function $h$ is either smooth or is an indicator function of a convex set with a smooth boundary whose curvature does not vanish.

math.PR↗
arXiv2022

Universality of Poisson limits for moduli of roots of Kac polynomials

We give a new proof of a recent resolution by Michelen and Sahasrabudhe of a conjecture of Shepp and Vanderbei that the moduli of roots of Gaussian Kac polynomials of degree $n$, centered at $1$ and rescaled by $n^2$, should form a Poisson point process. We use this new approach to verify a conjecture of Michelen and Sahasrabudhe that the Poisson statistics are in fact universal.

math.PR↗
arXiv2020

Recovering the lattice from its random perturbations

Given a $d$-dimensional Euclidean lattice we consider the random set obtained by adding an independent Gaussian vector to each of the lattice points. In this note we provide a simple procedure that recovers the lattice from a single realization of the random set.

math.PR↗
arXiv2019

Boundary Effects in the Discrete Bass Model

To study the effect of boundaries on diffusion of new products, we introduce two novel analytic tools: The indifference principle, which enables us to explicitly compute the aggregate diffusion on various networks, and the dominance principle, which enables us to rank the diffusion on different networks. Using these principles, we prove our main result that on a finite line, one-sided diffusion (i.e., when each consumer can only be influenced by her left neighbor) is strictly slower than two-sided diffusion (i.e., when each consumer can be influenced by her left and right neighbor). This is different from the periodic case of diffusion on a circle, where one-sided and two-sided diffusion are identical. We observe numerically similar results in higher dimensions.

cs.SI↗