arXiv subjects

Lior Alon

Publications and source records attributed to Lior Alon.

16 recordsLinked to original sources

arXiv2026

Wave Transport in Fourier Quasicrystals Revealed by Water Waves

Fourier quasicrystals are aperiodic structures whose diffraction spectrum consists not of a dense set of Bragg peaks, as in ordinary quasicrystals, but of isolated ones scattered across a discrete, nonperiodic set. This sparse reciprocal-space structure should leave wave transport largely undisturbed except at a few selected wavevectors. We put this prediction to the test using surface water waves scattering off a two-dimensional Fourier quasicrystal. Full-field measurements reveal three distinct transport regimes as the incident wavevector increases: transparency, selective scattering, and strong scattering. By reconstructing the structure factor from the measured wavefields, we directly relate these regimes to the underlying reciprocal-space structure. Our results establish Fourier quasicrystals as a physical platform in which wave transport can be controlled through the organization of diffraction peaks in reciprocal space.

cond-mat.soft↗

arXiv2025

Nodal Count for Orthogonally Invariant Ensembles

We investigate the nodal count of eigenvectors of random matrices interpreted as operators on signed complete graphs. Our focus is on orthogonally invariant ensembles, with particular attention to the Gaussian Orthogonal Ensemble (GOE). We establish that, as the matrix size tends to infinity, the distribution of nodal counts converges to the same limiting law as the eigenvalue distribution. In the GOE case, this limit is the semicircle law. This result refutes a conjecture, motivated by quantum chaos and quantum graphs, which predicted Gaussian behavior of the nodal count.

math-ph↗

arXiv2025

Periodic Hypersurfaces and Lee-Yang Polynomials

We study periodic measures on $\mathbb{R}^n$ whose Fourier transform is confined to a proper double cone, in the sense of Meyer's notion of lighthouse measures. Lee--Yang polynomials provide a natural family of examples: it follows from the work of Kurasov and Sarnak that the torus zero sets of such polynomials are hypersurfaces supporting directional lighthouse measures. We prove a rigidity theorem showing that, under mild assumptions, this is essentially the only possibility. Any periodic $C^{1+\epsilon}$ hypersurface supporting a directional lighthouse measure must arise as the torus zero set of an essentially Lee--Yang polynomial. The proof is based on the recent classification of one-dimensional Fourier quasicrystals and provides a geometric interpretation of this theory.

math.AG↗

arXiv2025

Smooth critical points of eigenvalues on the torus of magnetic perturbations of graphs

Motivated by the nodal distribution universality conjecture for discrete operators on graphs and by the spectral analysis of their maximal abelian covers, we consider a family of Hermitian matrices $h_{\alpha}$ obtained by varying the complex phases of individual matrix elements. This family is parametrized by a $\beta$-dimensional torus, where $\beta$ is the first Betti number of the underlying graph. The eigenvalues of each matrix are ordered, enabling us to treat the $k$-th eigenvalue $\lambda_k$ as a function on the torus. We classify the smooth critical points of $\lambda_k$, describe their structure and Morse index in terms of the support and nodal count, that is, the number of sign changes between adjacent vertices of the corresponding eigenvector. In general, the families under consideration exhibit critical submanifolds rather than isolated critical points. These critical manifolds appear frequently and cannot be removed through perturbations. We provide an algorithmic way of determining all critical submanifolds by investigating finitely many eigenvalue problems: the $2^\beta$ real symmetric matrices $h_\alpha$ in the family under consideration as well as their principal minors.

math-ph↗

arXiv2024

Higher Dimensional Fourier Quasicrystals from Lee-Yang Varieties

In this paper, we construct Fourier quasicrystals with unit masses in arbitrary dimensions. This generalizes a one-dimensional construction of Kurasov and Sarnak. To do this, we employ a class of complex algebraic varieties avoiding certain regions in $\mathbb{C}^n$, which generalize hypersurfaces defined by Lee-Yang polynomials. We show that these are Delone almost periodic sets that have at most finite intersection with every discrete periodic set.

math-ph↗

arXiv2024

Average Nodal Count and the Nodal Count Condition for Graphs

The nodal edge count of an eigenvector of the Laplacian of a graph is the number of edges on which it changes sign. This quantity extends to any real symmetric $n\times n$ matrix supported on a graph $G$ with $n$ vertices. The average nodal count, averaged over all eigenvectors of a given matrix, is known to be bounded between $\frac{n-1}{2}$ and $\frac{n-1}{2}+β(G)$, where $β(G)$ is the first Betti number of $G$ (a topological quantity), and it was believed that generically the average should be around $\frac{n-1}{2}+β(G)/2$. We prove that this is not the case: the average is bounded between $\frac{n-1}{2}+β(G)/n$ and $\frac{n-1}{2}+β(G)-β(G)/n$, and we provide graphs and matrices that attain the upper and lower bounds for any possible choice of $n$ and $β$. A natural condition on a matrix for defining the nodal count is that it has simple eigenvalues and non-vanishing eigenvectors. For any connected graph $G$, a generic real symmetric matrix supported on $G$ satisfies this nodal count condition. However, the situation for constant diagonal matrices is far more subtle. We completely characterize the graphs $G$ for which this condition is generically true, and show that if this is not the case, then any real symmetric matrix supported on $G$ with constant diagonal has a multiple eigenvalue or an eigenvector that vanishes somewhere. Finally, we discuss what can be said when this nodal count condition fails, and provide examples.

math-ph↗

arXiv2024

Nodal count for a random signing of a graph with disjoint cycles

Let $G$ be a simple, connected graph on $n$ vertices, and further assume that $G$ has disjoint cycles. Let $h$ be a real symmetric matrix supported on $G$ (for example, a discrete Schrödinger operator). The eigenvalues of $h$ are ordered increasingly, $λ_1 \le \cdots \le λ_n$, and if $ϕ$ is the eigenvector corresponding to $λ_k$, the nodal (edge) count $ν(h,k)$ is the number of edges $(rs)$ such that $ h_{rs}ϕ_{r}ϕ_{s}>0$. The nodal surplus is $σ(h,k)= ν(h,k) - (k-1)$. Let $h'$ be a random signing of $h$, that is a real symmetric matrix obtained from $h$ by changing the sign of some of its off-diagonal elements. If $h$ satisfies a certain generic condition, we show for each $k$ that the nodal surplus has a binomial distribution $σ(h',k)\sim Bin(β,\frac{1}{2})$. Part of the proof follows ideas developed by the first author together with Ram Band and Gregory Berkolaiko in a joint unpublished project studying a similar question on quantum graphs.

math-ph↗

arXiv2023

Gap distributions of Fourier quasicrystals via Lee-Yang polynomials

Recent work of Kurasov and Sarnak provides a method for constructing one-dimensional Fourier quasicrystals (FQ) from the torus zero sets of a special class of multivariate polynomials called Lee-Yang polynomials. In particular, they provided a non-periodic FQ with unit coefficients and uniformly discrete support, answering an open question posed by Meyer. Their method was later shown to generate all one-dimensional Fourier quasicrystals with $\mathbb{N}$-valued coefficients ($ \mathbb{N} $-FQ). In this paper, we characterize which Lee-Yang polynomials give rise to non-periodic $ \mathbb{N} $-FQs with unit coefficients and uniformly discrete support, and show that this property is generic among Lee-Yang polynomials. We also show that the infinite sequence of gaps between consecutive atoms of any $\mathbb{N}$-FQ has a well-defined distribution, which, under mild conditions, is absolutely continuous. This generalizes previously known results for the spectra of quantum graphs to arbitrary $\mathbb{N}$-FQs. Two extreme examples are presented: first, a sequence of $\mathbb{N}$-FQs whose gap distributions converge to a Poisson distribution. Second, a sequence of random Lee-Yang polynomials that results in random $\mathbb{N}$-FQs whose empirical gap distributions converge to that of a random unitary matrix (CUE).

math-ph↗

arXiv2023

Morse theory for discrete magnetic operators and nodal count distribution for graphs

Given a discrete Schrödinger operator $h$ on a finite connected graph $G$ of $n$ vertices, the nodal count $ϕ(h,k)$ denotes the number of edges on which the $k$-th eigenvector changes sign. A {\em signing} $h'$ of $h$ is any real symmetric matrix constructed by changing the sign of some off-diagonal entries of $h$, and its nodal count is defined according to the signing. The set of signings of $h$ lie in a naturally defined torus $\mathbb{T}_h$ of ``magnetic perturbations" of $h$. G. Berkolaiko discovered that every signing $h'$ of $h$ is a critical point of every eigenvalue $λ_k:\mathbb{T}_h \to \mathbb{R}$, with Morse index equal to the nodal surplus. We add further Morse theoretic information to this result. We show if $h_α \in \mathbb{T}_h$ is a critical point of $λ_k$ and the eigenvector vanishes at a single vertex $v$ of degree $d$, then the critical point lies in a nondegenerate critical submanifold of dimension $d+n-4$, closely related to the configuration space of a planar linkage. We compute its Morse index in terms of spectral data. The average nodal surplus distribution is the distribution of values of $ϕ(h',k)-(k-1)$, averaged over all signings $h'$ of $h$. If all critical points correspond to simple eigenvalues with nowhere-vanishing eigenvectors, then the average nodal surplus distribution is binomial. In general, we conjecture that the nodal surplus distribution converges to a Gaussian in a CLT fashion as the first Betti number of $G$ goes to infinity.

math-ph↗

arXiv2023

Every real-rooted exponential polynomial is the restriction of a Lee-Yang polynomial

A Lee-Yang polynomial $ p(z_{1},\ldots,z_{n}) $ is a polynomial that has no zeros in the polydisc $ \mathbb{D}^{n} $ and its inverse $ (\mathbb{C}\setminus\overline{\mathbb{D}})^{n} $. We show that any real-rooted exponential polynomial of the form $f(x) = \sum_{j=0}^s c_j e^{\lambda_j x}$ can be written as the restriction of a Lee-Yang polynomial to a positive line in the torus. Together with previous work by Olevskii and Ulanovskii, this implies that the Kurasov-Sarnak construction of $ \mathbb{N} $-valued Fourier quasicrystals from stable polynomials comprises every possible $ \mathbb{N} $-valued Fourier quasicrystal.

math.CV↗

arXiv2022

Generic Laplace eigenfunctions on metric graphs

It is known that up to certain pathologies, a compact metric graph with standard vertex conditions has a Baire-generic set of choices of edge lengths such that all Laplacian eigenvalues are simple and have eigenfunctions that do not vanish at the vertices. We provide a new notion of strong genericity, using subanalytic sets, that implies both Baire genericity and full Lebesgue measure. We show that the previous genericity results for metric graphs are strongly generic. In addition, we show that generically the derivative of an eigenfunction does not vanish at the vertices either. In fact, we show that generically an eigenfunction fails to satisfy any additional vertex condition. Finally, we show that any two different metric graphs with the same edge lengths do not share any non-zero eigenvalue, for a generic choice of lengths, except for a few explicit cases where the graphs have a common edge-reflection symmetry. The paper concludes by addressing three open conjectures for metric graphs that can benefit from the tools introduced in this paper.

math.CO↗

arXiv2022

Universality of nodal count distribution in large metric graphs

An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graph's non-trivial topology. This number, called the nodal surplus, is an integer between 0 and the graph's first Betti number $β$. We study the distribution of the nodal surplus values in the countably infinite set of the graph's eigenfunctions. We conjecture that this distribution converges to Gaussian for any sequence of graphs of growing $β$. We prove this conjecture for several special graph sequences and test it numerically for a variety of well-known graph families. Accurate computation of the distribution is made possible by a formula expressing the nodal surplus distribution as an integral over a high-dimensional torus.

math-ph↗

arXiv2020

Neumann Domains on Quantum Graphs

The Neumann points of an eigenfunction $f$ on a quantum (metric) graph are the interior zeros of $f'$. The Neumann domains of $f$ are the sub-graphs bounded by the Neumann points. Neumann points and Neumann domains are the counterparts of the well-studied nodal points and nodal domains. We prove some foundational results in this field: bounds on the number of Neumann points and properties of the probability distribution of this number. Two basic properties of Neumann domains are presented: the wavelength capacity and the spectral position. We state and prove bounds on those as well as key features of their probability distributions. To rigorously investigate those probabilities, we establish the notion of random variables for quantum graphs. In particular, we provide conditions for considering spectral functions of quantum graphs as random variables with respect to the natural density on $\mathbb{N}$.

math-ph↗

arXiv2020

Quantum graphs -- Generic eigenfunctions and their nodal count and Neumann count statistics

In this thesis, we study Laplacian eigenfunctions on metric graphs, also known as quantum graphs. We restrict the discussion to standard quantum graphs. These are finite connected metric graphs with functions that satisfy Neumann vertex conditions. The first goal of this thesis is the study of the nodal count problem. That is the number of points on which the $n$th eigenfunction vanishes. We provide a probabilistic setting using which we are able to define the nodal count\textquoteright s statistics. We show that the nodal count statistics admit a topological symmetry by which the first Betti number of the graph can be obtained. We revise a conjecture that predicts a universal Gaussian behavior of the nodal count statistics for large graphs and we prove it for a certain family of graphs. The second goal is to formulate and study the Neumann count, which is the number of local extrema of the $n$th eigenfunction. This counting problem is motivated by the Neumann partitions of planar domains, a novel concept in spectral geometry. We provide uniform bounds on the Neumann count and investigate its statistics. We show that the Neumann count provides complimentary geometrical information to that obtained from the nodal count. We show that for a certain family of growing tree graphs the Neumann count statistics approaches a Gaussian distribution. The third goal is a genericity result, which justifies the generality of the Neumann count discussion. To this day it was known that generically, eigenfunctions do not vanish on vertices. We generalize this result to derivatives at vertices as well. That is, generically, the derivatives of an eigenfunction on interior vertices do not vanish.

math-ph↗

arXiv2018

Neumann Domains on Graphs and Manifolds

The nodal set of a Laplacian eigenfunction forms a partition of the underlying manifold or graph. Another natural partition is based on the gradient vector field of the eigenfunction (on a manifold) or on the extremal points of the eigenfunction (on a graph). The submanifolds (or subgraphs) of this partition are called Neumann domains. This paper reviews the subject, as appears in a few recent works and points out some open questions and conjectures. The paper concerns both manifolds and metric graphs and the exposition allows for a comparison between the results obtained for each of them.

math.SP↗

arXiv2018

Nodal Statistics On Quantum Graphs

It has been suggested that the distribution of the suitably normalized number of zeros of Laplacian eigenfunctions contains information about the geometry of the underlying domain. We study this distribution (more precisely, the distribution of the "nodal surplus") for Laplacian eigenfunctions of a metric graph. The existence of the distribution is established, along with its symmetry. One consequence of the symmetry is that the graph's first Betti number can be recovered as twice the average nodal surplus of its eigenfunctions. Furthermore, for graphs with disjoint cycles it is proven that the distribution has a universal form --- it is binomial over the allowed range of values of the surplus. To prove the latter result, we introduce the notion of a local nodal surplus and study its symmetry and dependence properties, establishing that the local nodal surpluses of disjoint cycles behave like independent Bernoulli variables.

math-ph↗