arXiv subjects

Riccardo W. Maffucci

Publications and source records attributed to Riccardo W. Maffucci.

25 records · Page 2Linked to original sources

arXiv2021

Asymptotic distribution of Nodal Intersections for ARW against a Surface

We investigate Gaussian Laplacian eigenfunctions (Arithmetic Random Waves) on the three-dimensional standard flat torus, in particular the asymptotic distribution of the nodal intersection length against a fixed regular reference surface. Expectation and variance have been addressed by Maffucci (2019) who found that the expected length is proportional to the square root of the eigenvalue times the area of the surface, while the asymptotic variance only depends on the geometry of the surface, the projected lattice points being equidistributed on the two-dimensional unit sphere in the high-energy limit. He also noticed that there are ``special'' surfaces, so-called static, for which the variance is of smaller order; however he did not prescribe the precise asymptotic law in this case. In this paper, we study second order fluctuations of the nodal intersection length. Our first main result is a Central Limit Theorem for ``generic'' surfaces, while for static ones, a sphere or a hemisphere e.g., our main results are a non-Central Limit Theorem and a precise asymptotic law for the variance of the nodal intersection length, conditioned on the existence of so-called well-separated sequences of Laplacian eigenvalues. It turns out that, in this regime, the nodal area investigated by Cammarota (2019) is asymptotically fully correlated with the length of the nodal intersections against any sphere. The main ingredients for our proofs are the Kac-Rice formula for moments, the chaotic decomposition for square integrable functionals of Gaussian fields, and some arithmetic estimates that may be of independent interest.

math.PR↗

arXiv2021

Self-dual polyhedra of given degree sequence

Given vertex valencies admissible for a self-dual polyhedral graph, we describe an algorithm to explicitly construct such a polyhedron. Inputting in the algorithm permutations of the degree sequence can give rise to non-isomorphic graphs. As an application, we find as a function of $n\geq 3$ the minimal number of vertices for a self-dual polyhedron with at least one vertex of degree $i$ for each $3\leq i\leq n$, and construct such polyhedra. Moreover, we find a construction for non-self-dual polyhedral graphs of minimal order with at least one vertex of degree $i$ and at least one $i$-gonal face for each $3\leq i\leq n$.

math.CO↗

arXiv2021

On the limiting behaviour of arithmetic toral eigenfunctions

We consider a wide class of families $(F_m)_{m\in\mathbb{N}}$ of Gaussian fields on $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$ defined by \[F_m:x\mapsto \frac{1}{\sqrt{|Λ_m|}}\sum_{λ\inΛ_m}ζ_λe^{2πi\langle λ,x\rangle}\] where the $ζ_λ$'s are independent std. normals and $Λ_m$ is the set of solutions $λ\in\mathbb{Z}^d$ to $p(λ)=m$ for a fixed elliptic polynomial $p$ with integer coefficients. The case $p(x)=x_1^2+\dots+x_d^2$ is a random Laplace eigenfunction whose law is sometimes called the $\textit{arithmetic random wave}$, studied in the past by many authors. In contrast, we consider three classes of polynomials $p$: a certain family of positive definite quadratic forms in two variables, all positive definite quadratic forms in three variables except multiples of $x_1^2+x_2^2+x_3^2$, and a wide family of polynomials in many variables. For these classes of polynomials, we study the $(d-1)$-dimensional volume $\mathcal{V}_m$ of the zero set of $F_m$. We compute the asymptotics, as $m\to+\infty$ along certain sequences of integers, of the expectation and variance of $\mathcal{V}_m$. Moreover, we prove that in the same limit, $\frac{\mathcal{V}_m-\mathbb{E}[\mathcal{V}_m]}{\sqrt{\text{Var}(\mathcal{V}_m)}}$ converges to a std. normal. As in previous works, one reduces the problem of these asymptotics to the study of certain arithmetic properties of the sets of solutions to $p(λ)=m$. We need to study the number of such solutions for fixed $m$, the number of quadruples of solutions $(λ,μ,ν,ι)$ satisfying $λ+μ+ν+ι=0$, ($4$-correlations), and the rate of convergence of the counting measure of $Λ_m$ towards a certain limiting measure on the hypersurface $\{p(x)=1\}$. To this end, we use prior results on this topic but also prove a new estimate on correlations, of independent interest.

math.PR↗

arXiv2021

Constructing certain families of $\mathbf{3}$-polytopal graphs

Let $n\geq 3$ and $r_n$ be a $3$-polytopal graph such that for every $3\leq i\leq n$, $r_n$ has at least one vertex of degree $i$. We find the minimal vertex count for $r_n$. We then describe an algorithm to construct the graphs $r_n$. A dual statement may be formulated for faces of $3$-polytopes. The ideas behind the algorithm generalise readily to solve related problems. Moreover, given a $3$-polytope $t_l$ comprising a vertex of degree $i$ for all $3\leq i\leq l$, $l$ fixed, we define an algorithm to output for $n>l$ a $3$-polytope $t_n$ comprising a vertex of degree $i$, for all $3\leq i\leq n$, and such that the initial $t_l$ is a subgraph of $t_n$. The vertex count of $t_n$ is asymptotically optimal, in the sense that it matches the aforementioned minimal vertex count up to order of magnitude, as $n$ gets large. In fact, we only lose a small quantity on the coefficient of the second highest term, and this quantity may be taken as small as we please, with the tradeoff of first constructing an accordingly large auxiliary graph.

math.CO↗

arXiv2019

Coupling of stationary fields with application to arithmetic waves

In this paper we obtain a range of quantitative results of the following type: given two centered Gaussian fields with close covariance kernels we construct a coupling such that the fields are uniformly close on some compact with probability very close to one. As an application, we show that it is possible to couple arithmetic random waves so that they converge locally uniformly to the random plane wave and estimate the rate of convergence.

math.PR↗

arXiv2019

Intermediate and small scale limiting theorems for random fields

In this paper we study the nodal lines of random eigenfunctions of the Laplacian on the torus, the so called 'arithmetic waves'. To be more precise, we study the number of intersections of the nodal line with a straight interval in a given direction. We are interested in how this number depends on the length and direction of the interval and the distribution of spectral measure of the random wave. We analyse the second factorial moment in the short interval regime and the persistence probability in the long interval regime. We also study relations between the Cilleruelo and Cilleruelo-type fields. We give an explicit coupling between these fields which on mesoscopic scales preserves the structure of the nodal sets with probability close to one.

math.PR↗

arXiv2017

Random waves on $\mathbb{T}^3$: nodal area variance and lattice point correlations

We consider the ensemble of random Gaussian Laplace eigenfunctions on $\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$ (`$3d$ arithmetic random waves'), and study the distribution of their nodal surface area. The expected area is proportional to the square root of the eigenvalue, or `energy', of the eigenfunction. We show that the nodal area variance obeys an asymptotic law. The resulting asymptotic formula is closely related to the angular distribution and correlations of lattice points lying on spheres.

math.NT↗