Related papers: Counting open nodal lines of random waves on plana…

We investigate the number of nodal intersections of random Gaussian Laplace eigenfunctions on the standard two-dimensional flat torus ("arithmetic random waves") with a fixed real-analytic reference curve with nonvanishing curvature. The…

Assuming that a plane partition of the positive integer $n$ is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics…

In a closed manifold of positive dimension $n$, we estimate the expected volume and Euler characteristic for random submanifolds of codimension $r\in \{1,...,n\}$ in two different settings. On one hand, we consider a closed Riemannian…

In this work, we investigate the problem of finite time blow up as well as the upper bound estimates of lifespan for solutions to small-amplitude semilinear wave equations with time dependent damping and potential, and mixed nonlinearities…

Nodes are randomly distributed within an annulus (and then a shell) to form a point pattern of communication terminals which are linked stochastically according to the Rayleigh fading of radio-frequency data signals. We then present…

We prove two conjectures from [M. R. Douglas, B. Shiffman and S. Zelditch, Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics. J. Differential Geom. 72 (2006), no. 3, 381-427] concerning the expected number of…

We study the asymptotics of solutions to a particular class of systems of linear wave equations, namely, of silent equations. We obtain asymptotic estimates of all orders for the solutions, and show that solutions are uniquely determined by…

In 2017, Benatar and Maffucci arXiv:1708.07015 established an asymptotic law for the variance of the nodal surface of arithmetic random waves on the 3-torus in the high-energy limit. In a subsequent work, Cammarota arXiv:1708.07679 proved a…

In this paper, we consider the sublinear expectation on bounded random variables. With the notion of uncorrelatedness for random variables under the sublinear expectation, a weak law of large numbers is obtained. With the notion of…

I compute several terms of the asymptotic expansion of the number of connected labelled graphs with n nodes and m edges, for small k=m-n.

Linear statistics of random zero sets are integrals of smooth differential forms over the zero set and as such are smooth analogues of the volume of the random zero set inside a fixed domain. We derive an asymptotic expansion for the…

This paper is devoted to the study of the large scale geometry of the excursion set and nodal set of a planar smooth Gaussian field at criticality $\ell=\ell_c=0$. We prove that there exists $s_1>1$ such that with high probability,…

In this paper, we investigate the variance of the nodal length for band-limited spherical random waves. When the frequency window includes a number of eigenfunctions that grows linearly, the variance of the nodal length is linear with…

We show that the number $g_n$ of labelled series-parallel graphs on $n$ vertices is asymptotically $g_n \sim g\cdot n^{-5/2} \gamma^n n!$, where $\gamma$ and $g$ are explicit computable constants. We show that the number of edges in random…

A well-known conjecture asserts that, for any given positive real number $\lambda$ and nonnegative integer $m$, the proportion of positive integers $n \le x$ for which the interval $(n,n + \lambda\log n]$ contains exactly $m$ primes is…

We study the almost periods of the eigenmodes of flat planar manifolds in the high energy limit. We prove in particular that the Gaussian Arithmetic Random Waves replicate almost identically at a scale at most ${\ell}$n := n -- 1 2 exp…

We study free surface water waves in a 2-D symmetric triangular channel with sides that have a 45o slope. We develop models for small amplitude nonlinear waves, extending earlier studies that have considered the linearized problem. We see…

This is an addendum to the article of Qi Zhong cited above [arXiv:0705.2000]. It outlines how to apply the main result of that article to calculate the asymptotics of the expected energy of zeros of random polynomials on the Riemann sphere…

As a first step toward a fully two-dimensional asymptotic theory for the bifurcation of solitons from infinitesimal continuous waves, an analytical theory is presented for line solitons, whose envelope varies only along one direction, in…

We pursue the analysis made in [1] on the arithmetic area enclosed by m closed Brownian paths. We pay a particular attention to the random variable S{n1,n2, ...,n} (m) which is the arithmetic area of the set of points, also called winding…