Igor Wigman
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View article: The giant component of excursion sets of spherical Gaussian ensembles: existence, uniqueness, and volume concentration
The giant component of excursion sets of spherical Gaussian ensembles: existence, uniqueness, and volume concentration Open
We establish the existence and uniqueness of a well-concentrated giant component in the supercritical excursion sets of three important ensembles of spherical Gaussian random fields: Kostlan’s ensemble, band-limited ensembles, and the rand…
View article: Almost Sure GOE Fluctuations of Energy Levels for Hyperbolic Surfaces of High Genus
Almost Sure GOE Fluctuations of Energy Levels for Hyperbolic Surfaces of High Genus Open
We study the variance of a linear statistic of the Laplace eigenvalues on a hyperbolic surface, when the surface varies over the moduli space of all surfaces of fixed genus, sampled at random according to the Weil–Petersson measure. The en…
View article: Around the Gauss circle problem: Hardy's conjecture and the distribution of lattice points near circles
Around the Gauss circle problem: Hardy's conjecture and the distribution of lattice points near circles Open
Hardy conjectured that the error term arising from approximating the number of lattice points lying in a radius‐ disc by its area is . One source of support for this conjecture is a folklore heuristic that uses i.i.d. random variables to m…
View article: The giant component of excursion sets of spherical Gaussian ensembles: existence, uniqueness, and volume concentration
The giant component of excursion sets of spherical Gaussian ensembles: existence, uniqueness, and volume concentration Open
We establish the existence and uniqueness of a well-concentrated giant component in the supercritical excursion sets of three important ensembles of spherical Gaussian random fields: Kostlan's ensemble, band-limited ensembles, and the rand…
View article: On the nodal structures of random fields: a decade of results
On the nodal structures of random fields: a decade of results Open
We survey a decade worth of work pertaining to the nodal structures of random fields, with emphasis on the transformative techniques that shaped the field.
View article: Eigenvalue clusters for the hemisphere Laplacian with variable Robin condition
Eigenvalue clusters for the hemisphere Laplacian with variable Robin condition Open
We study the eigenvalue clusters of the Robin Laplacian on the 2-dimensional hemisphere with a variable Robin coefficient on the equator. The $\ell$'th cluster has $\ell+1$ eigenvalues. We determine the asymptotic density of eigenvalues in…
View article: Around the Gauss circle problem: Hardy's conjecture and the distribution of lattice points near circles
Around the Gauss circle problem: Hardy's conjecture and the distribution of lattice points near circles Open
Hardy conjectured that the error term arising from approximating the number of lattice points lying in a radius-$R$ disc by its area is $O(R^{1/2+o(1)})$. One source of support for this conjecture is a folklore heuristic that uses i.i.d. r…
View article: Almost sure GOE fluctuations of energy levels for hyperbolic surfaces of high genus
Almost sure GOE fluctuations of energy levels for hyperbolic surfaces of high genus Open
We study the variance of a linear statistic of the Laplace eigenvalues on a hyperbolic surface, when the surface varies over the moduli space of all surfaces of fixed genus, sampled at random according to the Weil-Petersson measure. The en…
View article: On the Central Limit Theorem for linear eigenvalue statistics on random surfaces of large genus
On the Central Limit Theorem for linear eigenvalue statistics on random surfaces of large genus Open
We study the fluctuations of smooth linear statistics of Laplace eigenvalues of compact hyperbolic surfaces lying in short energy windows, when averaged over the moduli space of surfaces of a given genus. The average is taken with respect …
View article: The expected nodal volume of non-Gaussian random band-limited functions, and their doubling index
The expected nodal volume of non-Gaussian random band-limited functions, and their doubling index Open
The asymptotic law for the expected nodal volume of random non-Gaussian monochromatic band-limited functions is determined in vast generality. Our methods combine microlocal analytic techniques and modern probability theory. A particularly…
View article: On the nodal structures of random fields -- a decade of results
On the nodal structures of random fields -- a decade of results Open
We survey a decade worth of work pertaining to the nodal structures of random fields, with emphasis on the transformative techniques that shaped the field.
View article: On the Robin spectrum for the equilateral triangle*
On the Robin spectrum for the equilateral triangle* Open
The equilateral triangle is one of the few planar domains where the Dirichlet and Neumann eigenvalue problems were explicitly determined, by Lamé in 1833, despite not admitting separation of variables. In this paper, we study the Robin spe…
View article: On the distribution of lattice points on hyperbolic circles
On the distribution of lattice points on hyperbolic circles Open
We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane $\mathbb{H}$. The angles of lattice points arising from the orbit of the modular group $PSL_{2}(\mathbb{Z})$, and lying on hyperbolic circl…
View article: The Robin problem on rectangles
The Robin problem on rectangles Open
We study the statistics and the arithmetic properties of the Robin spectrum of a rectangle. A number of results are obtained for the multiplicities in the spectrum depending on the Diophantine nature of the aspect ratio. In particular, it …
View article: The defect of toral Laplace eigenfunctions and arithmetic random waves
The defect of toral Laplace eigenfunctions and arithmetic random waves Open
We study the defect (or ‘signed area’) distribution of standard toral Laplace eigenfunctions restricted to shrinking balls of radius above the Planck scale, either for deterministic eigenfunctions averaged w.r.t. the spatial variable, or i…
View article: Expected nodal volume for non-Gaussian random band-limited functions
Expected nodal volume for non-Gaussian random band-limited functions Open
The asymptotic law for the expected nodal volume of random non-Gaussian monochromatic band-limited functions is determined in vast generality. Our methods combine microlocal analytic techniques and modern probability theory. A particularly…
View article: Nodal deficiency of random spherical harmonics in presence of boundary
Nodal deficiency of random spherical harmonics in presence of boundary Open
We consider a random Gaussian model of Laplace eigenfunctions on the hemisphere, satisfying the Dirichlet boundary conditions along the equator. For this model, we find a precise asymptotic law for the corresponding zero density functions,…
View article: Russo–Seymour–Welsh estimates for the Kostlan ensemble of random polynomials
Russo–Seymour–Welsh estimates for the Kostlan ensemble of random polynomials Open
Beginning with the predictions of Bogomolny–Schmit for the random plane wave, in recent years the deep connections between the level sets of smooth Gaussian random fields and percolation have become apparent. In classical percolation theor…
View article: On the distribution of lattice points on hyperbolic circles
On the distribution of lattice points on hyperbolic circles Open
We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane $\mathbb{H}$. The angles of lattice points arising from the orbit of the modular group $PSL_{2}(\mathbb{Z})$, and lying on hyperbolic circl…
View article: Points on nodal lines with given direction
Points on nodal lines with given direction Open
We study of the directional distribution function of nodal lines for eigenfunctions of the Laplacian on a planar domain. This quantity counts the number of points where the normal to the nodal line points in a given direction. We give uppe…
View article: No repulsion between critical points for planar Gaussian random fields
No repulsion between critical points for planar Gaussian random fields Open
We study the behaviour of the point process of critical points of isotropic stationary Gaussian fields. We compute the main term in the asymptotic expansion of the two-point correlation function near the diagonal. Our main result implies t…
View article: No repulsion between critical points for planar Gaussian random fields
No repulsion between critical points for planar Gaussian random fields Open
We study the behaviour of the point process of critical points of isotropic stationary Gaussian fields. We compute the main term in the asymptotic expansion of the two-point correlation function near the diagonal. Our main result implies t…
View article: Asymptotics for the expected number of nodal components for random lemniscates
Asymptotics for the expected number of nodal components for random lemniscates Open
We determine the true asymptotic behaviour for the expected number of connected components for a model of random lemniscates proposed recently by Lerario and Lundberg. These are defined as the subsets of the Riemann sphere, where the absol…
View article: CLT for mass distribution of toral Laplace eigenfunctions
CLT for mass distribution of toral Laplace eigenfunctions Open
We study the fine scale $L^{2}$-mass distribution of toral Laplace eigenfunctions with respect to random position, in $2$ and $3$ dimensions. In $2$d, under certain flatness assumptions on the Fourier coefficients and generic restrictions …
View article: Mean conservation of nodal volume and connectivity measures for Gaussian\n ensembles
Mean conservation of nodal volume and connectivity measures for Gaussian\n ensembles Open
We study in depth the nesting graph and volume distribution of the nodal\ndomains of a Gaussian field, which have been shown in previous works to exhibit\nasymptotic laws. A striking link is established between the asymptotic mean\nconnect…
View article: Topologies of Nodal Sets of Random Band‐Limited Functions
Topologies of Nodal Sets of Random Band‐Limited Functions Open
It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band‐limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particu…