arXiv subjects

Akshay Hegde

Publications and source records attributed to Akshay Hegde.

3 recordsLinked to original sources

arXiv2026

Excursion Fluctuations and Spectral Universality in Gaussian Fields

We study the large-scale spatial fluctuations of excursion volumes for a class of smooth stationary Gaussian fields. In the case of Berry's random wave model in dimension $d \geq 2$, we show that the spatial fluctuations for fixed $u>0$ converge to the fractional Gaussian field $(-\Delta)^{-1/4}W$ in the space of tempered distributions $\mathcal S'(\mathbb{R}^d)$, where $W$ is the $d$-dimensional Gaussian white noise. This explains the long-range correlations in the apparent filament structure of the Random Plane Wave model. For a class of smooth planar Gaussian fields whose spectral density has a power-law singularity at the origin, we prove convergence to fractional Gaussian fields with an index determined by the singularity exponent. More generally, the results illustrate that, for stationary random measures, large-scale spatial fluctuations are determined by the behaviour of the spectral measure density exponent near zero.

math.PR↗

arXiv2026

High local maxima of stationary smooth Gaussian fields

Consider the point process (in $\mathbb{R}^d$) of local maxima of smooth Gaussian fields, with sufficient decay of correlation at infinity, above a level $u$. We show that this point process, rescaled appropriately, converges weakly to a Poisson point process in the limit $u \to \infty$. Our proof relies on the classical observation that simple point processes are characterised by avoidance probabilities (i.e. $\mathbb{P}(\eta(B)=0)$ for a point process $\eta$ and Borel set $B$). Then we approximate avoidance probability with the excursion probability, where the latter is well studied. Second main result is a quantified version of the Poisson convergence of high local maxima of the Bargmann-Fock field in $\mathbb{R}^2$. We prove that, for Bargmann-Fock field in two dimensions, the total variation distance between a Poisson random variable and the number of local maxima of the field above a threshold $u$ in an $R \times R$ box in $\mathbb{R}^2$ decays like $\exp(- \beta u^2)$, for some fixed $\beta >0$. As an immediate consequence, when the level $u$ is a function of $R$ such that $u(R) \to \infty$ and $u(R)/ \sqrt{\log R} \to 0$ as $R \to \infty$, we have a quantitative central limit theorem for the number of high local maxima. The proof is based on the Chen-Stein method for quantitative Poisson approximation. We produce a close coupling of a stationary smooth field and its Palm version, which might be of independent interest.

math.PR↗

arXiv2023

On convergence of volume of level sets of stationary smooth Gaussian fields

We prove convergence of Hausdorff measure of level sets of smooth Gaussian fields when the levels converge. Given two coupled stationary fields $f_1, f_2$ , we estimate the difference of Hausdorff measure of level sets in expectation, in terms of $C^2$-fluctuations of the field $F=f_1-f_2$. The main idea in the proof is to represent difference in volume as an integral of mean curvature using the divergence theorem. This approach is different from using Kac-Rice type formula as main tool in the analysis.

math.PR↗