A law of large numbers concerning the distribution of critical points of random Fourier series
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| Format: | Preprint |
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2024
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| author | Fu, Qiangang "Brandon'' Nicolaescu, Liviu I. |
| author_facet | Fu, Qiangang "Brandon'' Nicolaescu, Liviu I. |
| contents | On the flat torus $\mathbb{T}^m=\mathbb{R}^m/\mathbb{Z}^m$ with angular coordinates $\vecθ$ we consider the random function $F_R=\mathfrak{a}\big(\, R^{-1} \sqrtΔ\,\big) W$, where $R>0$, $Δ$ is the Laplacian on this flat torus, $\mathfrak{a}$ is an even Schwartz function on $\mathbb{R}$ such that $\mathfrak{a}(0)>0$ and $W$ is the Gaussian white noise on $\mathbb{T}^m$ viewed as a random generalized function. For any $f\in C(\mathbb{T}^m)$ we set \[ Z_R(f):=\sum_{\nabla F_R(\vecθ)=0} f(\vecθ) \] We prove that if the support of $f$ is contained in a geodesic ball of $\mathbb{T}^m$, then the variance of $Z_R(f)$ is asymptotic to $const\times R^{m}$ as $R\to\infty$. We use this to prove that if $m\geq 2$, then as $N\to\infty$ the random measures $N^{-m}Z_N(-)$ converge a.s. to an explicit multiple of the volume measure on the flat torus. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_07690 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A law of large numbers concerning the distribution of critical points of random Fourier series Fu, Qiangang "Brandon'' Nicolaescu, Liviu I. Probability 60D05, 60F15, 60F17, On the flat torus $\mathbb{T}^m=\mathbb{R}^m/\mathbb{Z}^m$ with angular coordinates $\vecθ$ we consider the random function $F_R=\mathfrak{a}\big(\, R^{-1} \sqrtΔ\,\big) W$, where $R>0$, $Δ$ is the Laplacian on this flat torus, $\mathfrak{a}$ is an even Schwartz function on $\mathbb{R}$ such that $\mathfrak{a}(0)>0$ and $W$ is the Gaussian white noise on $\mathbb{T}^m$ viewed as a random generalized function. For any $f\in C(\mathbb{T}^m)$ we set \[ Z_R(f):=\sum_{\nabla F_R(\vecθ)=0} f(\vecθ) \] We prove that if the support of $f$ is contained in a geodesic ball of $\mathbb{T}^m$, then the variance of $Z_R(f)$ is asymptotic to $const\times R^{m}$ as $R\to\infty$. We use this to prove that if $m\geq 2$, then as $N\to\infty$ the random measures $N^{-m}Z_N(-)$ converge a.s. to an explicit multiple of the volume measure on the flat torus. |
| title | A law of large numbers concerning the distribution of critical points of random Fourier series |
| topic | Probability 60D05, 60F15, 60F17, |
| url | https://arxiv.org/abs/2412.07690 |