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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.16356 |
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| _version_ | 1866910015883837440 |
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| author | Monthus, Cecile |
| author_facet | Monthus, Cecile |
| contents | The statistical properties of non-linear observables of the fractal Gaussian field $ϕ(\vec x)$ of negative Hurst exponent $H<0$ in dimension $d$ are revisited with a focus on spatial-averaging observables and on the properties of the finite parts $ϕ_n(\vec x)$ of the ill-defined composite operators $ϕ^n(\vec x) $. For the special case $n=2$ of quadratic observables, explicit results include the cumulants of arbitrary order, the Lévy-Khintchine formula for the characteristic function and the anomalous large deviations properties. The case of observables of arbitrary order $n>2$ is analyzed via the Wiener-Ito chaos-expansion for functionals of the white noise: the multiple stochastic Ito integrals are useful to identify the finite parts $ϕ_n(\vec x)$ of the ill-defined composite operators $ϕ^n(\vec x) $ and to compute their correlations involving the Hurst exponents $H_n=nH$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16356 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Statistical properties of non-linear observables of fractal Gaussian fields with a focus on spatial-averaging observables and on composite operators Monthus, Cecile Statistical Mechanics Probability The statistical properties of non-linear observables of the fractal Gaussian field $ϕ(\vec x)$ of negative Hurst exponent $H<0$ in dimension $d$ are revisited with a focus on spatial-averaging observables and on the properties of the finite parts $ϕ_n(\vec x)$ of the ill-defined composite operators $ϕ^n(\vec x) $. For the special case $n=2$ of quadratic observables, explicit results include the cumulants of arbitrary order, the Lévy-Khintchine formula for the characteristic function and the anomalous large deviations properties. The case of observables of arbitrary order $n>2$ is analyzed via the Wiener-Ito chaos-expansion for functionals of the white noise: the multiple stochastic Ito integrals are useful to identify the finite parts $ϕ_n(\vec x)$ of the ill-defined composite operators $ϕ^n(\vec x) $ and to compute their correlations involving the Hurst exponents $H_n=nH$. |
| title | Statistical properties of non-linear observables of fractal Gaussian fields with a focus on spatial-averaging observables and on composite operators |
| topic | Statistical Mechanics Probability |
| url | https://arxiv.org/abs/2505.16356 |