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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2604.25078 |
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| _version_ | 1866910172773875712 |
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| author | Forrester, Peter J. |
| author_facet | Forrester, Peter J. |
| contents | The Riesz gas in one-dimension consists of particles interacting via a pair potential, ${\rm sgn}(s) |x - x'|^{-s}$, $s \ne 0$ and $-\log | x - x'|$ for $s=0$. In the infinite density limit, with the particle support the interval $[-1,1]$, we apply a functional derivative method due to Beenakker to compute the covariance of two smooth linear statistics for the Riesz gas with exponent $s \in (-1,1)$, $s \ne 0$. This we give in terms of a sum over Fourier components of the linear statistics with respect to a Gegenbauer polynomial $\{C_n^{(s/2)}(x) \}$ basis, which generalises a known form in the case $s=0$ involving a cosine expansion. For the power sum linear statistic, our general formula can be reduced to a product of gamma function form, and compared against recent exact results in the literature for this case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25078 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gegenbauer polynomials and fluctuation properties of the one-dimensional Riesz gas Forrester, Peter J. Mathematical Physics The Riesz gas in one-dimension consists of particles interacting via a pair potential, ${\rm sgn}(s) |x - x'|^{-s}$, $s \ne 0$ and $-\log | x - x'|$ for $s=0$. In the infinite density limit, with the particle support the interval $[-1,1]$, we apply a functional derivative method due to Beenakker to compute the covariance of two smooth linear statistics for the Riesz gas with exponent $s \in (-1,1)$, $s \ne 0$. This we give in terms of a sum over Fourier components of the linear statistics with respect to a Gegenbauer polynomial $\{C_n^{(s/2)}(x) \}$ basis, which generalises a known form in the case $s=0$ involving a cosine expansion. For the power sum linear statistic, our general formula can be reduced to a product of gamma function form, and compared against recent exact results in the literature for this case. |
| title | Gegenbauer polynomials and fluctuation properties of the one-dimensional Riesz gas |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2604.25078 |