Partition functions of two-dimensional Coulomb gases with circular root- and jump-type singularities
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866914070404268032 |
|---|---|
| author | Noda, Kohei |
| author_facet | Noda, Kohei |
| contents | In this paper, we study the random polynomial $p_n(ρ):=\prod_{j=1}^n (|z_j|-ρ)$, where the points $\{z_j\}_{j=1}^n$ are the eigenvalue moduli of random normal matrices with a radially symmetric potential. We establish precise large $n$ asymptotic expansions for the moment generating function \[ \mathbb{E}\!\left[e^{\tfrac{u}π\mathrm{Im}\log p_n(ρ)}\, e^{a\,\mathrm{Re}\log p_n(ρ)}\right], \qquad u\in\mathbb{R}, \; a>-1, \] where $ρ>0$ lies in the bulk of the spectral droplet. The asymptotic expansion is expressed in terms of parabolic cylinder functions, which confirms a conjecture of Byun and Charlier. This also provides the first free energy expansion of two-dimensional Coulomb gases with general circular root- and jump-type singularities. While the $a=0$ case has already been widely studied in the literature due to its relation to counting statistics, we also obtain new results for this special case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_00843 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Partition functions of two-dimensional Coulomb gases with circular root- and jump-type singularities Noda, Kohei Mathematical Physics Probability 1A60, 60B20, 60G55 In this paper, we study the random polynomial $p_n(ρ):=\prod_{j=1}^n (|z_j|-ρ)$, where the points $\{z_j\}_{j=1}^n$ are the eigenvalue moduli of random normal matrices with a radially symmetric potential. We establish precise large $n$ asymptotic expansions for the moment generating function \[ \mathbb{E}\!\left[e^{\tfrac{u}π\mathrm{Im}\log p_n(ρ)}\, e^{a\,\mathrm{Re}\log p_n(ρ)}\right], \qquad u\in\mathbb{R}, \; a>-1, \] where $ρ>0$ lies in the bulk of the spectral droplet. The asymptotic expansion is expressed in terms of parabolic cylinder functions, which confirms a conjecture of Byun and Charlier. This also provides the first free energy expansion of two-dimensional Coulomb gases with general circular root- and jump-type singularities. While the $a=0$ case has already been widely studied in the literature due to its relation to counting statistics, we also obtain new results for this special case. |
| title | Partition functions of two-dimensional Coulomb gases with circular root- and jump-type singularities |
| topic | Mathematical Physics Probability 1A60, 60B20, 60G55 |
| url | https://arxiv.org/abs/2510.00843 |