{"ID":23638584,"CreatedAt":"2026-09-18T07:21:59.209084423Z","UpdatedAt":"2026-09-18T07:21:59.209084423Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20675","arxiv_id":"2609.20675","title":"Limit theorems for Coulomb gases on a Jordan curve in an external potential","abstract":"We consider a Coulomb gas on a Jordan curve $γ$ in an external potential $V$ at inverse temperature $β\u003e0$ and obtain an asymptotic expansion of the free energy up to $o(1)$ and a central limit theorem for linear statistics. We focus on the one-cut regime, where the density of the weighted equilibrium measure of $γ$ in $V$ is strictly positive on $γ$. The constant term in the (normalized) expansion consists of two parts: the Fredholm determinant of a generalized Grunsky operator and the Dirichlet energy of the logarithm of the density of the weighted equilibrium measure of $γ$. The coefficient of the latter vanishes for $β=2$. The variance of the fluctuations of the linear statistics only depends on the Dirichlet energy of the test function and is therefore independent of $V$. Essential in our approach is that the generalized Grunsky operator and the accompanying equilibrium parametrization allow us to transport the particles on the curve in the external potential to a reference object in a way that preserves the equilibrium measure. In our setting, the unit circle is the natural reference object.","short_abstract":"We consider a Coulomb gas on a Jordan curve $γ$ in an external potential $V$ at inverse temperature $β\u003e0$ and obtain an asymptotic expansion of the free energy up to $o(1)$ and a central limit theorem for linear statistics. We focus on the one-cut regime, where the density of the weighted equilibrium measure of $γ$ in...","url_abs":"https://arxiv.org/abs/2609.20675","url_pdf":"https://arxiv.org/pdf/2609.20675v1","authors":"[\"Kurt Johansson\",\"Thomas Wolfs\"]","published":"2026-09-17T16:48:22Z","proceeding":"math.CV","tasks":"[\"math.CV\"]","methods":"[]","has_code":false}