Coulomb-Gas Asymptotics Beyond Weil–Petersson Quasicircles

Definitions and mathematical statement

A planar quasicircle is the image of the unit circle under a quasiconformal homeomorphism of ${\mathbb{C}}$. For a bounded planar quasicircle $\eta$, let $\Omega$ be its bounded complementary component and let $f:\mathbb D\to\Omega$ be a conformal bijection, where $\mathbb D=\{z\in{\mathbb{C}}:|z|<1\}$. A standard characterization of the Weil--Petersson property is \[ \int_{\mathbb D}\left|\frac{f''(z)}{f'(z)}\right|^2\,{\,\mathrm{d}} A(z)<\infty, \] where ${\,\mathrm{d}} A$ denotes planar area measure. This characterization specifies the terminology for bounded planar quasicircles; the supplied excerpt does not itself specify the ambient setting or a definition of the Weil--Petersson class.

Source question. Determine the Coulomb-gas asymptotics when the boundary curve $\eta$ is not a Weil--Petersson quasicircle. In particular, the supplied dated assessment identifies arbitrary quasicircles outside the Weil--Petersson class as a remaining target.

Unresolved formulation. The supplied record does not define the Coulomb-gas measure or partition function, the interaction strength, the reference measure, any external potential, the quantity whose asymptotics are requested, or the limiting regime and normalization. It also does not determine whether the question includes boundary curves that are not quasicircles at all. Consequently, the question cannot be converted into a uniquely specified asymptotic assertion from this record alone.

The original excerpt refers to an unspecified expression on the right-hand side of an omitted formula, describes that expression as potentially universal, and notes its appearance in small-$t$ heat-trace formulas with references labelled [2, 5]. It further states that the proof of an omitted Theorem 2 starts from an omitted Proposition 1 and uses careful asymptotic analysis of Grunsky coefficients. Those formulas, statements, and numbered references are not supplied, so their hypotheses and proposed asymptotic terms remain unresolved.

Short English statement

The problem asks how Coulomb-gas asymptotics extend beyond Weil--Petersson boundary curves, especially to arbitrary non-Weil--Petersson quasicircles. The supplied excerpt omits the gas model and the formula to be extended, so a precise asymptotic claim requires additional source context. The supplied partially solved status is dated curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005929$, title: Coulomb-Gas Asymptotics Beyond Weil--Petersson Quasicircles; exact source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2024; no individual proposer is supplied.
• Supplied report citation: Statistical Physics and Random Surfaces (2024), Oberwolfach Reports, pp. 1387--1444, DOI 10.4171/owr/2024/25. https://doi.org/10.4171/owr/2024/25. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Coulomb gas with corners. https://doi.org/10.1007/s00222-026-01417-5.
• Supplied literature source: Strong Szego on a Jordan curve. https://arxiv.org/abs/2110.11032.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified. The supplied assessment labels the problem partially solved and states that strong Szego and Coulomb-gas asymptotics cover Weil--Petersson curves and important non-Weil--Petersson piecewise-analytic curves with corners, while the question for arbitrary non-Weil--Petersson quasicircles remains open. The background expressly requests independent verification of these time-sensitive status and literature claims.
• The original excerpt cites references [2, 5] for small-$t$ heat-trace formulas, but their bibliographic identities are absent from the supplied record. The excerpt also mentions Theorem 2 and Proposition 1 without supplying their statements.