# Strong Szeg\H{o} theorem on a Jordan curve

@inproceedings{Johansson2021StrongST, title={Strong Szeg\H\{o\} theorem on a Jordan curve}, author={Kurt Johansson}, year={2021} }

Abstract. We consider certain determinants with respect to a sufficiently regular Jordan curve γ in the complex plane that generalize Toeplitz determinants which are obtained when the curve is the circle. This also corresponds to studying a planar Coulomb gas on the curve at inverse temperature β = 2. Under suitable assumptions on the curve we prove a strong Szegő type asymptotic formula as the size of the determinant grows. The resulting formula involves the Grunsky operator built from the…

## One Citation

The Loewner Energy via Moving Frames and Surfaces of Finite Renormalised Area Bounding Weil-Petersson Curves

- Mathematics
- 2021

We obtain a new formula for the Loewner energy of simple curves of the sphere as the renormalised energy of moving frames of the two domains of the sphere delimited by the given curve. We also…

## References

SHOWING 1-10 OF 25 REFERENCES

Multivariate normal approximation for traces of random unitary matrices

- Mathematics, PhysicsThe Annals of Probability
- 2021

In this article, we obtain a super-exponential rate of convergence in total variation between the traces of the first $m$ powers of an $n\times n$ random unitary matrices and a $2m$-dimensional…

The Loewner-Kufarev Energy and Foliations by Weil-Petersson Quasicircles.

- Mathematics, Physics
- 2020

We study foliations by non-smooth chord-arc Jordan curves of the twice punctured Riemann sphere $\mathbb C \smallsetminus \{0\}$ using the Loewner-Kufarev equation. We associate to such a foliation a…

Equivalent descriptions of the loewner energy

- Mathematics, PhysicsInventiones mathematicae
- 2019

Loewner’s equation provides a way to encode a simply connected domain or equivalently its uniformizing conformal map via a real-valued driving function of its boundary. The first main result of the…

Interplay Between Loewner and Dirichlet Energies via Conformal Welding and Flow-Lines

- Mathematics
- 2019

The Loewner energy of a Jordan curve is the Dirichlet energy of its Loewner driving term. It is finite if and only if the curve is a Weil–Petersson quasicircle. In this paper, we describe cutting and…

The Loewner Energy of Loops and Regularity of Driving Functions

- MathematicsInternational Mathematics Research Notices
- 2019

Loewner driving functions encode simple curves in 2D simply connected domains by real-valued functions. We prove that the Loewner driving function of a $C^{1,\beta }$ curve (differentiable…

The random matrix theory of the classical compact groups, Cambridge Tracts in Mathematics, 218

- 2019

Relative Szeg\H{o} asymptotics for Toeplitz determinants

- Mathematics, Physics
- 2016

We study the asymptotic behavior, as $n\to\infty$, of ratios of Toeplitz determinants $D_n(e^h d\mu)/D_n(d\mu)$ defined by a measure $\mu$ on the unit circle and a sufficiently smooth function $h$.…

Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model. Some history and some recent results

- Mathematics, Physics
- 2012

We review some history and some recent results concerning Toeplitz determinants and their applications. We discuss, in particular, the crucial role of the two-dimensional Ising model in stimulating…

Weil-Petersson metric on the universal Teichmüller space, Mem

- Amer. Math. Soc
- 2006