arXiv subjects

Fredrik Viklund

Publications and source records attributed to Fredrik Viklund.

At least 19 recordsLinked to original sources

arXiv2026

On the Loewner energy of a welding homeomorphism

To any Jordan curve one may associate a circle homeomorphism $φ: \mathbb S^1 \to \mathbb S^1$ via conformal welding. Through this correspondence, the Loewner energy $I^L$, also known as the universal Liouville action, is a Kähler potential for the unique homogeneous Kähler metric on the universal Teichmüller space. Despite this, explicit expressions for $I^L$ in terms of $φ$ alone do not seem to be available in the literature. In this paper, we obtain such formulas. For this, we introduce an operator ${\bf Λ}_φ$ defined using the Fourier coefficients of the function \[ (z,w) \mapsto \log \left|\frac{φ(z)-φ(w)}{z-w}\right|, \qquad (z,w) \in \mathbb{S}^1 \times \mathbb{S}^1. \] We relate ${\bf Λ}_φ$ to the single-layer potential and composition operator, and prove an analog of the classical Grunsky inequalities for quasisymmetric $φ$. We show moreover that $φ$ is Weil--Petersson if and only if ${\bf Λ}_φ$ is Hilbert--Schmidt, and we express $I^L$ as several related Fredholm determinants as well as a regularized Fredholm determinant. We also treat Schatten classes, and we obtain formulas in terms of Dirichlet integrals involving $\log φ'$ and in terms of the composition operator induced by $φ$.

math.CV↗

arXiv2026

Coulomb gas and the Grunsky operator on a Jordan domain with corners

Let $D$ be a Jordan domain of unit capacity. We study the partition function of a planar Coulomb gas in $D$ with a hard wall along $η= \partial D$, \[Z_{n}(D) =\frac 1{n!}\int_{D^n}\prod_{1\le k < \ell \le n}|z_k-z_\ell|^{2} \prod_{k=1}^n d^2z_k.\] We are interested in how the geometry of $η$ is reflected in the large $n$ behavior of $Z_n(D)$. We prove that $η$ is a Weil-Petersson quasicircle if and only if \[ \lim_{n \to \infty} \log \frac{Z_n(D)}{Z_n(\mathbb{D})} = -\frac{1}{12}I^L(η), \] where $I^L$ is the Loewner energy, $\mathbb{D}$ is the unit disc, and $\log Z_n(\mathbb{D}) = \log π^n/n!$. We next consider piecewise analytic $η$ with $m$ corners of interior opening angles $πα_p, p=1,\ldots, m$. Our main result is the asymptotic formula \[ \lim_{n\to\infty}\frac 1{\log n} \log \frac{Z_n(D)}{Z_n(\mathbb{D})} =-\frac 16\sum_{p=1}^m \left(α_p+\frac 1{α_p}-2 \right) \] which is consistent with physics predictions. The starting point of our analysis is an exact expression for $\log Z_{n}(D)$ in terms of a Fredholm determinant involving the truncated Grunsky operator for $D$. The proof of the main result is based on careful asymptotic analysis of the Grunsky coefficients. As further applications of our method we also study the Loewner energy and the related Fekete-Pommerenke energy, a quantity appearing in the analysis of Fekete points, for equipotentials approximating the boundary of a domain with corners. We formulate several conjectures and open problems.

math.CV↗

arXiv2026

Dyson Brownian motion on a Jordan curve

Zabrodin recently proposed a generalization of Dyson Brownian motion to a setting where the particles are confined to a smooth Jordan curve in the plane. In this paper, we discuss a rigorous construction of such a process on a rectifiable Jordan curve and study some of its basic properties. Under further smoothness assumptions, we derive the associated Fokker-Planck-Kolmogorov equation, prove convergence towards the stationary Coulomb gas distribution, study large deviations at low temperature, and derive the limiting mean-field McKean--Vlasov equation in the many-particle limit.

math.PR↗

arXiv2025

Loewner--Kufarev entropy and large deviations of the Hastings--Levitov model

We consider the Hastings--Levitov HL(0) model in the small particle scaling limit and prove a large deviation principle. The rate function is given by the relative entropy of the driving measure $ρ$ for the Loewner--Kufarev equation: \[ H(ρ) = \frac{1}{2π}\iint \barρ_t(θ) \log \barρ_t(θ) dθdt, \] whenever $ρ= \barρ_t dθdt/2π$ with $\int_{S^1} \barρ_t dθ/2π= 1$. We investigate the class of shapes that can be generated by finite entropy Loewner evolution and show that it contains all Weil-Petersson quasicircles, all Becker quasicircles, a Jordan curve with a cusp, and a non-simple curve. We also consider the problem of finding a measure of minimal entropy generating a given shape as well as a simplified version of the problem for a related transport equation.

math.PR↗

arXiv2025

Current expansion and couplings for Ising lattice gauge theory

In this note, we discuss a random current expansion and a switching lemma for Ising lattice gauge theory at all choices of inverse temperature $β$, leading to summation over surfaces. We also describe couplings of this expansion with other representations, including the high-temperature expansion and the cluster expansion. We deduce some simple consequences, including several expressions for the Wilson loop expectation (at any $β$), a new proof of the area law estimate for sufficiently small $ β$, and a proof of exponential decay of correlations for small and large $ β. $ We also derive a few results analogous to corresponding results for the Ising model. In particular, we show that the Wilson loop expectation is non-negative at any $β$ and give an alternative short proof of Griffith's second inequality and, as a consequence, show that the Wilson loop expectations are increasing in $ β$ for all $β$.

math.PR↗

arXiv2025

Planar Coulomb gas on a Jordan arc at any temperature

We study a planar Coulomb gas confined to a sufficiently smooth Jordan arc $γ$ in the complex plane, at inverse temperature $β> 0$. Let \[\bar{Z}_{n}^β(γ) = Z_{n}^β(γ)/\left(2 \textrm{cap}(γ)\right)^{βn^2/2+(1-β/2)n}\] be the normalized partition function. We compute the free energy as the number of particles tends to infinity, including the constant term: \[ \lim_{n\to \infty}\log \frac{\bar{Z}_{n}^β(γ)}{\bar{Z}_{n}^β([-1,1])} = \frac{1}{24}J^A(γ) + \frac{1}{8} \left( \sqrt{ \fracβ{2} } - \sqrt{ \frac{2}β} \right)^2 J^{F}(γ). \] Here $\bar{Z}_{n}^β([-1,1])$ is an explicit Selberg integral, $J^A(γ)$ is half the Loewner energy of a certain Jordan curve associated to $γ$ plus a covariance term, and $J^F(γ)$ is the Fekete energy, related to the zero temperature limit of the model. We also prove an asymptotic formula for the Laplace transform of linear statistics for sufficiently regular test functions. As a consequence, the centered empirical measure converges to a Gaussian field with explicit asymptotic mean, and asymptotic variance given by the Dirichlet energy of the bounded harmonic extension of the test function outside of the arc. A key tool in our analysis is the arc-Grunsky operator $B$ associated to $γ$, reminiscent to but different from the classical Grunsky operator. We derive several basic properties the arc-Grunsky operator, including an estimate analogous to the strengthened Grunsky inequality and the relation to the Dirichlet integral. In our proofs $J^A(γ)$ arises from the Fredholm determinant $\det(I+B)$ and a substantial part of the paper is devoted to relating this to the Loewner energy.

math.CV↗

arXiv2024

Wilson lines in the lattice Higgs model at strong coupling

We consider the 4D fixed length lattice Higgs model with Wilson action for the gauge field and structure group $\mathbb{Z}_n$. We study Wilson line observables in the strong coupling regime and compute their asymptotic behavior with error estimates. Our analysis is based on a high-temperature representation of the lattice Higgs measure combined with Poisson approximation. We also give a short proof of the folklore result that Wilson line (and loop) expectations exhibit perimeter law decay whenever the Higgs field coupling constant is positive.

math.PR↗

arXiv2024

Competitive Hele-Shaw flow and quadratic differentials

We introduce and investigate a generalization of the Hele-Shaw flow with injection where several droplets compete for space as they try to expand due to internal pressure while still preserving their topology. Droplets are described by their closed non-crossing interface curves in $\mathbb{C}$ or more generally in a Riemann surface of finite type. Our main focus is on stationary solutions which we show correspond to the critical vertical trajectories of a particular quadratic differential with second order poles at the source points. The quadratic differentials that arise in this way have a simple description in terms of their associated half-translation surfaces. Existence of stationary solutions is proved in some generality by solving an extremal problem involving an electrostatic energy functional, generalizing a classic problem studied by Teichmüller, Jenkins, Strebel and others. We study several special cases, including stationary Jordan curves on the Riemann sphere. We also introduce a discrete random version of the dynamics closely related to Propp's competitive erosion model, and conjecture that realizations of the lattice model will converge towards a corresponding solution to the competitive Hele-Shaw problem as the mesh size tends to zero.

math.CV↗

arXiv2024

Percolation for two-dimensional excursion clouds and the discrete Gaussian free field

We study percolative properties of excursion processes and the discrete Gaussian free field (dGFF) in the planar unit disk. We consider discrete excursion clouds, defined using random walks as a two-dimensional version of random interlacements, as well as its scaling limit, defined using Brownian motion. We prove that the critical parameters associated to vacant set percolation for the two models are the same and equal to $π/3.$ The value is obtained from a Schramm-Loewner evolution (SLE) computation. Via an isomorphism theorem, we use a generalization of the discrete result that also involves a loop soup (and an SLE computation) to show that the critical parameter associated to level set percolation for the dGFF is strictly positive and smaller than $\sqrt{π/2}.$ In particular this entails a strict inequality of the type $h_*<\sqrt{2u_*}$ between the critical percolation parameters of the dGFF and the two-dimensional excursion cloud. Similar strict inequalities are conjectured to hold in a general transient setup.

math.PR↗

arXiv2024

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

We study foliations by 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 function on the plane that describes the "local winding" along each leaf. Our main theorem is that this function has finite Dirichlet energy if and only if the Loewner driving measure $ρ$ has finite Loewner-Kufarev energy, defined by $$S(ρ) = \frac{1}{2}\iint_{S^1 \times \mathbb{R}} ν_t'(θ)^2 \, d θd t$$ whenever $ρ$ is of the form $ν_t(θ)^2 d θd t$, and set to $\infty$ otherwise. Moreover, if either of these two energies is finite they are equal up to a constant factor, and in this case, the foliation leaves are Weil-Petersson quasicircles. This duality between energies has several consequences. The first is that the Loewner-Kufarev energy is reversible, that is, invariant under inversion and time-reversal of the foliation. Furthermore, the Loewner energy of a Jordan curve can be expressed using the minimal Loewner-Kufarev energy of those measures that generate the curve as a leaf. This provides a new and quantitative characterization of Weil-Petersson quasicircles. Finally, we consider conformal distortion of the foliation and show that the Loewner-Kufarev energy satisfies an exact transformation law involving the Schwarzian derivative. The proof of our main theorem uses an isometry between the Dirichlet energy space on the unit disc and $L^2(2ρ)$ that we construct using Hadamard's variational formula expressed by means of the Loewner-Kufarev equation. Our results are related to $κ$-parameter duality and large deviations of Schramm-Loewner evolutions coupled with Gaussian random fields.

math.CV↗

arXiv2023

Free energy and quark potential in Ising lattice gauge theory via cluster expansion

We revisit the cluster expansion for Ising lattice gauge theory on $\mathbb{Z}^m, \, m \ge 3,$ with Wilson action, at a fixed inverse temperature $ β$ in the low-temperature regime. We prove existence and analyticity of the infinite volume limit of the free energy and compute the first few terms in its expansion in powers of $e^{-β}$. We further analyze Wilson loop expectations and derive an estimate that shows how the lattice scale geometry of a loop is reflected in the large $β$ asymptotic expansion. Specializing to axis parallel rectangular loops $γ_{T,R}$ with side-lengths $T$ and $R$, we consider the limiting function $$ V_β(R) := \lim_{T \to \infty} - \frac{1}{T} \log \, \langle W_{γ_{T,R}} \rangle_β, $$ known as the static quark potential in the physics literature. We verify existence of the limit (with an estimate on the convergence rate) and compute the first few terms in the expansion in powers of $e^{-β}$. As a consequence, a strong version of the perimeter law follows. We also treat $- \log \, \langle W_{γ_{T,R}} \rangle_β/ (T+R)$ as $T, R$ tend to infinity simultaneously and give analogous estimates.

math.PR↗

arXiv2022

Wilson loops in the abelian lattice Higgs model

We consider the lattice Higgs model on $\mathbb{Z}^4$, with structure group given by $ \mathbb{Z}_n $ for $ n \geq 2 $. We compute the expected value of the Wilson loop observable to leading order when the gauge coupling constant and hopping parameter are both sufficiently large. The leading order term is expressed in terms of a quantity arising from the related but much simpler $ \mathbb{Z}_n $ model, which reduces to the Ising model when $n=2$. As part of the proof, we construct a coupling between the lattice Higgs model and the $ \mathbb{Z}_n $ model.

math.PR↗

arXiv2021

Wilson loops in finite Abelian lattice gauge theories

We consider lattice gauge theories on $\mathbb{Z}^4$ with Wilson action and structure group $\mathbb{Z}_n$. We compute the expectation of Wilson loop observables to leading order in the weak coupling regime, extending and refining a recent result of Chatterjee. Our proofs use neither duality relations nor cluster expansion techniques.

math.PR↗

arXiv2020

Interplay between Loewner and Dirichlet energies via conformal welding and flow-lines

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 welding operations on finite Dirichlet energy functions defined in the plane, allowing expression of the Loewner energy in terms of Dirichlet energy dissipation. We show that the Loewner energy of a unit vector field flow-line is equal to the Dirichlet energy of the harmonically extended winding. We also give an identity involving a complex-valued function of finite Dirichlet energy that expresses the welding and flow-line identities simultaneously. As applications, we prove that arclength isometric welding of two domains is sub-additive in the energy, and that the energy of equipotentials in a simply connected domain is monotone. Our main identities can be viewed as action functional analogs of both the welding and flow-line couplings of Schramm-Loewner evolution curves with the Gaussian free field.

math.CV↗

arXiv2019

Transition probabilities for infinite two-sided loop-erased random walks

The infinite two-sided loop-erased random walk (LERW) is a measure on infinite self-avoiding walks that can be viewed as giving the law of the `middle part' of an infinite LERW loop going through 0 and infinity. In this note we derive expressions for transition probabilities for this model in dimensions two and up. In the plane, the formula can be further expressed in terms of a Laplacian with signed weights acting on certain discrete harmonic functions at the tips of the walk, and taking a determinant. The discrete harmonic functions are closely related to a discrete version of the complex square-root.

math.PR↗

arXiv2019

Asymptotic analysis of Dotsenko-Fateev integrals

We develop a method for evaluating asymptotics of certain contour integrals that appear in Conformal Field Theory under the name of Dotsenko-Fateev integrals and which are natural generalizations of the classical hypergeometric functions. We illustrate the method by establishing a number of estimates that are useful in the context of martingale observables for multiple Schramm-Loewner evolution processes.

math-ph↗

arXiv2019

Dimension of two-valued sets via imaginary chaos

Two-valued sets are local sets of the two-dimensional Gaussian free field (GFF) that can be thought of as representing all points of the domain that may be connected to the boundary by a curve on which the GFF takes values only in [-a,b]. Two-valued sets exist whenever $a+b\geq 2λ$, where $λ$ depends explicitly on the normalization of the GFF. We prove that the almost sure Hausdorff dimension of the two-valued set $A_{-a,b}$ equals $d=2-2λ^2/(a+b)^2$. For the two-point estimate, we use the real part of a "vertex field" built from the purely imaginary Gaussian multiplicative chaos. We also construct a non-trivial $d$-dimensional measure supported on $A_{-a,b}$ and discuss its relation with the $d$-dimensional conformal Minkowski content for $A_{-a,b}$.

math.PR↗

arXiv2019

Remarks on the regularity of quasislits

A quasislit is the image of a vertical line segment [0, iy], y > 0, under a quasiconformal homeomorphism of the upper half-plane fixing infinity. Quasislits correspond precisely to curves generated by the Loewner equation with a driving function in the Lip-1/2 class. It is known that a quasislit is contained in a cone depending only on its Loewner driving function Lip-1/2 seminorm, s. In this note we use the Loewner equation to give quantitative estimates on the opening angle of this cone in the full range s <4. The estimate is shown to be sharp for small s. As consequences, we derive explicit Hölder exponents for s < 4 as well as estimates on winding rates. We also relate quantitatively the Lip-1/2 seminorm with the quasiconformal dilatation and discuss the optimal regularity of quasislits achievable through reparametrization.

math.CV↗