arXiv subjects

Janne Junnila

Publications and source records attributed to Janne Junnila.

11 recordsLinked to original sources

arXiv2025

Piecewise geodesic Jordan curves II: Loewner energy, projective structures, and accessory parameters

In this paper we consider Jordan curves on the Riemann sphere passing through $n \ge 3$ given points. We show that in each relative isotopy class of such curves, there exists a unique curve that minimizes the Loewner energy. These curves have the property that each arc between two consecutive points is a hyperbolic geodesic in the domain bounded by the other arcs. This geodesic property lets us define a complex projective structure whose holonomy lies in $\mathrm{PSL}(2,\mathbb{R})$. We show that the quadratic differential comparing this projective structure to the trivial projective structure on the sphere has simple poles whose residues (accessory parameters) are given by the Wirtinger derivatives of the minimal Loewner energy. This is reminiscent of Polyakov's conjecture for Fuchsian projective structures, proven by Takhtajan and Zograf. Finally, we show that the projective structures we obtain are related to Fuchsian projective structures through $π$-grafting.

math.CV↗

arXiv2025

Noise-like analytic properties of imaginary chaos

In this note we continue the study of imaginary multiplicative chaos $μ_β:= \exp(i βΓ)$, where $Γ$ is a two-dimensional continuum Gaussian free field. We concentrate here on the fine-scale analytic properties of $|μ_β(Q(x,r))|$ as $r \to 0$, where $Q(x,r)$ is a square of side-length $2r$ centred at $x$. More precisely, we prove monofractality of this process, a law of the iterated logarithm as $r \to 0$ and analyse its exceptional points, which have a close connection to fast points of Brownian motion. Some of the technical ideas developed to address these questions also help us pin down the exact Besov regularity of imaginary chaos, a question left open in [JSW20]. All the mentioned properties illustrate the noise-like behaviour of the imaginary chaos. We conclude by proving that the processes $x \mapsto |μ_β(Q(x,r))|^2$, when normalised additively and multiplicatively, converge as $r \to 0$ in law, but not in probability, to white noise; this suggests that all the information of the multiplicative chaos is contained in the angular parts of $μ_β(Q(x,r))$.

math.PR↗

arXiv2024

The density of imaginary multiplicative chaos is positive

Consider a log-correlated Gaussian field $Γ$ and its associated imaginary multiplicative chaos $:e^{i βΓ}:$ where $β$ is a real parameter. In [AJJ22], we showed that for any nonzero test function $f$, the law of $\int f :e^{i βΓ}:$ possesses a smooth density with respect to Lebesgue measure on $\mathbb{C}$. In this note, we show that this density is strictly positive everywhere on $\mathbb{C}$. Our simple and direct strategy could be useful for studying other functionals on Gaussian spaces.

math.PR↗

arXiv2023

Multiplicative chaos measures from thick points of log-correlated fields

We prove that multiplicative chaos measures can be constructed from extreme level sets or thick points of the underlying logarithmically correlated field. We develop a method which covers the whole subcritical phase and only requires asymptotics of suitable exponential moments for the field. As an application, we establish these estimates hold for the logarithm of the absolute value of the characteristic polynomial of a Haar distributed random unitary matrix (CUE), using known asymptotics for Toeplitz determinant with (merging) Fisher-Hartwig singularities. Hence, this proves a conjecture of Fyodorov and Keating concerning the fluctuations of the volume of thick points of the CUE characteristic polynomial.

math.PR↗

arXiv2022

Density of imaginary multiplicative chaos via Malliavin calculus

We consider the imaginary Gaussian multiplicative chaos, i.e. the complex Wick exponential $μ_β:= :e^{iβΓ(x)}:$ for a log-correlated Gaussian field $Γ$ in $d \geq 1$ dimensions. We prove a basic density result, showing that for any nonzero continuous test function $f$, the complex-valued random variable $μ_β(f)$ has a smooth density w.r.t. the Lebesgue measure on $\mathbb{C}$. As a corollary, we deduce that the negative moments of imaginary chaos on the unit circle do not correspond to the analytic continuation of the Fyodorov-Bouchaud formula, even when well-defined. Somewhat surprisingly, basic density results are not easy to prove for imaginary chaos and one of the main contributions of the article is introducing Malliavin calculus to the study of (complex) multiplicative chaos. To apply Malliavin calculus to imaginary chaos, we develop a new decomposition theorem for non-degenerate log-correlated fields via a small detour to operator theory, and obtain small ball probabilities for Sobolev norms of imaginary chaos.

math.PR↗

arXiv2020

Reconstructing the base field from imaginary multiplicative chaos

We show that the imaginary multiplicative chaos $\exp(iβΓ)$ determines the gradient of the underlying field $Γ$ for all log-correlated Gaussian fields with covariance of the form $-\log |x-y| + g(x,y)$ with mild regularity conditions on $g$, for all $d \geq 2$ and for all $β\in (0,\sqrt{d})$. In particular, we show that the 2D continuum zero boundary Gaussian free field is measurable w.r.t. its imaginary chaos.

math.PR↗

arXiv2019

On the regularity of complex multiplicative chaos

Denote by $μ_β="\exp(βX)"$ the Gaussian multiplicative chaos which is defined using a log-correlated Gaussian field $X$ on a domain $U\subset\mathbb{R}^d$. The case $β\in\mathbb{R}$ has been studied quite intensively, and then $μ_β$ is a random measure on $U$. It is known that $μ_β$ can also be defined for complex values $β$ lying in certain subdomain of $\mathbb{C}$, and then the realizations of $μ_β$ are random generalized functions on $U$. In this note we complement the results of Junnila et al. (where the case of purely imaginary $β$ was considered) by studying the Besov-regularity of $μ_β$ and the finiteness of moments for general complex values of $β$.

math.PR↗

arXiv2019

Decompositions of log-correlated fields with applications

In this article we establish novel decompositions of Gaussian fields taking values in suitable spaces of generalized functions, and then use these decompositions to prove results about Gaussian multiplicative chaos. We prove two decomposition theorems. The first one is a global one and says that if the difference between the covariance kernels of two Gaussian fields, taking values in some Sobolev space, has suitable Sobolev regularity, then these fields differ by a Hölder continuous Gaussian process. Our second decomposition theorem is more specialized and is in the setting of Gaussian fields whose covariance kernel has a logarithmic singularity on the diagonal -- or log-correlated Gaussian fields. The theorem states that any log-correlated Gaussian field $X$ can be decomposed locally into a sum of a Hölder continuous function and an independent almost $\star$-scale invariant field (a special class of stationary log-correlated fields with 'cone-like' white noise representations). This decomposition holds whenever the term $g$ in the covariance kernel $C_X(x,y)=\log(1/|x-y|)+g(x,y)$ has locally $H^{d+\varepsilon}$ Sobolev smoothness. We use these decompositions to extend several results that have been known basically only for $\star$-scale invariant fields to general log-correlated fields. These include the existence of critical multiplicative chaos, analytic continuation of the subcritical chaos in the so-called inverse temperature parameter $β$, as well as generalised Onsager-type covariance inequalities which play a role in the study of imaginary multiplicative chaos.

math.PR↗

arXiv2018

Imaginary multiplicative chaos: Moments, regularity and connections to the Ising model

In this article we study imaginary Gaussian multiplicative chaos -- namely a family of random generalized functions which can formally be written as $e^{i X(x)}$, where $X$ is a log-correlated real-valued Gaussian field on $\mathbb{R}^d$, i.e. it has a logarithmic singularity on the diagonal of its covariance. We study basic analytic properties of these random generalized functions, such as what spaces of distributions do these objects live in, along with their basic stochastic properties, such as moment and tail estimates. After this, we discuss connections between imaginary multiplicative chaos and the critical planar Ising model, namely that the scaling limit of the spin field of the so called critical planar XOR-Ising model can be expressed in terms of the cosine of the Gaussian free field, i.e. the real part of an imaginary multiplicative chaos distribution. Moreover, if one adds a magnetic perturbation to the XOR-Ising model, then the scaling limit of the spin field can be expressed in terms of the cosine of the sine-Gordon field, which can also be viewed as the real part of an imaginary multiplicative chaos distribution. The first sections of the article have been written in the style of a review, and we hope that the text will also serve as an introduction to imaginary chaos for an uninitiated reader.

math.PR↗

arXiv2016

On the multiplicative chaos of non-Gaussian log-correlated fields

We study non-Gaussian log-correlated multiplicative chaos, where the random field is defined as a sum of independent fields that satisfy suitable moment and regularity conditions. The convergence, existence of moments and analyticity with respect to the inverse temperature are proven for the resulting chaos in the full subcritical range. These results are generalizations of the corresponding theorems for Gaussian multiplicative chaos. A basic example where our results apply is the non-Gaussian Fourier series \[\sum_{k=1}^\infty \frac{1}{\sqrt{k}}(A_k \cos(2πk x) + B_k \sin(2πk x)),\] where $A_k$ and $B_k$ are i.i.d. random variables.

math.PR↗

arXiv2015

The uniqueness of the Gaussian multiplicative chaos revisited

We consider Gaussian multiplicative chaos measures defined in a general setting of metric measure spaces. Uniqueness results are obtained, verifying that different sequences of approximating Gaussian fields lead to the same chaos measure. Specialized to Euclidean spaces, our setup covers both the subcritical chaos and the critical chaos, actually extending to all non-atomic Gaussian chaos measures.

math.PR↗