arXiv subjects

Juhan Aru

Publications and source records attributed to Juhan Aru.

At least 19 recordsLinked to original sources

arXiv2026

Renormalised two-point functions of CLE$_4$ gaskets

We first consider nested CLE$_4$ in a simply-connected domain and compute some exact renormalised probabilities: the probability that two points belong to the same CLE$_4$ gasket and the probability that two points belong to the outermost CLE$_4$ gasket. The resulting conformally covariant formulas have a non-trivial modular dependence, expressed explicitly in terms of Jacobi theta functions. We are guided by the conformal field theory of the Ashkin-Teller (AT) model, but our proofs are purely probabilistic using Brownian loop soups, gauge-twisted Gaussian free fields (GFFs) and the level line geometry of the 2D continuum GFF. More generally, we also calculate renormalised probabilities that two points belong to CLE$_4$ gaskets sampled in alternation with certain two-valued sets of the Gaussian free field. These quantities correspond to the two-point function of the conjectured scaling limit of the AT single spins on the critical line. At the decoupling point, our results recover the Ising model correlations; they also suggest a CLE$_4$-based continuum FK representation of the AT single-spin fields along a whole segment of the critical line.

math.PR↗

arXiv2026

SLE and its partition function in multiply connected domains via the Gaussian Free Field and restriction measures

One way to uniquely define Schramm-Loewner Evolution (SLE) in multiply connected domains is to use the restriction property. This gives an implicit definition of a $σ$-finite measure on curves; yet it is in general not clear how to construct such measures nor whether the mass of these measures, called the partition function, is finite. We provide an explicit construction of the such conformal restriction SLEs in multiply connected domains when $κ= 4$ using the Gaussian Free Field (GFF). In particular, both when the target points of the curve are on the same or on distinct boundary components, we show that there is a mixture of laws of level lines of GFFs that satisfies the restriction property. This allows us to give an expression for the partition function of $\mathrm{SLE}_4$ on multiply connected domains and shows that the partition function is finite, answering the question raised in [Lawler, J. Stat. Phys. 2009]. In a second part, we provide a second construction of $\mathrm{SLE}_κ$ in multiply-connected domains for the whole range $κ\in (8/3,4]$; specific, however, to the case of the two target points belonging to the same boundary components. This is inspired by [Werner, Wu, Electron. J. Probab. 2013] and consists of a mixture of laws on curves obtained by following $\mathrm{CLE}_κ$ loops and restriction hulls attached to parts of the boundary of the domain. In this case as well, we obtain as a corollary the finiteness of the partition function for this type of $\mathrm{SLE}_κ$.

math.PR↗

arXiv2025

The Spherical model and the large-$N$ limit of the Spin $O(N)$ model via the Gaussian free field

We revisit the relation between the spherical model of Berlin-Kac and the spin $O(N)$ model in the limit $N \to \infty$, and explain how they are connected via the discrete Gaussian free field (GFF). Using probabilistic limit theorems and concentration of measure, we prove that the infinite-volume limit of the spherical model on a $d$-dimensional torus is a massive GFF in the high-temperature regime, a standard GFF at the critical temperature, and a standard GFF plus a Rademacher random constant in the low-temperature regime. The proof at the critical temperature appears to be new and relies on a fine analysis of the zero-average Green's function on the torus. We study the spin $O(N)$ model in the double limit of spin dimensionality and torus size. Sending $N \to \infty$ first, and then the torus size to infinity, we show that the different spin coordinates become i.i.d. fields, distributed as a massive GFF in the high-temperature regime, a standard GFF at the critical temperature, and a standard GFF plus a Gaussian random constant in the low-temperature regime. In particular, although the limiting free energies per site of the two models agree at all temperatures, their finite-dimensional laws still differ in terms of their zero modes in the low-temperature regime.

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↗

arXiv2024

A martingale-type characterisation of the Gaussian free field and fractional Gaussian free fields

We establish a martingale-type characterisations for the continuum Gaussian free field (GFF) and for fractional Gaussian free fields (FGFs), using their connection to the stochastic heat equation and to fractional stochastic heat equations. The main theorem on the GFF generalizes previous results of similar flavour and the characterisation theorems on the FGFs are new. The proof strategy is to link the resampling dynamics coming from a martingale-type of decomposition property to the stationary dynamics of the desired field, i.e. to the (fractional) stochastic heat equation.

math.PR↗

arXiv2024

A characterisation of the continuum Gaussian free field in $d \geq 2$ dimensions

We prove that under certain mild moment and continuity assumptions, the $d$-dimensional Gaussian free field is the only stochastic process in $d\geq 2$ that is translation invariant, exhibits a certain scaling, and satisfies the usual domain Markov property. Our proof is based on a decomposition of the underlying functional space in terms of radial processes and spherical harmonics.

math.PR↗

arXiv2024

Noise-like analytic properties of imaginary chaos

In this note we continue the study of imaginary multiplicative chaos $\mu_\beta := \exp(i \beta \Gamma)$, where $\Gamma$ is a two-dimensional continuum Gaussian free field. We concentrate here on the fine-scale analytic properties of $|\mu_\beta(Q(x,r))|$ 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 $x \mapsto |\mu_\beta(Q(x,r))|^2$, 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 $\mu_\beta(Q(x,r))$.

math.PR↗

arXiv2023

Excursion decomposition of the 2D continuum GFF

In this note we show that the 2D continuum Gaussian free field (GFF) admits an excursion decomposition that is on the one hand similar to the classical excursion decomposition of the Brownian motion, and on the other hand can be seen as an FK representation of the continuum GFF. In particular, 2D continuum GFF can be written as an infinite sum of disjoint positive and negative sign excursions, which are given by Minkowski content measures of clusters of a critical 2D Brownian loop soup with i.i.d. signs. Although the 2D continuum GFF is not even a signed measure, we show that the decomposition to positive and negative parts is unique under natural conditions.

math.PR↗

arXiv2022

Percolation for 2D classical Heisenberg model and exit sets of vector valued GFF

Our motivation in this paper is twofold. First, we study the geometry of a class of exploration sets, called exit sets, which are naturally associated with a 2D vector-valued GFF : $ϕ: Z^2 \to R^N, N\geq 1$. We prove that, somewhat surprisingly, these sets are a.s. degenerate as long as $N\geq 2$, while they are conjectured to be macroscopic and fractal when $N=1$. This analysis allows us, when $N\geq 2$, to understand the percolation properties of the level sets of $\{\|ϕ(x)\|, x\in Z^2\}$ and leads us to our second main motivation in this work: if one projects a spin $O(N+1)$ model (classical Heisenberg model is $N=2$) down to a spin $O(N)$ model, we end up with a spin $O(N)$ in a quenched disorder given by random conductances on $Z^2$. Using the exit sets of the $N$-vector-valued GFF, we obtain a local and geometric description of this random disorder in the limit $β\to \infty$. This allows us to revisit a series of celebrated works by Patrascioiu and Seiler ([PS92, PS93, PS02]) which argued against Polyakov's prediction that spin $O(N+1)$ model is massive at all temperatures when $N\geq 2$ ([Pol75]). We make part of their arguments rigorous and more importantly we provide the following counter-example: we build ergodic environments of (arbitrary) high conductances with (arbitrary) small and disconnected regions of low conductances in which, despite the predominance of high conductances, the $XY$ model remains massive. Of independent interest, we prove that at high $β$, the transverse fluctuations of a classical Heisenberg model are given by a $N=2$ vectorial GFF. This is implicit in [Pol75] but we give here the first (non-trivial) rigorous proof. Also, independently of the recent work [DF22], we show that two-point correlation functions of the spin $O(N)$ model are given in terms of certain percolation events in the cable graph for any $N\geq 1$.

math.PR↗

arXiv2022

Thick points of the planar GFF are totally disconnected for all $γ\ne 0$

We prove that the set of $γ$-thick points of a planar Gaussian free field (GFF) with Dirichlet boundary conditions is a.s. totally disconnected for all $γ\neq 0$. Our proof relies on the coupling between a GFF and the nested CLE$_4$. In particular, we show that the thick points of the GFF are the same as those of the weighted CLE$_4$ nesting field and establish the almost sure total disconnectedness of the complement of a nested CLE$_κ$, $κ\in (8/3,4]$. As a corollary we see that the set of singular points for supercritical LQG metrics is a.s. totally disconnected.

math.PR↗

arXiv2022

Mating of trees for critical Liouville quantum gravity

In a groundbreaking work, Duplantier, Miller and Sheffield showed that subcritical Liouville quantum gravity (LQG) coupled with Schramm-Loewner evolutions (SLE) can be described by the mating of two continuum random trees. In this paper, we consider the counterpart of their result for critical LQG and SLE, i.e., for the case when $γ^2=κ=16/κ=4$. We prove that as one sends $κ\downarrow 4$ in the subcritical setting, the space-filling SLE$_κ$ in a disk degenerates to the CLE$_4$ exploration introduced by Werner and Wu, along with a collection of i.i.d.\ coin tosses indexed by the branch points of the exploration. Furthermore, in the $κ=16/γ^2\downarrow 4$ limit, the pair of continuum random trees collapse into a single continuum random tree, and we observe that upon applying an appropriate affine transform to the encoding Brownian motions before taking the limit, we get convergence to a pair of independent Brownian motions $(A,B)$. The Brownian motion $A$ encodes the LQG distance from the CLE loops to the boundary of the disk, while the Brownian motion $B$ encodes the boundary lengths of the CLE$_4$ loops. In contrast to the subcritical setting, $(A,B)$ does not determine the CLE-decorated LQG surface.

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↗

arXiv2020

Extremal distance and conformal radius of a CLE_4 loop

Consider CLE$_4$ in the unit disk and let $\ell$ be the loop of the CLE$_4$ surrounding the origin. Schramm, Sheffield and Wilson determined the law of the conformal radius seen from the origin of the domain surrounded by $\ell$. We complement their result by determining the law of the extremal distance between $\ell$ and the boundary of the unit disk. More surprisingly, we also compute the joint law of these conformal radius and extremal distance. This law involves first and last hitting times of a one-dimensional Brownian motion. Similar techniques also allow us to determine joint laws of some extremal distances in a critical Brownian loop-soup cluster.

math.PR↗

arXiv2020

Gaussian multiplicative chaos through the lens of the 2D Gaussian free field

The aim of this review-style paper is to provide a concise, self-contained and unified presentation of the construction and main properties of Gaussian multiplicative chaos (GMC) measures for log-correlated fields in 2D in the subcritical regime. By considering the case of the 2D Gaussian free field, we review convergence, uniqueness and characterisations of the measures; revisit Kahane's convexity inequalities and existence and scaling of moments; discuss the measurability of the underlying field with respect to the GMC measure and present a KPZ relation for scaling exponents.

math.PR↗

arXiv2019

First passage sets of the 2D continuum Gaussian free field

We introduce the first passage set (FPS) of constant level $-a$ of the two-dimensional continuum Gaussian free field (GFF) on finitely connected domains. Informally, it is the set of points in the domain that can be connected to the boundary by a path on which the GFF does not go below $-a$. It is, thus, the two-dimensional analogue of the first hitting time of $-a$ by a one-dimensional Brownian motion. We provide an axiomatic characterization of the FPS, a continuum construction using level lines, and study its properties: it is a fractal set of zero Lebesgue measure and Minkowski dimension 2 that is coupled with the GFF $Φ$ as a local set $A$ so that $Φ+a$ restricted to $A$ is a positive measure. One of the highlights of this paper is identifying this measure as a Minkowski content measure in the non-integer gauge $r \mapsto \vert\log(r)\vert^{1/2}r^{2}$, by using Gaussian multiplicative chaos theory.

math.PR↗

arXiv2019

Critical Liouville measure as a limit of subcritical measures

We study how the Gaussian multiplicative chaos (GMC) measures $μ^γ$ corresponding to the 2D Gaussian free field change when $γ$ approaches the critical parameter $2$. In particular, we show that as $γ\to 2^{-}$, $(2-γ)^{-1}μ^γ$ converges in probability to $2μ'$, where $μ'$ is the critical GMC measure.

math.PR↗

arXiv2018

Exceptional graphs for the random walk

If $\mathcal{W}$ is the simple random walk on the square lattice $\mathbb{Z}^2$, then $\mathcal{W}$ induces a random walk $\mathcal{W}_G$ on any spanning subgraph $G\subset \mathbb{Z}^2$ of the lattice as follows: viewing $\mathcal{W}$ as a uniformly random infinite word on the alphabet $\{\mathbf{x}, -\mathbf{x}, \mathbf{y}, -\mathbf{y} \}$, the walk $\mathcal{W}_G$ starts at the origin and follows the directions specified by $\mathcal{W}$, only accepting steps of $\mathcal{W}$ along which the walk $\mathcal{W}_G$ does not exit $G$. For any fixed subgraph $G \subset \mathbb{Z}^2$, the walk $\mathcal{W}_G$ is distributed as the simple random walk on $G$, and hence $\mathcal{W}_G$ is almost surely recurrent in the sense that $\mathcal{W}_G$ visits every site reachable from the origin in $G$ infinitely often. This fact naturally leads us to ask the following: does $\mathcal{W}$ almost surely have the property that $\mathcal{W}_G$ is recurrent for \emph{every} subgraph $G \subset \mathbb{Z}^2$? We answer this question negatively, demonstrating that exceptional subgraphs exist almost surely. In fact, we show more to be true: exceptional subgraphs continue to exist almost surely for a countable collection of independent simple random walks, but on the other hand, there are almost surely no exceptional subgraphs for a branching random walk.

math.PR↗