arXiv subjects

Titus Lupu

Publications and source records attributed to Titus Lupu.

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

Three-dimensional Brownian loop soup clusters

We study Brownian loop soup clusters in $\mathbb{R}^3$ for an arbitrary intensity $α>0$. We show the existence of a phase transition for the presence of unbounded clusters and study its basic properties. In particular, we show that, when $α$ is sufficiently large, almost surely all the loops are connected into a single cluster. Such a phenomenon is not observed in discrete percolation-type models. In addition, we prove the existence of a one-arm exponent and compare the clusters with the finite-range system obtained by imposing lower and upper bounds on the diameter of the loops. Finally, we provide a toolbox concerning the Brownian loop measure in $\mathbb{R}^d$, $d \ge 3$. In particular, we derive decomposition formulas by rerooting the loops in specific ways and show that the loop measure is conformally invariant, generalising results of [Lup18] in dimension 1 and [LW04] in dimension 2.

math.PR↗

arXiv2025

Intensity doubling for Brownian loop-soups in high dimensions

We derive an intensity doubling feature of critical Brownian loop-soups on the cable-graphs of ${\mathbb Z}^d$ for $d \ge 7$ that can be described as follows: In the box $[-N, N]^d$ (and with a probability that goes to $1$ as $N$ goes to infinity), the set of all clusters of Brownian loops that do contain proper self-avoiding cycles of diameter comparable to $N$ can be decomposed into two identically distributed families: (a) The collection of clusters that do contain a large Brownian loop from the loop-soup (and therefore do automatically contain such a large cycle) (b) The collection of clusters that contain no macroscopic loop from the loop-soup (more specifically, no loop of diameter greater than $N^{\beta}$ when $\beta > 4/ (d-2)$ is fixed) but nevertheless contain a large cycle. In particular, due to the fact that these two families are asymptotically identically distributed, large cycles formed in case (b) by chains of small Brownian loops (i.e., all of diameter much smaller than $N$) will look like large Brownian loops themselves, and form a second independent "ghost" critical loop-soup in the scaling limit. Reformulated in terms of the Gaussian free field on such cable-graphs, this shows that large cycles in the collection of its sign clusters will converge in the scaling limit to a Brownian loop-soup with twice the usual critical intensity. This result had been conjectured by the first author in arXiv:2209.07901 [math.PR] ; our proof builds heavily on the second author's switching property for such loop-soups from arXiv:2502.06754 [math.PR] .

math.PR↗

arXiv2025

Relation between Wick powers and excursion clusters of the 2D GFF

We study the decomposition of the Wick powers of the continuum GFF in dimension $2$ via the first passage sets (FPS) and the excursion clusters (sign components) of the GFF. These sets are non-thin for the GFF, that is to say the field has non-trivial restriction to such a set, which is a measure, negative or positive depending on the sign. In this work we show that all the odd Wick powers of the GFF can be restricted to the FPS and the excursion clusters, and the restrictions are generalized functions supported on these fractal sets. By contrast, the restriction of an even Wick power to an FPS or excursion cluster is diverging, and to get something converging an additional compensation is required, which is provided by a smooth function living outside of the set and blowing up in a non-integrable way when approaching the set. We further provide expressions of restricted odd Wick powers and restricted-compensated even Wick powers as limits of functions living outside the FPS/excursion cluster. Then, we study the $\varepsilon$-neighborhoods, in the sense of conformal radius, of first passage sets and excursion clusters. We show that such $\varepsilon$-neighborhoods admit asymptotic expansions in $L^2$ into half-integer powers $\vert\log \varepsilon\vert^{-(n+1/2)}$, $n\in\mathbb{N}$, of $1/\vert\log \varepsilon\vert$. The coefficients of the expansion involve the restrictions of the odd Wick powers. By contrast, the even Wick powers do not appear in the expansion. Our expansion is reminiscent of Le Gall's expansion for the Wiener sausage in dimension 2, with however some important differences. The most important one is that the powers of $1/\vert\log \varepsilon\vert$ are different. In the case of the Wiener sausage the powers are integer, $\vert\log \varepsilon\vert^{-n}$, $n\in\mathbb{N}\setminus \{0\}$.

math.PR↗

arXiv2024

The height gap of planar Brownian motion is $\frac{5}{\pi}$

We show that the occupation measure of planar Brownian motion exhibits a constant height gap of $5/\pi$ across its outer boundary. This property bears similarities with the celebrated results of Schramm--Sheffield [18] and Miller--Sheffield [12] concerning the height gap of the Gaussian free field across SLE$_4$/CLE$_4$ curves. Heuristically, our result can also be thought of as the $\theta \to 0^+$ limit of the height gap property of a field built out of a Brownian loop soup with subcritical intensity $\theta>0$, proved in our recent paper [3]. To obtain the explicit value of the height gap, we rely on the computation by Garban and Trujillo Ferreras [1] of the expected area of the domain delimited by the outer boundary of a Brownian bridge.

math.PR↗

arXiv2023

An equivalence between gauge-twisted and topologically conditioned scalar Gaussian free fields

We study on the metric graphs two types of scalar Gaussian free fields (GFF), the usual one and the one twisted by a $\{-1,1\}$-valued gauge field. We show that the latter can be obtained, up to an additional deterministic transformation, by conditioning the first on a topological event. This event is that all the sign clusters of the field should be trivial for the gauge field, that is to say should not contain loops with holonomy $-1$. We also express the probability of this topological event as a ratio of two determinants of Laplacians to the power $1/2$, the usual Laplacian and the gauge-twisted Laplacian. As an example, this gives on annular planar domains the probability that no sign cluster of the metric graph GFF surrounds the inner hole of the domain. Based on our result on the metric graph, and on previous works by Werner and Cai-Ding on the clusters of the metric graph GFF in high dimension, we formulate an intensity doubling conjecture. According to it, if the space dimension is high enough, the cycles in the sign clusters of the metric graph GFF converge in the scaling limit to a Brownian loop soup of intensity parameter $α= 1 = 2\times \dfrac{1}{2}$, which is the double of the intensity parameter appearing in isomorphism theorems.

math.PR↗

arXiv2023

Conformally invariant fields out of Brownian loop soups

Consider a Brownian loop soup $\mathcal{L}_D^θ$ with subcritical intensity $θ\in (0,1/2]$ in some 2D bounded simply connected domain. We define and study the properties of a conformally invariant field $h_θ$ naturally associated to $\mathcal{L}_D^θ$. Informally, this field is a signed version of the local time of $\mathcal{L}_D^θ$ to the power $1-θ$. When $θ=1/2$, $h_θ$ is a Gaussian free field (GFF) in $D$. Our construction of $h_θ$ relies on the multiplicative chaos $\mathcal{M}_γ$ associated with $\mathcal{L}_D^θ$, as introduced in [ABJL23]. Assigning independent symmetric signs to each cluster, we restrict $\mathcal{M}_γ$ to positive clusters. We prove that, when $θ=1/2$, the resulting measure $\mathcal{M}_γ^+$ corresponds to the exponential of $γ$ times a GFF. At this intensity, the GFF can be recovered by differentiating at $γ=0$ the measure $\mathcal{M}_γ^+$. When $θ<1/2$, we show that $\mathcal{M}_γ^+$ has a nondegenerate fractional derivative at $γ=0$ defining a random generalised function $h_θ$. We establish a result which is analoguous to the recent work [ALS23] in the GFF case ($θ=1/2$), but for $h_θ$ with $θ\in (0,1/2]$. Relying on the companion article [JLQ23], we prove that each cluster of $\mathcal{L}_D^θ$ possesses a nondegenerate Minkowski content in some non-explicit gauge function $r \mapsto r^2 |\log r|^{1-θ+o(1)}$. We then prove that $h_θ$ agrees a.s. with the sum of the Minkowski content of each cluster multiplied by its sign. We further extend the couplings between CLE$_4$, SLE$_4$ and the GFF to $h_θ$ for $θ\in(0,1/2]$. We show that the (non-nested) CLE$_κ$ loops form level lines for $h_θ$ and that there exists a constant height gap between the values of the field on either side of the CLE loops.

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↗

arXiv2023

Crossing exponent in the Brownian loop soup

We study the clusters of loops in a Brownian loop soup in some bounded two-dimensional domain with subcritical intensity $θ\in (0,1/2]$. We obtain an exact expression for the asymptotic probability of the existence of a cluster crossing a given annulus of radii $r$ and $r^s$ as $r \to 0$ ($s >1$ fixed). Relying on this result, we then show that the probability for a macroscopic cluster to hit a given disc of radius $r$ decays like $|\log r|^{-1+θ+ o(1)}$ as $r \to 0$. Finally, we characterise the polar sets of clusters, i.e. sets that are not hit by the closure of any cluster, in terms of $\log^α$-capacity. This paper reveals a connection between the 1D and 2D Brownian loop soups. This connection in turn implies the existence of a second critical intensity $θ= 1$ that describes a phase transition in the percolative behaviour of large loops on a logarithmic scale targeting an interior point of the domain.

math.PR↗

arXiv2022

Multiplicative chaos of the Brownian loop soup

We construct a measure on the thick points of a Brownian loop soup in a bounded domain D of the plane with given intensity $θ>0$, which is formally obtained by exponentiating the square root of its occupation field. The measure is constructed via a regularisation procedure, in which loops are killed at a fix rate, allowing us to make use of the Brownian multiplicative chaos measures previously considered in [BBK94, AHS20, Jeg20a], or via a discrete loop soup approximation. At the critical intensity $θ= 1/2$, it is shown that this measure coincides with the hyperbolic cosine of the Gaussian free field, which is closely related to Liouville measure. This allows us to draw several conclusions which elucidate connections between Brownian multiplicative chaos, Gaussian free field and Liouville measure. For instance, it is shown that Liouville-typical points are of infinite loop multiplicity, with the relative contribution of each loop to the overall thickness of the point being described by the Poisson--Dirichlet distribution with parameter $θ= 1/2$. Conversely, the Brownian chaos associated to each loop describes its microscopic contribution to Liouville measure. Along the way, our proof reveals a surprising exact integrability of the multiplicative chaos associated to a killed Brownian loop soup. We also obtain some estimates on the discrete and continuous loop soups which may be of independent interest.

math.PR↗

arXiv2021

Isomorphisms of $β$-Dyson's Brownian motion with Brownian local time

We show that the Brydges-Fröhlich-Spencer-Dynkin and the Le Jan's isomorphisms between the Gaussian free fields and the occupation times of symmetric Markov processes generalize to the $β$-Dyson's Brownian motion. For $β\in\{1,2,4\}$ this is a consequence of the Gaussian case, however the relation holds for general $β$. We further raise the question whether there is an analogue of $β$-Dyson's Brownian motion on general electrical networks, interpolating and extrapolating the fields of eigenvalues in matrix-valued Gaussian free fields. In the case $n=2$ we give a simple construction.

math.PR↗

arXiv2021

Topological expansion in isomorphism theorems between matrix-valued fields and random walks

We consider Gaussian fields of real symmetric, complex Hermitian or quaternionic Hermitian matrices over an electrical network, and describe how the isomorphisms between these fields and random walks give rise to topological expansions encoded by ribbon graphs. We further consider matrix-valued Gaussian fields twisted by an orthogonal, unitary or symplectic connection. In this case the isomorphisms involve traces of holonomies of the connection along random walk loops parametrized by boundary cycles of ribbon graphs.

math.PR↗

arXiv2021

Inverting the Ray-Knight identity on the line

Using a divergent Bass-Burdzy flow we construct a self-repelling one-dimensional diffusion. Heuristically, it can be interpreted as a solution to an SDE with a singular drift involving a derivative of the local time. We show that this self-repelling diffusion inverts the second Ray-Knight identity on the line. The proof goes through an approximation by a self-repelling jump processes that has been previously shown by the authors to invert the Ray-Knight identity in the discrete.

math.PR↗

arXiv2021

A level line of the Gaussian free field with measure-valued boundary conditions

In this article, we construct samples of SLE-like curves out of samples of CLE and Poisson point process of Brownian excursions. We show that the law of these curves depends continuously on the intensity measure of the Brownian excursions. Using such construction of curves, we extend the notion of level lines of GFF to the case when the boundary condition is measure-valued.

math.PR↗

arXiv2020

From loop clusters and random interlacement to the free field

It was shown by Le Jan that the occupation field of a Poisson ensemble of Markov loops ("loop soup") of parameter one-half associated to a transient symmetric Markov jump process on a network is half the square of the Gaussian free field on this network. We construct a coupling between these loops and the free field such that an additional constraint holds: the sign of the free field is constant on each cluster of loops. As a consequence of our coupling we deduce that the loop clusters of parameter one-half do not percolate on periodic lattices. We also construct a coupling between the random interlacement on $\mathbb{Z}^{d}$, $d\geq 3$, introduced by Sznitman, and the Gaussian free field on the lattice such that the set of vertices visited by the interlacement is contained in a level set of the free field. We deduce an inequality between the critical level for the percolation by level sets of the free field and the critical parameter for the percolation of the vacant set of the random interlacement. Both in the case of loops and of the random interlacement, the couplings are constructed by replacing discrete graphs by metric graphs. Le Jan's and Sznitman's isomorphism theorems between the Gaussian free field and the occupation field of trajectories can be extended to the metric graph setting on which the intermediate value principle for continuous fields holds.

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↗

arXiv2019

Convergence of the two-dimensional random walk loop soup clusters to CLE

We consider the random walk loop soup on the discrete half-plane corresponding to a central charge c in (0, 1]. We look at the clusters of discrete loops and show that the scaling limit of the outer boundaries of outermost clusters is the CLE(kappa) loop ensemble, with the same relation between kappa and c as in the continuum Brownian setting.

math.PR↗

arXiv2019

Fine mesh limit of the VRJP in dimension one and Bass-Burdzy flow

We introduce a continuous space limit of the Vertex Reinforced Jump Process (VRJP) in dimension one, which we call Linearly Reinforced Motion (LRM) on $\R$. It is constructed out of a convergent Bass-Burdzy flow. The proof goes through the representation of the VRJP as a mixture of Markov jump processes. As a by-product this gives a representation in terms of a mixture of diffusions of the LRM and of the Bass-Burdzy flow itself. We also show that our continuous space limit can be obtained out of the Edge Reinforced Random Walk (ERRW), since the ERRW and the VRJP are known to be closely related. Compared to the discrete space processes, the LRM has an additional symmetry in the initial local times (initial occupation profile): changing them amounts to a deterministic change of the space and time scales.

math.PR↗