arXiv subjects

Konstantinos Kavvadias

Publications and source records attributed to Konstantinos Kavvadias.

14 recordsLinked to original sources

arXiv2026

The conformally invariant metric on CLE$_4$ I: subsequential limits of the non-simple CLE graph metric

We consider the conformal loop ensemble (CLE) with the parameter $κ=4$, the critical value at or below which the loops are simple and do not intersect each other or the domain boundary. We show that the loops of a CLE$_4$ uniquely determine a conformally invariant, local, and geodesic metric so that the metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. This metric was previously constructed in unpublished work of Sheffield, Watson, and Wu. Our approach differs in that we show that the metric arises as the renormalized limit of the graph metric on CLE$_κ$ loops as $κ\downarrow 4$. In this first paper in a series of three, we prove that the subsequential limits exist and define a non-trivial conformally invariant metric on CLE$_4$ which is local and such that the metric ball growth from the boundary is given by the uniform exploration of Werner and Wu. In subsequent work, we will show that the subsequential limit exists as a true limit.

math.PR↗

arXiv2026

The conformally invariant metric on CLE$_4$ II: existence of geodesics

We continue our study of the conformal loop ensemble (CLE) with parameter $κ=4$, the critical threshold at or below which the loops are simple and disjoint, touching neither each other nor the domain boundary. This paper is the second in a series of three establishing that the loops of a CLE$_4$ uniquely determine a conformally invariant, local, and geodesic metric such that the metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this second paper, we prove the existence of geodesics, showing that any geodesic between two loops is supported on the CLE$_4$ loops (off a set of Hausdorff dimension zero) and does not intersect the domain boundary. Along the way, we establish sharp quantitative estimates for the CLE$_4$ metric geometry, including exponential tail bounds for rectangle distances and multi-scale four-arm SLE$_4$ non-intersection bounds for metric balls.

math.PR↗

arXiv2026

The conformally invariant metric on CLE$_4$ III: uniqueness

This paper is the third and final article in a series of papers constructing the canonical conformally invariant metric on the set of loops of the conformal loop ensemble (CLE) with critical parameter $κ=4$. The previous two articles construct, as a subsequential limit of the renormalized graph metric on the loops of CLE$_κ$ as $κ\downarrow 4$, a conformally invariant, local metric on the loops of a CLE$_4$ whose metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this paper, we establish that this metric is uniquely characterized by its properties, as are its geodesics, and that it is a measurable function of the CLE$_4$. In particular, we show that the renormalized CLE$_κ$ graph metric converges as $κ\downarrow 4$ without passing to a subsequence. A key step in the proof is to show that the metric is determined by the geodesics from each loop to the domain boundary, which are in turn determined by the uniform exploration; this representation will have important applications in future work.

math.PR↗

arXiv2026

Convergence of the Subcritical $γ$-LQG Metric to the Critical 2-LQG Metric as $γ\to 2$

We show that the $γ$-LQG metric (appropriately re-normalized) for $γ\in (0,2)$ converges in law to the critical 2-LQG metric (appropriately re-normalized) with respect to the local uniform topology of continuous functions on $\mathbb{C} \times \mathbb{C}$. Combining with the results in arXiv:2509.15544, we obtain that the law of the $γ$-LQG metric (appropriately re-normalized) is continuous in $γ\in [0,2]$ with respect to the local uniform topology of continuous functions on $\mathbb{C} \times \mathbb{C}$, where the 0-LQG metric corresponds to the Euclidean metric on $\mathbb{C}$.

math.PR↗

arXiv2026

Strong confluence of geodesics in Liouville quantum gravity

$γ$-Liouville quantum gravity ($γ$-LQG) constitutes a family of planar random geometries whose geodesics exhibit intricate fractal behaviour. As is observed in various planar models of random geometry as part of the phenomenon of geodesic confluence, geodesics in $γ$-LQG tend to merge with each other. In particular, in Gwynne-Miller '19, it was established that in $γ$-LQG, geodesics targeted to a fixed point do coalesce in the sense that any two such geodesics almost surely merge before reaching their common target. However, in view of the randomness inherent to the geometry, it is a priori possible that while geodesics targeted to a fixed point do coalesce, there exists a sequence of geodesics $P_n$ converging to an exceptional geodesic $P$ as $n\rightarrow \infty$ such that $P_n$ does not overlap with $P$ for any $n$. In this paper, we prove that this is not possible, thereby establishing a strong confluence statement for $γ$-LQG for all $γ\in (0,2)$. This extends the results obtained in Miller-Qian '20 for $γ=\sqrt{8/3}$ to all subcritical values of $γ$. We discuss applications to the study of geodesic stars and geodesic networks and include a list of open questions.

math.PR↗

arXiv2026

Uniformity of extremal behaviour along geodesics in Liouville quantum gravity

In random geometry, geodesics often have a tendency to coalesce together and traverse regions highly singular relative to typical environments. In this paper, working with the model of Liouville quantum gravity, we develop a technique which yields zero-one laws for many extremal statistics measured along interior segments of the geodesic. Namely, working with statistics such as the Euclidean dimension, optimal Hölder continuity exponents with respect to the Euclidean metric and the minimal/maximal encountered thickness for the underlying GFF, we obtain zero-one laws for the above for segments in the bulk of geodesics, thereby upgrading the results of Gwynne-Pfeffer-Sheffield '22. The primary technique used, developed in Bhatia-Kavvadias '25, is to lay down a large family of small-scale "typical" and well-behaved geodesics along interior segments of a long geodesic.

math.PR↗

arXiv2026

Regularity of the SLE$_4$ uniformizing map and the SLE$_8$ trace

We show that the modulus of continuity of the SLE$_4$ uniformizing map is given by $(\log δ^{-1})^{-1/3+o(1)}$ as $δ\to 0$. As a consequence of our analysis, we show that the Jones-Smirnov condition for conformal removability (with quasihyperbolic geodesics) does not hold for SLE$_4$. We also show that the modulus of continuity for SLE$_8$ with the capacity time parameterization is given by $(\log δ^{-1})^{-1/4+o(1)}$ as $δ\to 0$, proving a conjecture of Alvisio and Lawler.

math.PR↗

arXiv2026

Conformal removability of non-simple Schramm-Loewner evolutions

We consider the Schramm-Loewner evolution (SLE$_κ$) for $κ\in (4,8)$, which is the regime that the curve is self-intersecting but not space-filling. We let ${\mathcal K}$ be the set of $κ\in (4,8)$ for which the adjacency graph of connected components of the complement of an SLE$_κ$ is a.s. connected, meaning that for every pair of complementary components $U, V$ there exist complementary components $U_1,\ldots,U_n$ with $U_1 = U$, $U_n = V$, and $\partial U_i \cap \partial U_{i+1} \neq \emptyset$ for each $1 \leq i \leq n-1$. It was proved by Gwynne and Pfeffer that this set is non-empty. We show that the range of an SLE$_κ$ for $κ\in {\mathcal K}$ is a.s. conformally removable, which answers a question of Sheffield. As a step in the proof, we construct the canonical conformally covariant volume measure on the cut points of an SLE$_κ$ for $κ\in (4,8)$ and establish a precise upper bound on the measure that it assigns to any Borel set in terms of its diameter.

math.PR↗

arXiv2026

Two-sided heat kernel bounds for $\sqrt{8/3}$-Liouville Brownian motion

Liouville Brownian motion (LBM) is the canonical diffusion process on a Liouville quantum gravity (LQG) surface. In this work, we establish upper and lower bounds for the heat kernel for LBM when $γ=\sqrt{8/3}$ in terms of the $\sqrt{8/3}$-LQG metric which are sharp up to a polylogarithmic factor in the exponential.

math.PR↗

arXiv2025

Continuity in $ξ$ and $ξ\to \infty$ tightness of the LQG metric

We show that the law of the $γ$-LQG metric (appropriately renormalized) is continuous in $γ\in (0,2)$ with respect to the local uniform topology of metrics on $\mathbf{C} \times \mathbf{C}$ whenever $γ$ lies on compact subsets of $(0,2)$. Moreover we show that as $γ\to 0$, the $γ$-LQG metric (appropriately renormalized) converges to the Euclidean metric with respect to the local uniform topology of metrics on $\mathbf{C} \times \mathbf{C}$. More generally, we show that the law of the LQG metric with parameter $ξ>0$ (appropriately renormalized) is tight with respect to the topology on lower semicontinuous functions on $\mathbf{C} \times \mathbf{C}$ whenever $ξ$ lies on compact subsets of $(0,\infty)$, and any subsequential limit in law is non-trivial almost surely. If in addition we assume that the limit satisfies the triangle inequality almost surely, then it has the law of an LQG metric with an appropriate parameter $ξ$. Finally we examine the limit as $ξ\to \infty$, which is a regime that has not been studied before. More precisely we show that if $D_h^ξ$ denotes the LQG metric with parameter $ξ>0$ (appropriately renormalized) associated with the whole-plane GFF $h$, the family of metrics $(D_h^ξ)^{1 / ξ}$ is tight as $ξ\to \infty$ and any subsequential limit is non-trivial almost surely. If in addition we assume that the subsequential limit satisfies the triangle inequality almost surely, then the limit is a metric almost surely.

math.PR↗

arXiv2024

SLE$_κ(ρ)$ processes in the light cone regime on Liouville quantum gravity

We study the relationship between certain SLE$_κ(ρ)$ processes, which are variants of the Schramm-Loewner evolution with parameter $κ$ in which one keeps track of an extra marked point, and Liouville quantum gravity (LQG). These processes are defined whenever $ρ> -2-κ/2$ and in this work we will focus on the light cone regime, meaning that $κ\in (0,4)$ and $\max(κ/2-4,-2-κ/2) < ρ< -2$. Such processes are self-intersecting even though ordinary SLE$_κ$ curves are simple for $κ\in (0,4)$. We show that such a process drawn on top of an independent $\sqrtκ$-LQG surface called a weight $(ρ+4)$-quantum wedge can be represented as a gluing of a pair of trees which are described by the two coordinate functions of a correlated $α$-stable Lévy process with $α= 1-2(ρ+2)/κ$. Combined with another work, this shows that bipolar oriented random planar maps with large faces can be identified in the scaling limit with an SLE$_κ(κ-4)$ curve on an independent $\sqrtκ$-LQG surface for $κ\in (4/3,2)$.

math.PR↗

arXiv2024

Connectivity of the adjacency graph of complementary components of the SLE fan

Suppose that $h$ is an instance of the Gaussian free field (GFF) on a simply connected domain $D \subseteq {\mathbf C}$ and $x,y \in \partial D$ are distinct. Fix $κ\in (0,4)$ and for each $θ\in {\mathbf R}$ let $η_θ$ be the flow line of $h$ from $x$ to $y$. Recall that for $θ_1 < θ_2$ the fan ${\mathbf F}(θ_1,θ_2)$ of flow lines of $h$ from $x$ to $y$ is the closure of the union of $η_θ$ as $θ$ varies in any fixed countable dense subset of $[θ_1,θ_2]$. We show that the adjacency graph of components of $D \setminus {\mathbf F}(θ_1,θ_2)$ is a.s. connected, meaning it a.s. holds that for every pair $U,V$ of components there exist components $U_1,\ldots,U_n$ so that $U_1 = U$, $U_n = V$, and $\partial U_i \cap \partial U_{i+1} \neq \emptyset$ for each $1 \leq i \leq n-1$. We further show that ${\mathbf F}(θ_1,θ_2)$ a.s. determines the flow lines used in its construction. That is, for each $θ\in [θ_1,θ_2]$ we prove that $η_θ$ is a.s. determined by ${\mathbf F}(θ_1,θ_2)$ as a set.

math.PR↗

arXiv2022

Conformal removability of SLE$_4$

We consider the Schramm-Loewner evolution (SLE$_κ$) with $κ=4$, the critical value of $κ> 0$ at or below which SLE$_κ$ is a simple curve and above which it is self-intersecting. We show that the range of an SLE$_4$ curve is a.s. conformally removable, answering a question posed by Sheffield. Such curves arise as the conformal welding of a pair of independent critical ($γ=2$) Liouville quantum gravity (LQG) surfaces along their boundaries and our result implies that this conformal welding is unique. In order to establish this result, we give a new sufficient condition for a set $X \subseteq {\mathbf C}$ to be conformally removable which applies in the case that $X$ is not necessarily the boundary of a simply connected domain.

math.PR↗

arXiv2022

Bipolar oriented random planar maps with large faces and exotic SLE$_κ(ρ)$ processes

We consider bipolar oriented random planar maps with heavy-tailed face degrees. We show for each $α\in (1,2)$ that if the face degree is in the domain of attraction of an $α$-stable Lévy process, the corresponding random planar map has an infinite volume limit in the Benjamini-Schramm topology. We also show in the limit that the properly rescaled contour functions associated with the northwest and southeast trees converge in law to a certain correlated pair of $α$-stable Lévy processes. Combined with other work, this allows us to identify the scaling limit of the planar map with an SLE$_κ(ρ)$ process with $ρ= κ-4 < -2$ on $\sqrtκ$-Liouville quantum gravity for $κ\in (4/3,2)$ where $α, κ$ are related by $α= 4/κ-1$.

math.PR↗