arXiv subjects

Ewain Gwynne

Publications and source records attributed to Ewain Gwynne.

At least 19 recordsLinked to original sources

arXiv2026

Directed distances in spanning-tree-decorated planar maps: exact exponent, scaling limit and universality

We define a natural orientation on a spanning-tree-decorated planar map whereby, roughly speaking, each directed edge in the map is oriented to match the direction of the contour exploration of the spanning tree. We study directed distances (lengths of shortest directed paths) with respect to this orientation. We construct the Busemann function which measures directed distances to $\infty$ along a natural interface in the uniform infinite spanning-tree-decorated map. We show that this Busemann function, re-scaled appropriately, converges in law to a $3/2$-stable Lévy process. We also show that in a uniform spanning-tree-decorated map with $n$ edges, directed distances are typically of order $n^{1/3}$. Using a strong coupling argument, we deduce analogous statements for directed distances in other random planar maps in the $\sqrt 2$-Liouville quantum gravity (LQG) universality class, including uniform meandric systems and mated-CRT maps for $γ=\sqrt 2$. These results give the scaling dimension for a hypothetical directed version of the $\sqrt 2$-LQG metric. Our proof strategy is inspired by work of Borga and Gwynne (2025) on directed distances in bipolar-oriented triangulations.

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arXiv2026

Inverting the operation of conditioning a branching process on extinction

It is well-known that conditioning a supercritical (multi-type) branching process on the event that it eventually becomes extinct yields a subcritical branching process. We study the corresponding inverse problem: given a subcritical branching process, does there exist a supercritical branching process with the property that when we condition it on extinction, we get back the original subcritical branching process? We show that such a supercritical branching process (which we call a conjugate branching process) exists under mild hypotheses on the original subcritical branching process. We also show by example that if there are at least two types, then the conjugate branching process is not necessarily unique. Our results are relevant to the problem of constructing natural random planar maps whose scaling limit is given by supercritical Liouville quantum gravity. Moreover, conjugate branching processes can also be used to give alternative evolutionary hypotheses in cancer modeling.

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arXiv2026

Tightness of exponential metrics for log-correlated Gaussian fields in arbitrary dimension

We prove the tightness of a natural approximation scheme for an analog of the Liouville quantum gravity metric on $\mathbb R^d$ for arbitrary $d\geq 2$. More precisely, let $\{h_n\}_{n\geq 1}$ be a suitable sequence of Gaussian random functions which approximates a log-correlated Gaussian field on $\mathbb R^d$. Consider the family of random metrics on $\mathbb R^d$ obtained by weighting the lengths of paths by $e^{ξh_n}$, where $ξ> 0$ is a parameter. We prove that if $ξ$ belongs to the subcritical phase (which is defined by the condition that the distance exponent $Q(ξ)$ is greater than $\sqrt{2d}$), then after appropriate re-scaling, these metrics are tight and that every subsequential limit is a metric on $\mathbb R^d$ which induces the Euclidean topology. We include a substantial list of open problems.

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arXiv2026

Random walk reflected off of infinity, with applications to uniform spanning forests and supercritical Liouville quantum gravity

Let $\mathcal G$ be an infinite graph -- not necessarily one-ended -- on which the simple random walk is transient. We define a variant of the continuous-time random walk on $\mathcal G$ which reaches $\infty$ in finite time and "reflects off of $\infty$" infinitely many times. We show that the Aldous-Broder algorithm for the random walk reflected off of $\infty$ gives the free uniform spanning forest (FUSF) on $\mathcal G$. Furthermore, Wilson's algorithm for the random walk reflected off of $\infty$ gives the FUSF on $\mathcal G$ on the event that the FUSF is connected, but not in general. We also apply the theory of random walk reflected off of $\infty$ to study random planar maps in the universality class of supercritical Liouville quantum gravity (LQG), equivalently LQG with central charge in $(1,25)$. Such random planar maps are infinite, with uncountably many ends. We define a version of the Tutte embedding for such maps under which they conjecturally converge to LQG. We also make several conjectures regarding the qualitative behavior of stochastic processes on such maps -- including the FUSF and critical percolation.

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arXiv2026

Dimension lower bounds in random geometry via Lipschitz functions

We prove lower bounds for the Hausdorff dimensions of various natural sets associated with the Liouville quantum gravity (LQG) metric. We prove that the set of 3-star points (i.e., starting points of three disjoint geodesics) has Hausdorff dimension at least two with respect to the LQG metric, which is conjectured to be optimal. Our proof works for a general class of planar length metrics which also includes, e.g., Kendall's Poisson roads metric. We additionally prove a dimension lower bound of one for the set of 2-star points intersected with the boundary and for the metric net intersected with the boundary, as well as a dimension lower bound of two for the intersection of two metric nets. In the particular setting of LQG, we obtain sharper lower bounds for the Hausdorff dimensions of the set of 2-star points and the LQG metric net, with respect to both the Euclidean metric and the LQG metric. Our proofs are primarily topological. The key idea is to express the sets of interest in terms of non-constancy sets of Lipschitz functions.

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arXiv2026

CaTherine wheels from trees and Liouville quantum gravity

A CaTherine wheel is a space-filling curve $f : S^1\to S^2$ such that for every closed interval $J\subset S^1$, $f(J)$ is homeomorphic to a closed disk and $f(\partial J)$ is contained in $\partial f(J)$. A CaTherine wheel gives rise to a pair of disjoint, dense topological trees in $S^2$ which roughly speaking lie to the left and right of $f$. We give necessary and sufficient conditions for a topological tree in $S^2$ to arise as one of these trees for some CaTherine wheel $f$. We apply this result to show that there is a unique CaTherine wheel corresponding to the geodesic tree rooted at $\infty$ for the $γ$-Liouville quantum gravity (LQG) metric, for $γ\in (0,2)$. In other words, we construct the space-filling curve which is the contour exploration of the LQG geodesic tree.

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arXiv2026

Percolation of thick points of the log-correlated Gaussian field in high dimensions

We prove that the set of thick points of the log-correlated Gaussian field contains an unbounded path in sufficiently high dimensions. This contrasts with the two-dimensional case, where Aru, Papon, and Powell (2023) showed that the set of thick points is totally disconnected. This result has an interesting implication for the exponential metric of the log-correlated Gaussian field: in sufficiently high dimensions, when the parameter $ξ$ is large, the set-to-set distance exponent (if it exists) is negative. This suggests that a new phase may emerge for the exponential metric, which does not appear in two dimensions. In addition, we establish similar results for the set of thick points of the branching random walk. As an intermediate result, we also prove that the critical probability for fractal percolation converges to 0 as $d \to \infty$.

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arXiv2025

Joint scaling limit of a bipolar-oriented triangulation and its dual in the peanosphere sense

Kenyon, Miller, Sheffield, and Wilson (2015) showed how to encode a random bipolar-oriented planar map by means of a random walk with a certain step size distribution. Using this encoding together with the mating-of-trees construction of Liouville quantum gravity (LQG) due to Duplantier, Miller, and Sheffield (2014), they proved that random bipolar-oriented planar maps converge in the scaling limit to a $\sqrt{4/3}$-LQG surface decorated by an independent SLE$_{12}$ in the peanosphere sense, meaning that the height functions of a particular pair of trees on the maps converge in the scaling limit to the correlated planar Brownian motion which encodes the SLE-decorated LQG surface. We improve this convergence result by proving that the pair of height functions for an infinite-volume random bipolar-oriented triangulation and the pair of height functions for its dual map converge jointly in law in the scaling limit to the two planar Brownian motions which encode the same $\sqrt{4/3}$-LQG surface decorated by both an SLE$_{12}$ curve and the ``dual'' SLE$_{12}$ curve which travels in a direction perpendicular (in the sense of imaginary geometry) to the original curve. This confirms a conjecture of Kenyon, Miller, Sheffield, and Wilson (2015). Our paper is the starting point of recent works connecting LQG and random permutons such as the Baxter permuton.

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arXiv2025

Directed distances in bipolar-oriented triangulations: exact exponents and scaling limits

We study longest and shortest directed paths in the following natural model of directed random planar maps: the uniform infinite bipolar-oriented triangulation (UIBOT), which is the local limit of uniform bipolar-oriented triangulations around a typical edge. We construct the Busemann function which measures directed distance to $\infty$ along a natural interface in the UIBOT. We show that in the case of longest (resp.\ shortest) directed paths, this Busemann function converges in the scaling limit to a $2/3$-stable Lévy process (resp.\ a $4/3$-stable Lévy process). We also prove up-to-constants bounds for directed distances in finite bipolar-oriented triangulations sampled from a Boltzmann distribution, and for size-$n$ cells in the UIBOT. These bounds imply that in a typical subset of the UIBOT with $n$ edges, longest directed path lengths are of order $n^{3/4}$ and shortest directed path lengths are of order $n^{3/8}$. These results give the scaling dimensions for discretizations of the (hypothetical) $\sqrt{4/3}$-directed Liouville quantum gravity metrics. The main external input in our proof is the bijection of Kenyon-Miller-Sheffield-Wilson (2015). We do not use any continuum theory. We expect that our techniques can also be applied to prove similar results for directed distances in other random planar map models and for longest increasing subsequences in pattern-avoiding permutations.

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arXiv2025

Random walk on sphere packings and Delaunay triangulations in arbitrary dimension

We prove that random walks on a family of tilings of d-dimensional Euclidean space, with a canonical choice of conductances, converge to Brownian motion modulo time parameterization. This class of tilings includes Delaunay triangulations (the dual of Voronoi tesselations) and sphere packings. Our regularity assumptions are deterministic and mild. For example, our results apply to Delaunay triangulations with vertices sampled from a d-dimensional Gaussian multiplicative chaos measure. As part of our proof, we establish the uniform convergence of certain finite volume schemes for the Laplace equation, with quantitative bounds on the rate of convergence. In the special case of two dimensions, we give a new, short proof of the main result of Gurel-Gurevich--Jerison--Nachmias (2020).

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arXiv2025

Area measures and branched polymers in supercritical Liouville quantum gravity

We study Liouville quantum gravity (LQG) in the supercritical (a.k.a. strongly coupled) phase, which has background charge $Q \in (0,2)$ and central charge $\mathbf{c}_{\mathrm{L}} = 1+6Q^2 \in (1,25)$. Recent works have shown how to define LQG in this phase as a planar random geometry associated with a variant of the Gaussian free field, which exhibits "infinite spikes." In contrast, a number of results from physics, dating back to the 1980s, suggest that supercritical LQG surfaces should behave like "branched polymers": i.e., they should look like the continuum random tree. We prove a result which reconciles these two descriptions of supercritical LQG. More precisely, we show that for a family of random planar maps with boundary in the universality class of supercritical LQG, if we condition on the (small probability) event that the planar map is finite, then the scaling limit is the continuum random tree. We also show that there does not exist any locally finite measure associated with supercritical LQG which is locally determined by the field and satisfies the LQG coordinate change formula. Our proofs are based on a branching process description of supercritical LQG which comes from its coupling with CLE$_4$ (Ang and Gwynne, arXiv:2308.11832).

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arXiv2025

Permutons, meanders, and SLE-decorated Liouville quantum gravity

We study a class of random permutons which can be constructed from a pair of space-filling Schramm-Loewner evolution (SLE) curves on a Liouville quantum gravity (LQG) surface. This class includes the skew Brownian permutons introduced by Borga (2021), which describe the scaling limit of various types of random pattern-avoiding permutations. Another interesting permuton in our class is the meandric permuton, which corresponds to two independent SLE$_8$ curves on a $γ$-LQG surface with $γ= \sqrt{\frac13 \left( 17 - \sqrt{145} \right)}$. Building on work by Di Francesco, Golinelli, and Guitter (2000), we conjecture that the meandric permuton describes the scaling limit of uniform meandric permutations, i.e., the permutations induced by a simple loop in the plane which crosses a line a specified number of times. We show that for any sequence of random permutations which converges to one of the above random permutons, the length of the longest increasing subsequence is sublinear. This proves that the length of the longest increasing subsequence is sublinear for Baxter, strong-Baxter, and semi-Baxter permutations and leads to the conjecture that the same is true for meandric permutations. We also prove that the closed support of each of the random permutons in our class has Hausdorff dimension one. Finally, we prove a re-rooting invariance property for the meandric permuton and write down a formula for its expected pattern densities in terms of LQG correlation functions (which are known explicitly) and the probability that an SLE$_8$ hits a given set of points in numerical order (which is not known explicitly). We conclude with a list of open problems.

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arXiv2025

Supercritical Liouville quantum gravity and CLE$_4$

We establish the first relationship between Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG) in the supercritical (a.k.a. strongly coupled) phase, which corresponds to central charge values $\mathbf c_{\mathrm L} \in (1,25)$ or equivalently to complex values of $γ$ with $|γ|=2$. More precisely, we introduce a canonical supercritical LQG surface with the topology of the disk. We then show that for each $\mathbf c_{\mathrm L} \in (1,25)$ there is a coupling of this LQG surface with a conformal loop ensemble with parameter $κ=4$ (CLE$_4$) wherein the LQG surfaces parametrized by the regions enclosed by the CLE$_4$ loops are conditionally independent supercritical LQG disks given their boundary lengths. In this coupling, the CLE$_4$ is neither determined by nor independent from the LQG. Guided by our coupling result, we exhibit a combinatorially natural family of loop-decorated random planar maps whose scaling limit we conjecture to be the supercritical LQG disk coupled to CLE$_4$. We include a substantial list of open problems.

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arXiv2025

Liouville Brownian motion and quantum cones in dimension $d > 2$

For $d > 2$ and $γ\in (0, \sqrt{2d})$, we study the Liouville Brownian motion associated with the whole-space log-correlated Gaussian field in $\mathbb{R}^d$. We compute its spectral dimension, i.e., the short-time asymptotics of the heat kernel along the diagonal, which, in contrast to the two-dimensional case, depends on both $γ$ and on the thickness of the starting point. Furthermore, for even dimensions $d > 2$, we show that the spherical average process of the whole-space log-correlated Gaussian field in $\mathbb{R}^d$ can be identified with the integral of a stationary Gaussian Markov process of order $(d-2)/2$. Exploiting this representation, we construct the higher-dimensional analogue of the $β$-quantum cone for $β\in (-\infty, Q)$, with $Q = d/γ+ γ/2$. Lastly, for $α= Q - \sqrt{Q^2-4}$, we prove that the law of the $d$-dimensional $α$-quantum cone is invariant under shifts along the trajectories of the associated Liouville Brownian motion.

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arXiv2024

Critical Liouville quantum gravity and CLE$_4$

Consider a critical ($γ=2$) Liouville quantum gravity (LQG) disk together with an independent conformal loop ensemble (CLE) with parameter $κ=4$. We show that the critical LQG surfaces parametrized by the regions enclosed by the CLE$_4$ loops are conditionally independent critical LQG disks given the LQG lengths of the loops. We also show that the joint law of the LQG lengths of the loops is described in terms of the jumps of a certain $3/2$-stable process. Our proofs are via a limiting argument based on the analogous statements for $γ\in (\sqrt{8/3},2)$ and $κ= γ^2 \in (8/3,4)$ which were proven by Miller, Sheffield, and Werner (2020). Our results are used in the construction of a coupling of supercritical LQG with CLE$_4$ in another paper by the same authors.

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arXiv2024

Harmonic balls in Liouville quantum gravity

Harmonic balls are domains which satisfy the mean-value property for harmonic functions. We establish the existence and uniqueness of harmonic balls on Liouville quantum gravity (LQG) surfaces using the obstacle problem formulation of Hele-Shaw flow. We show that LQG harmonic balls are neither Lipschitz domains nor LQG metric balls, and that the boundaries of their complementary connected components are Jordan curves. We conjecture that LQG harmonic balls are the scaling limit of internal diffusion limited aggregation (IDLA) on random planar maps. In a companion paper, we prove this in the special case of mated-CRT maps.

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arXiv2024

A support theorem for exponential metrics of log-correlated Gaussian fields in arbitrary dimension

Let $h$ be a log-correlated Gaussian field on $\R^d$, let $γ\in (0,\sqrt{2d}),$ let $μ_h$ be the $γ$-Gaussian multiplicative chaos measure, and let $D_h$ be an exponential metric associated with $h$ satisfying certain natural axioms. In the special case when $d=2$, this corresponds to the Liouville quantum gravity (LQG) measure and metric. We show that the closed support of the law of $(D_h,μ_h)$ includes all length metrics and probability measures on $\R^d$. That is, if $\mathfrak d$ is any length metric on $\R^d$ and $\mathfrak m$ is any probability measure on $\R^d$, then with positive probability $(D_h , μ_h)$ is close to $(\mathfrak d , \mathfrak m)$ with respect to the uniform distance and the Prokhorov distance. Key ingredients include a scaling limit theorem for a first passage percolation type model associated with $h$, a special version of the white noise decomposition of $h$ in arbitrary dimension, and an approximation property by conformally flat Riemannian metrics in the uniform sense. Our results provide a robust tool to show that the LQG measure and metric, and its higher dimensional analogs, satisfy certain properties with positive probability.

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arXiv2024

Scaling limits of planar maps under the Smith embedding

The Smith embedding of a finite planar map with two marked vertices, possibly with conductances on the edges, is a way of representing the map as a tiling of a finite cylinder by rectangles. In this embedding, each edge of the planar map corresponds to a rectangle, and each vertex corresponds to a horizontal segment. Given a sequence of finite planar maps embedded in an infinite cylinder, such that the random walk on both the map and its planar dual converges to Brownian motion modulo time change, we prove that the a priori embedding is close to an affine transformation of the Smith embedding at large scales. By applying this result, we prove that the Smith embeddings of mated-CRT maps with the sphere topology converge to $γ$-Liouville quantum gravity ($γ$-LQG).

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