arXiv subjects
Publications and source records attributed to Ewain Gwynne.
We present a proof of the folklore result that any length metric on $\mathbb R^d$ can be approximated by conformally flat Riemannian distance functions in the uniform distance. This result is used to study Liouville quantum gravity in another paper by the same authors.
Permutons constructed from a Liouville quantum gravity surface and a pair of space-filling Schramm-Loewner evolutions (SLEs) have been shown -- or are conjectured -- to describe the scaling limit of various natural models of random constrained permutations. We prove that, in two distinct and natural settings, these permutons uniquely determine, modulo rotation, scaling, translation and reflection, both the Liouville quantum gravity surface and the pair of space-filling SLEs used in their construction. In other words, the Liouville quantum gravity surface and the pair of space-filling SLEs can be deterministically reconstructed from the permuton. Our results cover the cases of the skew Brownian permutons, the universal limits of pattern-avoiding permutations, and the meandric permuton, which is the conjectural permuton limit of permutations obtained from uniform meanders. In the course of the proof, we give a detailed description of how the support of the permuton relates to the multiple points of the two space-filling SLEs.
We investigate the notion of curvature in the context of Liouville quantum gravity (LQG) surfaces. We define the Gaussian curvature for LQG, which we conjecture is the scaling limit of discrete curvature on random planar maps. Motivated by this, we study asymptotics for the discrete curvature of $ε$-mated CRT maps. More precisely, we prove that the discrete curvature integrated against a $C_c^2$ test function is of order $ε^{o(1)},$ which is consistent with our scaling limit conjecture. On the other hand, we prove the total discrete curvature on a fixed space-filling SLE segment scaled by $ε^{\frac{1}{4}}$ converges in distribution to an explicit random variable.
There are many deep and useful theorems relating Schramm-Loewner evolution (SLE$_κ$) and Liouville quantum gravity ($γ$-LQG) in the case when the parameters satisfy $κ\in \{γ^2, 16/γ^2\}$. Roughly speaking, these theorems say that the SLE$_κ$ curve cuts the $γ$-LQG surface into two or more independent $γ$-LQG surfaces. We extend these theorems to the case when $κ\not\in \{γ^2, 16/γ^2\}$. Roughly speaking we show that if we have an appropriate variant of SLE$_κ$ and an independent $γ$-LQG disk, then the SLE curve cuts the LQG disk into two or more $γ$-LQG surfaces which are conditionally independent given the values along the SLE curve of a certain collection of auxiliary imaginary geometry fields, viewed modulo conformal coordinate change. These fields are sampled independently from the SLE and the LQG and have the property that that the sum of the central charges associated with the SLE$_κ$ curve, the $γ$-LQG surface, and the auxiliary fields is 26. This condition on the central charge is natural from the perspective of bosonic string theory. We also prove analogous statements when the SLE curve is replaced by, e.g., an LQG metric ball or a Brownian motion path. Statements of this type were conjectured by Sheffield and are continuum analogs of certain Markov properties of random planar maps decorated by two or more statistical physics models. We include a substantial list of open problems.
We prove a shape theorem for internal diffusion limited aggregation on mated-CRT maps, a family of random planar maps which approximate Liouville quantum gravity (LQG) surfaces. The limit is an LQG harmonic ball, which we constructed in a companion paper. We also prove an analogous result for the divisible sandpile.
The Brownian separable permutons are a one-parameter family -- indexed by $p\in(0,1)$ -- of universal limits of random constrained permutations. We show that for each $p\in (0,1)$, there are explicit constants $1/2 < α_*(p) \leq β^*(p) < 1$ such that the length of the longest increasing subsequence in a random permutation of size $n$ sampled from the Brownian separable permuton is between $n^{α_*(p) - o(1)}$ and $n^{β^*(p) + o(1)}$ with probability tending to 1 as $n\to\infty$. In the symmetric case $p=1/2$, we have $α_*(p) \approx 0.812$ and $β^*(p)\approx 0.975$. We present numerical simulations which suggest that the lower bound $α_*(p)$ is close to optimal in the whole range $p\in(0,1)$. Our results work equally well for the closely related Brownian cographons. In this setting, we show that for each $p\in (0,1)$, the size of the largest clique (resp. independent set) in a random graph on $n$ vertices sampled from the Brownian cographon is between $n^{α_*(p) - o(1)}$ and $n^{β^*(p) + o(1)}$ (resp. $n^{α_*(1-p) - o(1)}$ and $n^{β^*(1-p) + o(1)}$) with probability tending to 1 as $n\to\infty$. Our proofs are based on the analysis of a fragmentation process embedded in a Brownian excursion introduced by Bertoin (2002). We expect that our techniques can be extended to prove similar bounds for uniform separable permutations and uniform cographs.
A Liouville quantum gravity (LQG) surface is a natural random two-dimensional surface, initially formulated as a random measure space and later as a random metric space. We show that the LQG measure can be recovered as the Minkowski measure with respect to the LQG metric, answering a question of Gwynne and Miller (arXiv:1905.00383). As a consequence, we prove that the metric structure of a $γ$-LQG surface determines its conformal structure for every $γ\in (0,2)$. Our primary tool is the continuum mating-of-trees theory for space-filling SLE. In the course of our proof, we also establish a Hölder continuity result for space-filling SLE with respect to the LQG metric.
A meandric system of size $n$ is the set of loops formed from two arc diagrams (non-crossing perfect matchings) on $\{1,\dots,2n\}$, one drawn above the real line and the other below the real line. A uniform random meandric system can be viewed as a random planar map decorated by a Hamiltonian path (corresponding to the real line) and a collection of loops (formed by the arcs). Based on physics heuristics and numerical evidence, we conjecture that the scaling limit of this decorated random planar map is given by an independent triple consisting of a Liouville quantum gravity (LQG) sphere with parameter $γ=\sqrt 2$, a Schramm-Loewner evolution (SLE) curve with parameter $κ=8$, and a conformal loop ensemble (CLE) with parameter $κ=6$. We prove several rigorous results which are consistent with this conjecture. In particular, a uniform meandric system admits loops of nearly macroscopic graph-distance diameter with high probability. Furthermore, a.s., the uniform infinite meandric system with boundary has no infinite path. But, a.s., its boundary-modified version has a unique infinite path whose scaling limit is conjectured to be chordal SLE$_6$.
Let $h$ be the planar Gaussian free field and let $D_h$ be a supercritical Liouville quantum gravity (LQG) metric associated with $h$. Such metrics arise as subsequential scaling limits of supercritical Liouville first passage percolation (Ding-Gwynne, 2020) and correspond to values of the matter central charge $\mathbf{c}_{\mathrm M} \in (1,25)$. We show that a.s. the boundary of each complementary connected component of a $D_h$-metric ball is a Jordan curve and is compact and finite-dimensional with respect to $D_h$. This is in contrast to the \emph{whole} boundary of the $D_h$-metric ball, which is non-compact and infinite-dimensional with respect to $D_h$ (Pfeffer, 2021). Using our regularity results for boundaries of complementary connected components of $D_h$-metric balls, we extend the confluence of geodesics results of Gwynne-Miller (2019) to the case of supercritical Liouville quantum gravity. These results show that two $D_h$-geodesics with the same starting point and different target points coincide for a non-trivial initial time interval.
We study the Loewner evolution whose driving function is $W_t = B_t^1 + i B_t^2$, where $(B^1,B^2)$ is a pair of Brownian motions with a given covariance matrix. This model can be thought of as a generalization of Schramm-Loewner evolution (SLE) with complex parameter values. We show that our Loewner evolutions behave very differently from ordinary SLE. For example, if neither $B^1$ nor $B^2$ is identically equal to zero, then the set of points disconnected from $\infty$ by the Loewner hull has non-empty interior at each time. We also show that our model exhibits three phases analogous to the phases of SLE: a phase where the hulls have zero Lebesgue measure, a phase where points are swallowed but not hit by the hulls, and a phase where the hulls are space-filling. The phase boundaries are expressed in terms of the signs of explicit integrals. These boundaries have a simple closed form when the correlation of the two Brownian motions is zero.
Liouville quantum gravity (LQG) is a one-parameter family of models of random fractal surfaces which first appeared in the physics literature in the 1980s. Recent works have constructed a metric (distance function) on an LQG surface. We give an overview of the construction of this metric and discuss some of its most important properties, such as the behavior of geodesics and the KPZ formula. We also discuss some of the main techniques for proving statements about the LQG metric, give examples of their use, and discuss some open problems.
We survey the theory and applications of mating-of-trees bijections for random planar maps and their continuum analog: the mating-of-trees theorem of Duplantier, Miller, and Sheffield (2014). The latter theorem gives an encoding of a Liouville quantum gravity (LQG) surface decorated by a Schramm-Loewner evolution (SLE) curve in terms of a pair of correlated linear Brownian motions. We assume minimal familiarity with the theory of SLE and LQG. Mating-of-trees theory enables one to reduce problems about SLE and LQG to problems about Brownian motion and leads to deep rigorous connections between random planar maps and LQG. Applications discussed in this article include scaling limit results for various functionals of decorated random planar maps, estimates for graph distances and random walk on (not necessarily uniform) random planar maps, computations of the Hausdorff dimensions of sets associated with SLE, scaling limit results for random planar maps conformally embedded in the plane, and special symmetries for $\sqrt{8/3}$-LQG which allow one to prove its equivalence with the Brownian map.
We show that for each ${\mathbf c}_{\mathrm M} \in [1,25)$, there is a unique metric associated with Liouville quantum gravity (LQG) with matter central charge ${\mathbf c}_{\mathrm M}$. An earlier series of works by Ding-Dubédat-Dunlap-Falconet, Gwynne-Miller, and others showed that such a metric exists and is unique in the subcritical case ${\mathbf c}_{\mathrm M} \in (-\infty,1)$, which corresponds to coupling constant $γ\in (0,2)$. The critical case ${\mathbf c}_{\mathrm M} = 1$ corresponds to $γ=2$ and the supercritical case ${\mathbf c}_{\mathrm M} \in (1,25)$ corresponds to $γ\in \mathbb C$ with $|γ| = 2$. Our metric is constructed as the limit of an approximation procedure called Liouville first passage percolation, which was previously shown to be tight for $\mathbf c_{\mathrm M} \in [1,25)$ by Ding and Gwynne (2020). In this paper, we show that the subsequential limit is uniquely characterized by a natural list of axioms. This extends the characterization of the LQG metric proven by Gwynne and Miller (2019) for $\mathbf c_{\mathrm M} \in (-\infty,1)$ to the full parameter range $\mathbf c_{\mathrm M} \in (-\infty,25)$. Our argument is substantially different from the proof of the characterization of the LQG metric for $\mathbf c_{\mathrm M} \in (-\infty,1)$. In particular, the core part of the argument is simpler and does not use confluence of geodesics.
We prove a scaling limit result for random walk on certain random planar maps with its natural time parametrization. In particular, we show that for $γ\in (0,2)$, the random walk on the mated-CRT map with parameter $γ$ converges to $γ$-Liouville Brownian motion, the natural quantum time parametrization of Brownian motion on a $γ$-Liouville quantum gravity (LQG) surface. Our result applies if the mated-CRT map is embedded into the plane via the embedding which comes from SLE / LQG theory or via the Tutte embedding (a.k.a. the harmonic or barycentric embedding). In both cases, the convergence is with respect to the local uniform topology on curves and it holds in the quenched sense, i.e., the conditional law of the walk given the map converges. Previous work by Gwynne, Miller, and Sheffield (2017) showed that the random walk on the mated-CRT map converges to Brownian motion modulo time parametrization. This is the first work to show the convergence of the parametrized walk. As an intermediate result of independent interest, we derive an axiomatic characterisation of Liouville Brownian motion, for which the notion of Revuz measure of a Markov process plays a crucial role.
Recent works have shown that there is a canonical way to to assign a metric (distance function) to a Liouville quantum gravity (LQG) surface for any parameter $γ\in (0,2)$. We establish a strong confluence property for LQG geodesics, which generalizes a result proven by Angel, Kolesnik and Miermont for the Brownian map. Using this property, we also establish zero-one laws for the Hausdorff dimensions of geodesics, metric ball boundaries, and metric nets w.r.t. the Euclidean or LQG metric. In the case of a metric ball boundary, our result combined with earlier work of Gwynne (2020) gives a formula for the a.s. Hausdorff dimension for the boundary of the metric ball stopped when it hits a fixed point in terms of the Hausdorff dimension of the whole LQG surface. We also show that the Hausdorff dimension of the metric ball boundary is carried by points which are not on the boundary of any complementary connected component of the ball.
Discrete Liouville first passage percolation (LFPP) with parameter $ξ> 0$ is the random metric on a sub-graph of $\mathbb Z^2$ obtained by assigning each vertex $z$ a weight of $e^{ξh(z)}$, where $h$ is the discrete Gaussian free field. We show that the distance exponent for discrete LFPP is strictly positive for all $ξ> 0$. More precisely, the discrete LFPP distance between the inner and outer boundaries of a discrete annulus of size $2^n$ is typically at least $2^{αn}$ for an exponent $α> 0$ depending on $ξ$. This is a crucial input in the proof that LFPP admits non-trivial subsequential scaling limits for all $ξ> 0$ and also has theoretical implications for the study of distances in Liouville quantum gravity.
Liouville first passage percolation (LFPP) with parameter $ξ> 0$ is the family of random distance functions $\{D_h^ε\}_{ε>0}$ on the plane obtained by integrating $e^{ξh_ε}$ along paths, where $\{h_ε\}_{ε>0}$ is a smooth mollification of the planar Gaussian free field. Recent works have shown that for all $ξ> 0$ the LFPP metrics, appropriately re-scaled, admit non-trivial subsequential limiting metrics. In the case when $ξ< ξ_{\mathrm{crit}} \approx 0.41$, it has been shown that the subsequential limit is unique and defines a metric on $γ$-Liouville quantum gravity $γ= γ(ξ) \in (0,2)$. We prove that for all $ξ> 0$, each possible subsequential limiting metric is nearly bi-Lipschitz equivalent to the LFPP metric $D_h^ε$ when $ε$ is small, even if $ε$ does not belong to the appropriate subsequence. Using this result, we obtain bounds for the scaling constants for LFPP which are sharp up to polylogarithmic factors. We also prove that any two subsequential limiting metrics are bi-Lipschitz equivalent. Our results are an input in subsequent work which shows that the subsequential limits of LFPP induce the same topology as the Euclidean metric when $ξ= ξ_{\mathrm{crit}}$. We expect that our results will also have several other uses, e.g., in the proof that the subsequential limit is unique when $ξ\geq ξ_{\mathrm{crit}}$.
We show that every possible metric associated with critical ($γ=2$) Liouville quantum gravity (LQG) induces the same topology on the plane as the Euclidean metric. More precisely, we show that the optimal modulus of continuity of the critical LQG metric with respect to the Euclidean metric is a power of $1/\log(1/|\cdot|)$. Our result applies to every possible subsequential limit of critical Liouville first passage percolation, a natural approximation scheme for the LQG metric which was recently shown to be tight.