arXiv subjects

Ewain Gwynne

Publications and source records attributed to Ewain Gwynne.

At least 73 records · Page 4Linked to original sources

arXiv2020

Confluence of geodesics in Liouville quantum gravity for $γ\in (0,2)$

We prove that for any metric which one can associate with a Liouville quantum gravity (LQG) surface for $γ\in (0,2)$ satisfying certain natural axioms, its geodesics exhibit the following confluence property. For any fixed point $z$, a.s.\ any two $γ$-LQG geodesics started from distinct points other than $z$ must merge into each other and subsequently coincide until they reach $z$. This is analogous to the confluence of geodesics property for the Brownian map proven by Le Gall (2010). Our results apply for the subsequential limits of Liouville first passage percolation and are an important input in the proof of the existence and uniqueness of the LQG metric for all $γ\in (0,2)$.

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arXiv2019

KPZ formulas for the Liouville quantum gravity metric

Let $γ\in (0,2)$, let $h$ be the planar Gaussian free field, and let $D_h$ be the associated $γ$-Liouville quantum gravity (LQG) metric. We prove that for any random Borel set $X \subset \mathbb{C}$ which is independent from $h$, the Hausdorff dimensions of $X$ with respect to the Euclidean metric and with respect to the $γ$-LQG metric $D_h$ are a.s. related by the (geometric) KPZ formula. As a corollary, we deduce that the Hausdorff dimension of the continuum $γ$-LQG metric is equal to the exponent $d_γ> 2$ studied by Ding and Gwynne (2018), which describes distances in discrete approximations of $γ$-LQG such as random planar maps. We also derive "worst-case" bounds relating the Euclidean and $γ$-LQG dimensions of $X$ when $X$ and $h$ are not necessarily independent, which answers a question posed by Aru (2015). Using these bounds, we obtain an upper bound for the Euclidean Hausdorff dimension of a $γ$-LQG geodesic which equals $1.312\dots$ when $γ= \sqrt{8/3}$; and an upper bound of $1.9428\dots$ for the Euclidean Hausdorff dimension of a connected component of the boundary of a $\sqrt{8/3}$-LQG metric ball. We use the axiomatic definition of the $γ$-LQG metric, so the paper can be understood by readers with minimal background knowledge beyond a basic level of familiarity with the Gaussian free field.

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arXiv2019

Liouville quantum gravity with matter central charge in $(1,25)$: a probabilistic approach

There is a substantial literature concerning Liouville quantum gravity (LQG) in two dimensions with conformal matter field of central charge ${\mathbf{c}}_{\mathrm M}\in(-\infty,1]$. Via the DDK ansatz, LQG can equivalently be described as the random geometry obtained by exponentiating $γ$ times a variant of the planar Gaussian free field (GFF), where $γ\in(0,2]$ satisfies $\mathbf c_{\mathrm M}=25-6(2/γ+γ/2)^2$. Physics considerations suggest that LQG should also make sense in the regime when $\mathbf c_{\mathrm M}>1$. However, the behavior in this regime is rather mysterious in part because the corresponding value of $γ$ is complex, so analytic continuations of various formulas give complex answers which are difficult to interpret in a probabilistic setting. We introduce and study a discretization of LQG which makes sense for all values of $\mathbf c_{\mathrm M}\in(-\infty,25)$. Our discretization consists of a random planar map, defined as the adjacency graph of a tiling of the plane by dyadic squares which all have approximately the same "LQG size" with respect to the GFF. We prove that several formulas for dimension-related quantities are still valid for $\mathbf c_{\mathrm M}\in(1,25)$, with the caveat that the dimension is infinite when the formulas give a complex answer. In particular, we prove an extension of the (geometric) KPZ formula for $\mathbf c_{\mathrm M}\in(1,25)$, which gives a finite quantum dimension iff the Euclidean dimension is at most $(25-\mathbf c_{\mathrm M})/12$. We also show that the graph distance between typical points with respect to our discrete model grows polynomially whereas the cardinality of a graph distance ball of radius $r$ grows faster than any power of $r$ (which suggests that the Hausdorff dimension of LQG is infinite for $\mathbf c_{\mathrm M}\in(1,25)$). We include a substantial list of open problems.

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arXiv2019

Conformal invariance of CLE$_κ$ on the Riemann sphere for $κ\in (4,8)$

The conformal loop ensemble (CLE) is the canonical conformally invariant probability measure on non-crossing loops in a simply connected domain in $\mathbb C$ and is indexed by a parameter $κ\in (8/3,8)$. We consider CLE$_κ$ on the whole-plane in the regime in which the loops are self-intersecting ($κ\in (4,8)$) and show that it is invariant under the inversion map $z \mapsto 1/z$. This shows that whole-plane CLE$_κ$ for $κ\in (4,8)$ defines a conformally invariant measure on loops on the Riemann sphere. The analogous statement in the regime in which the loops are simple ($κ\in (8/3,4]$) was proven by Kemppainen and Werner and together with the present work covers the entire range $κ\in (8/3,8)$ for which CLE$_κ$ is defined. As an intermediate step in the proof, we show that CLE$_κ$ for $κ\in (4,8)$ on an annulus, with any specified number of inner-boundary-surrounding loops, is well-defined and conformally invariant.

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arXiv2019

Dimension transformation formula for conformal maps into the complement of an SLE curve

We prove a formula relating the Hausdorff dimension of a deterministic Borel subset of $\mathbb R$ and the Hausdorff dimension of its image under a conformal map from the upper half-plane to a complementary connected component of an SLE$_κ$ curve for $κ\not =4$. Our proof is based on the relationship between SLE and Liouville quantum gravity together with the one-dimensional KPZ formula of Rhodes-Vargas (2011) and the KPZ formula of Gwynne-Holden-Miller (2015). As an intermediate step we prove a KPZ formula which relates the Euclidean dimension of a subset of an SLE$_κ$ curve for $κ\in (0,4)\cup(4,8)$ and the dimension of the same set with respect to the $γ$-quantum natural parameterization of the curve induced by an independent Gaussian free field, $γ= \sqrt κ\wedge (4/\sqrtκ)$.

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arXiv2019

Convergence of the self-avoiding walk on random quadrangulations to SLE$_{8/3}$ on $\sqrt{8/3}$-Liouville quantum gravity

We prove that a uniform infinite quadrangulation of the half-plane decorated by a self-avoiding walk (SAW) converges in the scaling limit to the metric gluing of two independent Brownian half-planes identified along their positive boundary rays. Combined with other work of the authors, this implies the convergence of the SAW on a random quadrangulation to SLE$_{8/3}$ on a certain $\sqrt{8/3}$-Liouville quantum gravity surface. The topology of convergence is the local Gromov-Hausdorff-Prokhorov-uniform topology, the natural generalization of the local Gromov-Hausdorff topology to curve-decorated metric measure spaces. We also prove analogous scaling limit results for uniform infinite quadrangulations of the whole plane decorated by either a one-sided or two-sided SAW. Our proof uses only the peeling procedure for random quadrangulations and some basic properties of the Brownian half-plane, so can be read without any knowledge of SLE or LQG.

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arXiv2019

The Tutte embedding of the Poisson-Voronoi tessellation of the Brownian disk converges to $\sqrt{8/3}$-Liouville quantum gravity

Recent works have shown that an instance of a Brownian surface (such as the Brownian map or Brownian disk) a.s. has a canonical conformal structure under which it is equivalent to a $\sqrt{8/3}$-Liouville quantum gravity (LQG) surface. In particular, Brownian motion on a Brownian surface is well-defined. The construction in these works is indirect, however, and leaves open a basic question: is Brownian motion on a Brownian surface the limit of simple random walk on increasingly fine discretizations of that surface, the way Brownian motion on $\mathbb R^2$ is the $ε\to 0$ limit of simple random walk on $ε\mathbb Z^2$? We answer this question affirmatively by showing that Brownian motion on a Brownian surface is (up to time change) the $λ\to \infty$ limit of simple random walk on the Voronoi tessellation induced by a Poisson point process whose intensity is $λ$ times the associated area measure. Among other things, this implies that as $λ\to \infty$ the Tutte embedding (a.k.a. harmonic embedding) of the discretized Brownian disk converges to the canonical conformal embedding of the continuum Brownian disk, which in turn corresponds to $\sqrt{8/3}$-LQG. Along the way, we obtain other independently interesting facts about conformal embeddings of Brownian surfaces, including information about the Euclidean shapes of embedded metric balls and Voronoi cells. For example, we derive moment estimates that imply, in a certain precise sense, that these shapes are unlikely to be very long and thin.

math.PR↗

arXiv2019

Connectivity properties of the adjacency graph of SLE$_κ$ bubbles for $κ\in (4,8)$

We study the adjacency graph of bubbles---i.e., complementary connected components---of an SLE$_κ$ curve for $κ\in (4,8)$, with two such bubbles considered to be adjacent if their boundaries intersect. We show that this adjacency graph is a.s. connected for $κ\in (4,κ_0]$, where $κ_0 \approx 5.6158$ is defined explicitly. This gives a partial answer to a problem posed by Duplantier, Miller and Sheffield (2014). Our proof in fact yields a stronger connectivity result for $κ\in (4,κ_0]$, which says that there is a Markovian way of finding a path from any fixed bubble to $\infty$. We also show that there is a (non-explicit) $κ_1 \in (κ_0, 8)$ such that this stronger condition does not hold for $κ\in [κ_1,8)$. Our proofs are based on an encoding of SLE$_κ$ in terms of a pair of independent $κ/4$-stable processes, which allows us to reduce our problem to a problem about stable processes. In fact, due to this encoding, our results can be re-phrased as statements about the connectivity of the adjacency graph of loops when one glues together an independent pair of so-called $κ/4$-stable looptrees, as studied, e.g., by Curien and Kortchemski (2014). The above encoding comes from the theory of Liouville quantum gravity (LQG), but the paper can be read without any knowledge of LQG if one takes the encoding as a black box.

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arXiv2019

An almost sure KPZ relation for SLE and Brownian motion

The peanosphere construction of Duplantier, Miller, and Sheffield provides a means of representing a $γ$-Liouville quantum gravity (LQG) surface, $γ\in (0,2)$, decorated with a space-filling form of Schramm's SLE$_κ$, $κ= 16/γ^2 \in (4,\infty)$, $η$ as a gluing of a pair of trees which are encoded by a correlated two-dimensional Brownian motion $Z$. We prove a KPZ-type formula which relates the Hausdorff dimension of any Borel subset $A$ of the range of $η$ which can be defined as a function of $η$ (modulo time parameterization) to the Hausdorff dimension of the corresponding time set $η^{-1}(A)$. This result serves to reduce the problem of computing the Hausdorff dimension of any set associated with an SLE, CLE, or related processes in the interior of a domain to the problem of computing the Hausdorff dimension of a certain set associated with a Brownian motion. For many natural examples, the associated Brownian motion set is well-known. As corollaries, we obtain new proofs of the Hausdorff dimensions of the SLE$_κ$ curve for $κ\not=4$; the double points and cut points of SLE$_κ$ for $κ>4$; and the intersection of two flow lines of a Gaussian free field. We also obtain the Hausdorff dimension of the set of $m$-tuple points of space-filling SLE$_κ$ for $κ>4$ and $m \geq 3$ by computing the Hausdorff dimension of the so-called $(m-2)$-tuple $π/2$-cone times of a correlated planar Brownian motion.

math.PR↗

arXiv2019

Bounds for distances and geodesic dimension in Liouville first passage percolation

For $ξ\geq 0$, Liouville first passage percolation (LFPP) is the random metric on $\varepsilon \mathbb Z^2$ obtained by weighting each vertex by $\varepsilon e^{ξh_\varepsilon(z)}$, where $h_\varepsilon(z)$ is the average of the whole-plane Gaussian free field $h$ over the circle $\partial B_\varepsilon(z)$. Ding and Gwynne (2018) showed that for $γ\in (0,2)$, LFPP with parameter $ξ= γ/d_γ$ is related to $γ$-Liouville quantum gravity (LQG), where $d_γ$ is the $γ$-LQG dimension exponent. For $ξ> 2/d_2$, LFPP is instead expected to be related to LQG with central charge greater than 1. We prove several estimates for LFPP distances for general $ξ\geq 0$. For $ξ\leq 2/d_2$, this leads to new bounds for $d_γ$ which improve on the best previously known upper (resp.\ lower) bounds for $d_γ$ in the case when $γ> \sqrt{8/3}$ (resp.\ $γ\in (0.4981, \sqrt{8/3})$). These bounds are consistent with the Watabiki (1993) prediction for $d_γ$. However, for $ξ> 1/\sqrt 3$ (or equivalently for LQG with central charge larger than 17) our bounds are inconsistent with the analytic continuation of Watabiki's prediction to the $ξ>2/d_2$ regime. We also obtain an upper bound for the Euclidean dimension of LFPP geodesics.

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arXiv2019

Harmonic functions on mated-CRT maps

A mated-CRT map is a random planar map obtained as a discretized mating of correlated continuum random trees. Mated-CRT maps provide a coarse-grained approximation of many other natural random planar map models (e.g., uniform triangulations and spanning tree-weighted maps), and are closely related to $γ$-Liouville quantum gravity (LQG) for $γ\in (0,2)$ if we take the correlation to be $-\cos(πγ^2/4)$. We prove estimates for the Dirichlet energy and the modulus of continuity of a large class of discrete harmonic functions on mated-CRT maps, which provide a general toolbox for the study of the quantitative properties random walk and discrete conformal embeddings for these maps. For example, our results give an independent proof that the simple random walk on the mated-CRT map is recurrent, and a polynomial upper bound for the maximum length of the edges of the mated-CRT map under a version of the Tutte embedding. Our results are also used in other work by the first two authors which shows that for a class of random planar maps --- including mated-CRT maps and the UIPT --- the spectral dimension is two (i.e., the return probability of the simple random walk to its starting point after $n$ steps is $n^{-1+o_n(1)}$) and the typical exit time of the walk from a graph-distance ball is bounded below by the volume of the ball, up to a polylogarithmic factor.

math.PR↗

arXiv2019

The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds

We prove that for each $γ\in (0,2)$, there is an exponent $d_γ> 2$, the "fractal dimension of $γ$-Liouville quantum gravity (LQG)", which describes the ball volume growth exponent for certain random planar maps in the $γ$-LQG universality class, the exponent for the Liouville heat kernel, and exponents for various continuum approximations of $γ$-LQG distances such as Liouville graph distance and Liouville first passage percolation. We also show that $d_γ$ is a continuous, strictly increasing function of $γ$ and prove upper and lower bounds for $d_γ$ which in some cases greatly improve on previously known bounds for the aforementioned exponents. For example, for $γ=\sqrt 2$ (which corresponds to spanning-tree weighted planar maps) our bounds give $3.4641 \leq d_{\sqrt 2} \leq 3.63299$ and in the limiting case we get $4.77485 \leq \lim_{γ\rightarrow 2^-} d_γ\leq 4.89898$.

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arXiv2018

Metric gluing of Brownian and $\sqrt{8/3}$-Liouville quantum gravity surfaces

In a recent series of works, Miller and Sheffield constructed a metric on $\sqrt{8/3}$-Liouville quantum gravity (LQG) under which $\sqrt{8/3}$-LQG surfaces (e.g., the LQG sphere, wedge, cone, and disk) are isometric to their Brownian surface counterparts (e.g., the Brownian map, half-plane, plane, and disk). We identify the metric gluings of certain collections of independent $\sqrt{8/3}$-LQG surfaces with boundaries identified together according to LQG length along their boundaries. Our results imply in particular that the metric gluing of two independent instances of the Brownian half-plane along their positive boundaries is isometric to a certain LQG wedge decorated by an independent chordal SLE$_{8/3}$ curve. If one identifies the two sides of the boundary of a single Brownian half-plane, one obtains a certain LQG cone decorated by an independent whole-plane SLE$_{8/3}$. If one identifies the entire boundaries of two Brownian half-planes, one obtains a different LQG cone and the interface between them is a two-sided variant of whole-plane SLE$_{8/3}$. Combined with another work of the authors, the present work identifies the scaling limit of self-avoiding walk on random quadrangulations with SLE$_{8/3}$ on $\sqrt{8/3}$-LQG.

math.PR↗

arXiv2018

A distance exponent for Liouville quantum gravity

Let $γ\in (0,2)$ and let $h$ be the random distribution on $\mathbb C$ which describes a $γ$-Liouville quantum gravity (LQG) cone. Also let $κ= 16/γ^2 >4$ and let $η$ be a whole-plane space-filling SLE$_κ$ curve sampled independent from $h$ and parametrized by $γ$-quantum mass with respect to $h$. We study a family $\{\mathcal G^ε\}_{ε>0}$ of planar maps associated with $(h, η)$ called the \textit{LQG structure graphs} (a.k.a.\ \textit{mated-CRT maps}) which we conjecture converge in probability in the scaling limit with respect to the Gromov-Hausdorff topology to a random metric space associated with $γ$-LQG. In particular, $\mathcal G^ε$ is the graph whose vertex set is $ε\mathbb Z$, with two such vertices $x_1,x_2\in ε\mathbb Z$ connected by an edge if and only if the corresponding curve segments $η([x_1-ε, x_1])$ and $η([x_2-ε,x_2])$ share a non-trivial boundary arc. Due to the peanosphere description of SLE-decorated LQG due to Duplantier, Miller, and Sheffield (2014), the graph $\mathcal G^ε$ can equivalently be expressed as an explicit functional of a correlated two-dimensional Brownian motion, so can be studied without any reference to SLE or LQG. We prove non-trivial upper and lower bounds for the cardinality of a graph-distance ball of radius $n$ in $\mathcal G^ε$ which are consistent with the prediction of Watabiki (1993) for the Hausdorff dimension of LQG. Using subadditivity arguments, we also prove that there is an exponent $χ> 0$ for which the expected graph distance between generic points in the subgraph of $\mathcal G^ε$ corresponding to the segment $η([0,1])$ is of order $ε^{-χ+ o_ε(1)}$, and this distance is extremely unlikely to be larger than $ε^{-χ+ o_ε(1)}$.

math.PR↗

arXiv2018

Chordal SLE$_6$ explorations of a quantum disk

We consider a particular type of $\sqrt{8/3}$-Liouville quantum gravity surface called a doubly marked quantum disk (equivalently, a Brownian disk) decorated by an independent chordal SLE$_6$ curve $η$ between its marked boundary points. We obtain descriptions of the law of the quantum surfaces parameterized by the complementary connected components of $η([0,t])$ for each time $t \geq 0$ as well as the law of the left/right $\sqrt{8/3}$-quantum boundary length process for $η$.

math.PR↗

arXiv2018

Convergence of the free Boltzmann quadrangulation with simple boundary to the Brownian disk

We prove that the free Boltzmann quadrangulation with simple boundary and fixed perimeter, equipped with its graph metric, natural area measure, and the path which traces its boundary converges in the scaling limit to the free Boltzmann Brownian disk. The topology of convergence is the so-called Gromov-Hausdorff-Prokhorov-uniform (GHPU) topology, the natural analog of the Gromov-Hausdorff topology for curve-decorated metric measure spaces. From this we deduce that a random quadrangulation of the sphere decorated by a $2l$-step self-avoiding loop converges in law in the GHPU topology to the random curve-decorated metric measure space obtained by gluing together two independent Brownian disks along their boundaries.

math.PR↗

arXiv2017

Active spanning trees with bending energy on planar maps and SLE-decorated Liouville quantum gravity for $κ> 8$

We introduce a two-parameter family of probability measures on spanning trees of a planar map. One of the parameters controls the activity of the spanning tree and the other is a measure of its bending energy. When the bending parameter is 1, we recover the active spanning tree model, which is closely related to the critical Fortuin--Kasteleyn model. A random planar map decorated by a spanning tree sampled from our model can be encoded by means of a generalized version of Sheffield's hamburger-cheeseburger bijection. Using this encoding, we prove that for a range of parameter values (including the ones corresponding to maps decorated by an active spanning tree), the infinite-volume limit of spanning-tree-decorated planar maps sampled from our model converges in the peanosphere sense, upon rescaling, to an SLE$_κ$-decorated $γ$-Liouville quantum cone with $κ> 8$ and $γ= 4/\sqrtκ\in (0,\sqrt 2)$.

math.PR↗

arXiv2017

Almost sure multifractal spectrum of SLE

Suppose that $η$ is a Schramm-Loewner evolution (SLE$_κ$) in a smoothly bounded simply connected domain $D \subset {\mathbb C}$ and that $ϕ$ is a conformal map from ${\mathbb D}$ to a connected component of $D \setminus η([0,t])$ for some $t>0$. The multifractal spectrum of $η$ is the function $(-1,1) \to [0,\infty)$ which, for each $s \in (-1,1)$, gives the Hausdorff dimension of the set of points $x \in \partial {\mathbb D}$ such that $|ϕ'( (1-ε) x)| = ε^{-s+o(1)}$ as $ε\to 0$. We rigorously compute the a.s. multifractal spectrum of SLE, confirming a prediction due to Duplantier. As corollaries, we confirm a conjecture made by Beliaev and Smirnov for the a.s. bulk integral means spectrum of SLE and we obtain a new derivation of the a.s. Hausdorff dimension of the SLE curve for $κ\leq 4$. Our results also hold for the SLE$_κ(\underline ρ)$ processes with general vectors of weight $\underlineρ$.

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