Existence and uniqueness of the
Liouville quantum gravity metric for
We show that for each , there is a unique metric associated with -Liouville quantum gravity (LQG). More precisely, we show that for the whole-plane Gaussian free field (GFF) , there is a unique random metric on which is characterized by a certain list of axioms: it is locally determined by and it transforms appropriately when either adding a continuous function to or applying a conformal automorphism of (i.e., a complex affine transformation). Metrics associated with other variants of the GFF can be constructed using local absolute continuity.
The -LQG metric can be constructed explicitly as the scaling limit of Liouville first passage percolation (LFPP), the random metric obtained by exponentiating a mollified version of the GFF. Earlier work by Ding, Dubédat, Dunlap, and Falconet (2019) showed that LFPP admits non-trivial subsequential limits. This paper shows that the subsequential limit is unique and satisfies our list of axioms. In the case when , our metric coincides with the -LQG metric constructed in previous work by Miller and Sheffield, which in turn is equivalent to the Brownian map for a certain variant of the GFF. For general , we conjecture that our metric is the Gromov-Hausdorff limit of appropriate weighted random planar map models, equipped with their graph distance. We include a substantial list of open problems.
- 1 Introduction
- 2 Preliminaries
- 3 The optimal bi-Lipschitz constant
- 4 Independence along a geodesic
5 Forcing a geodesic to take a shortcut
- 5.1 Outline of the proof of Proposition 4.3
- 5.2 Proof of Proposition 4.3 assuming the existence of events and functions
- 5.3 Building a tube which contains a shortcut with positive probability
- 5.4 Building a tube which contains a shortcut with high probability
- 5.5 Definition of the event and the bump functions
- 5.6 Subtracting a bump function to move a geodesic
- 6 Proof of Theorem 1.9
- 7 Open problems
Fix , let be an open domain, and let be the Gaussian free field (GFF) on , or some minor variant thereof. The -Liouville quantum gravity (LQG) surface described by is formally the random two-dimensional Riemannian manifold with metric tensor
where is the Euclidean Riemannian metric tensor.
LQG surfaces were first introduced non-rigorously in the physics literature by Polyakov [Pol81a, Pol81b] as a canonical model of a random Riemannian metric on . Another motivation to study LQG surfaces is that they describe the scaling limit of random planar maps. The special case when (called “pure gravity”) corresponds to uniformly random planar maps, including uniform triangulations, quadrangulations, etc. Other values of (sometimes referred to as “gravity coupled to matter”) correspond to random planar maps weighted by the partition function of an appropriate statistical mechanics model on the map, for example the uniform spanning tree for or the Ising model for .
The definition (1.1) of LQG does not make literal sense since is only a distribution, not a function, so it does not have well-defined pointwise values and cannot be exponentiated. Nevertheless, it is known that one can make sense of the associated volume form (where denotes Lebesgue measure) as a random measure on via various regularization procedures [Kah85, DS11, RV14]. One such regularization procedure is as follows. For and , let be the heat kernel, and note that approximates a point mass at when is small. For , we define a mollified version of the GFF by
By [DS11, Proposition 2.1], the metric is conformally covariant: if is a conformal map and we set
then a.s. for each Borel set . This leads one to define a -LQG surface as an equivalence class of pairs , with two such pairs and declared to be equivalent if there is a conformal map for which and are related as in (1.4). We think of two equivalent pairs as representing different parameterizations of the same random surface. The conformal covariance property of says that this measure is intrinsic to the quantum surface — it does not depend on the particular equivalence class representative.
In order for -LQG to be a reasonable model of a “random two-dimensional Riemannian manifold”, one also needs a random metric on which is in some sense obtained by exponentiating and which satisfies a conformal covariance property analogous to that of the -LQG area measure. Moreover, this metric should be the scaling limit of the graph distance on random planar maps with respect to the Gromov-Hausdorff topology. Constructing a metric on -LQG is a much more difficult problem than constructing the measure . Indeed, any natural regularization schemes for LQG distances involves minimizing over a large collection of paths, which results in a substantial degree of non-linearity.
Prior to this work, a -LQG metric has only been constructed in the special case when in a series of works by Miller and Sheffield [MS15b, MS16b, MS16c]. In this case, for certain special choices of the pair , the random metric space agrees in law with a Brownian surface, such as the Brownian map [Le 13, Mie13] or the Brownian disk [BM17]. These Brownian surfaces are continuum random metric spaces which arise as the scaling limits of uniform random planar maps with respect to the Gromov-Hausdorff topology. Miller and Sheffield’s construction of the -LQG metric does not use a direct regularization of the field . Instead, they first construct a candidate for -LQG metric balls using a process called quantum Loewner evolution, which is built out of the Schramm-Loewner evolution with parameter (), then show that there is a metric with the desired balls.
In this paper, we will construct a -LQG metric for all via an explicit regularization procedure analogous to (1.3). We will also show that this metric is uniquely characterized by a list of natural properties that any reasonable notion of a metric on -LQG should satisfy, so is in some sense the only “correct” metric on -LQG. For simplicity, we will mostly restrict attention to the whole-plane case, but metrics associated with GFF’s on other domains can be easily constructed via restriction and/or absolute continuity (see Remark 1.5). In contrast to [MS15b, MS16b, MS16c], the present work will make no use of . Furthermore, we do not a priori have an ambient metric space to compare to (such as the Brownian map in the case ) and we do not have any sort of exact solvability, i.e., we do not know the exact laws of any observables related to the metric.
We now describe how our metric is constructed. It is shown in [DG18], building on [DZZ18, GHS17], that for each , there is an exponent which describes distances in various discrete approximations of -LQG and which should be the Hausdorff dimension of the continuum -LQG metric (it will be proven that this is indeed the case in [GP19a]). The value of is not known explicitly except in the case when , in which case we know that (see Problem 7.1). We refer to [DG18, GP19b] for bounds for and some speculation about its possible value. For , we define
We say that a random distribution on is a whole plane GFF plus a continuous function if there exists a coupling of with a random continuous function such that the law of is that of a whole-plane GFF. We similarly define a whole-plane GFF plus a bounded continuous function, except we require that is bounded.111The reason why we sometimes restrict to bounded continuous functions is to ensure that the convolution with the whole-plane heat kernel is finite (so is defined) and that the results about subsequential limits of LFPP in [DDDF19, DFG19] are applicable. Note that the whole-plane GFF is defined only modulo a global additive constant, but these definitions does not depend on the choice of additive constant.
If is a whole-plane GFF plus a bounded continuous function, we define for and as in (1.2). For and , we define the -LFPP metric by
where the infimum is over all piecewise continuously differentiable paths from to . One should think of LFPP as the metric analog of the approximations of the LQG measure in (1.3).222One can also consider other variants of LFPP, defined using different approximations of the GFF, but we consider here since this is the approximation for which tightness is proven in [DDDF19]. If we knew tightness (and some basic properties of the subsequential limiting metrics) for LFPP defined using a different approximation of the GFF, then Theorem 1.8 below would show that these variants of LFPP also converge to the -LQG metric. The intuitive reason why we look at instead of to define the metric is as follows. By (1.3), we can scale LQG areas by a factor of by adding to the field. By (1.6), this results in scaling distances by , which is consistent with the fact that the “dimension” should be the exponent relating the scaling of areas and distances.
Let be the median of the -distance between the left and right boundaries of the unit square in the case when is a whole-plane GFF normalized so that its circle average333See [DS11, Section 3.1] for the basic properties of the circle average process. Even though we define LFPP using truncation with the heat kernel, we will always fix the additive constant for the whole-plane GFF using the circle average. over is zero. We do not know the value of explicitly, but see Corollary 1.11. It was shown by Ding, Dubedát, Dunlap, and Falconet [DDDF19] that the laws of the metrics are tight w.r.t. the local uniform topology on , and every possible subsequential limit induces the Euclidean topology on . Subsequently, it was shown by Dubedát, Falconet, Gwynne, Pfeffer, and Sun [DFG19], using a general theorem from [GM19c], that every subsequential limit can be realized as a measurable function of , so in fact the metrics admit subsequential limits in probability. One of the main results of this paper gives the uniqueness of this subsequential limit.
Theorem 1.1 (Convergence of LFPP).
The random metrics converge in probability w.r.t. the local uniform topology on to a random metric on which is a.s. determined by .
It is natural to define the limiting metric from Theorem 1.1 to be the -LQG metric associated with . However, this definition is not entirely satisfactory since it is a priori possible that there are other natural ways to construct a metric on -LQG which do not yield the same result as the one in Theorem 1.1. For example, Theorem 1.1 does not yet tell us that the limit of LFPP coincides with the metric of [MS15b, MS16b, MS16c] in the case when .
We will therefore define a -LQG metric in terms of a list of axioms (see Section 1.2 just below). We will show that (a) the metric of Theorem 1.1 satisfies these axioms and (b) there is at most one metric satisfying these axioms for each . Taken together, these statements tell us that the metric of Theorem 1.1 is the only reasonable metric that one can put on -LQG. Our axiomatic characterization is similar to the axiomatic characterization of the -LQG measure from [Sha14].
An important feature of our proofs is that they can be read with essentially no knowledge of the (substantial) existing literature on LQG. Aside from basic properties of the GFF (as discussed, e.g., in [She07] and the introductory sections of [SS13, MS16d, MS17]), the only prior works which this paper relies on are [DDDF19, GM19c, DFG19, GM19a]. All of the results which we need from these papers are reviewed in Section 2.
Our results open up many important new research directions in the theory of LQG. We have included in Section 7 a substantial list of open problems related to the -LQG metric.
Acknowledgments. We thank Jian Ding, Julien Dubédat, Alex Dunlap, Hugo Falconet, Josh Pfeffer, Scott Sheffield, and Xin Sun for helpful discussions. JM was supported by ERC Starting Grant 804166.
1.2 Axiomatic characterization of the -LQG metric
To state our list of axioms precisely, we will need some preliminary definitions concerning metric spaces. In what follows, we let be a metric space.
For a curve , the -length of is defined by
where the supremum is over all partitions of . Note that the -length of a curve may be infinite.
For , the internal metric of on is defined by
where the infimum is over all paths in from to . Then is a metric on , except that it is allowed to take infinite values.
We say that is a length space if for each and each , there exists a curve of -length at most from to .
A continuous metric on an open domain is a metric on which induces the Euclidean topology on , i.e., the identity map is a homeomorphism. We equip the space of continuous metrics on with the local uniform topology for functions from to and the associated Borel -algebra. We allow a continuous metric to satisfy if and are in different connected components of . In this case, in order to have w.r.t. the local uniform topology we require that for large enough , if and only if .
Let be the space of distributions (generalized functions) on , equipped with the usual weak topology. For , a (strong) -Liouville quantum gravity (LQG) metric is a measurable function from to the space of continuous metrics on such that the following is true whenever is a whole-plane GFF plus a continuous function.
Length space. Almost surely, is a length space, i.e., the -distance between any two points of is the infimum of the -lengths of -continuous paths (equivalently, Euclidean continuous paths) between the two points.
Locality. Let be a deterministic open set. The internal metric is a.s. determined by .
Weyl scaling. Let be as in (1.5) and for each continuous function , define
where the infimum is over all continuous paths from to parameterized by -length. Then a.s. for every continuous function .
Coordinate change for translation and scaling. For each fixed deterministic and , a.s.
Let us briefly discuss why the above axioms are natural. Recall that -LQG should be the random Riemannian metric with metric tensor . Axiom I is simply the LQG analog of the statement that for a true Riemannian metric, the distance between two points can be defined as the infima of the lengths of paths connecting them. In a similar vein, Axiom II corresponds to the fact that for a smooth Riemannian metric, the lengths of paths are determined locally by the Riemannian metric tensor. Axiom III is just expressing the fact that the metric is obtained by exponentiating , so adding a continuous function to results in re-scaling the metric length measure on paths by .
Axiom IV is the metric analog of the conformal coordinate change formula (1.4) for the -LQG area measure, but restricted to translations and scalings. This axiom together with Corollary 1.3 says that depends only on the LQG surface , not on the particular choice of parameterization. We will prove a conformal covariance property for the -LQG metric w.r.t. conformal automorphisms between arbitrary domains, directly analogous to the conformal covariance of the -LQG area measure, in [GM19b].
Theorem 1.2 (Existence and uniqueness of the LQG metric).
Fix . There is a -LQG metric such that the limiting metric of Theorem 1.1 is a.s. equal to whenever is a whole-plane GFF plus a bounded continuous function. Furthermore, the -LQG metric is unique in the following sense. If and are two -LQG metrics, then there is a deterministic constant such that if is a whole-plane GFF plus a continuous function, then a.s. .
Theorem 1.2 justifies us in referring to the -LQG metric. Technically speaking there is a one-parameter family of such metrics, which differ by a global deterministic multiplicative constant. But, one can fix the constant in various ways to get a single canonically defined metric. For example, we can require that the median distance between the left and right boundaries of the unit square is 1 for the metric associated with a whole-plane GFF normalized so that its circle average over is zero (the limiting metric in Theorem 1.1 has this normalization).
In Axiom IV in the definition of a strong -LQG metric, we did not require that the metric is invariant under rotations of . It turns out that rotational invariance is implied by the other axioms. See Remark 1.6 below for an intuitive explanation of why this is the case.
Corollary 1.3 (Rotational invariance).
If and is a -LQG metric then is rotationally invariant, i.e., if with and is a whole-plane GFF plus a continuous function, then a.s. for all .
Define . It is easily verified that is a strong LQG metric, so Theorem 1.2 implies that there is a deterministic constant such that a.s. whenever is a whole-plane GFF plus a continuous function. To check that , consider a whole-plane GFF normalized so that its circle average over is . Then the law of is rotationally invariant, so for every . Therefore . ∎
It is easy to check that the metric constructed in [MS15b, MS16b, MS16c] satisfies the axioms for a -LQG metric; see [GMS18, Section 2.5] for a careful explanation of why this is the case. Consequently, Theorem 1.2 implies the following.
The present work does not use the results of [MS15b, MS16b, MS16c], but also does not supersede these results. Indeed, without these works it is not at all clear how to link the -LQG metric constructed in the present article to Brownian surfaces, and thereby to uniform random planar maps.
There are a number of properties of the -LQG metric which are already known. The optimal Hölder exponents between and the Euclidean metric, in both directions, as well as moment bounds for various distance quantities are established in [DFG19] (see also Section 2.4). Confluence properties for -geodesics analogous to the ones known for the Brownian map [Le 10] are proven in [GM19a] (see also Section 2.5). It is shown in [MQ18] that -geodesics are conformally removable and their laws are mutually singular with respect to Schramm-Loewner evolution curves. The forthcoming work [GP19a] will show that satisfies a version of the KPZ formula [KPZ88, DS11].
Remark 1.5 (Metrics associated with other fields).
Theorem 1.2 gives us a canonical -LQG metric associated with a whole-plane GFF plus a continuous function. It is not hard to see that one can also define the metric if is equal to a whole-plane GFF plus a continuous function plus a finite number of logarithmic singularities of the form for and ; see [DFG19, Theorem 1.10 and Proposition 3.17].
We can also define metrics associated with GFF’s on proper sub-domains of . To this end, let be open and let be a whole-plane GFF. Due to Axiom II, we can define for each open set the metric as a measurable function of . We can write , where is a zero-boundary GFF on and is a random harmonic function on independent from . In the notation (1.8), we define
It is easily seen from Axioms II and III that is a measurable function of ; see [GM19a, Remark 1.2]. This defines the -LQG metric for a zero-boundary GFF. Using Axiom III, we can similarly define the metric in the case when is a zero-boundary GFF plus a continuous function on . It is shown in [GM19b] that this metric satisfies a conformal coordinate change relation analogous to the one satisfied by the -LQG measure (as discussed just below (1.4)).
Remark 1.6 (Why rotational invariance is unnecessary).
At a first glance, it may seem surprising that one does not need rotational invariance to uniquely characterize the LQG metric in Theorem 1.2. Indeed, one can define variants of LFPP which are not rotationally invariant by working with a stretched version of the Euclidean metric. For example, for a given one can replace (1.6) by
where the infimum is over all piecewise continuously differentiable paths from to . The arguments of this paper and its predecessors apply verbatim with in place of . In particular, converges in probability to (a deterministic constant times) the -LQG metric and hence satisfies the rotational invariance property of Corollary 1.3. This is despite the fact that the metrics (1.11) do not satisfy this rotational invariance property.
Here is an intuitive explanation for this phenomenon. First, we note that is bi-Lipschitz equivalent with respect to for each , with a deterministic bi-Lipschitz constants. Therefore in a subsequential limit as , we obtain two metrics and which are bi-Lipschitz equivalent with deterministic bi-Lipschitz constants. Suppose that is a -geodesic connecting and . Using the confluence of geodesics results from [GM19a], one can show that (very roughly speaking) for distinct times , the restrictions of to small neighborhoods of and are approximately independent; see the outline of Section 4 in Section 1.5 below for details. Moreover, since is a fractal type curve, it has no local notion of direction, so one expects that the law of restricted to a small neighborhood of does not depend very strongly on or on the endpoints of . If we fix and let be equally spaced times, we can approximate the -length of by
The above considerations suggest that each of the random variables has approximately the same distribution and is bounded above and below by deterministic constants times . From law of large numbers type considerations, it follows that the -length of is a deterministic constant times the -length of , where the constant does not depend on the endpoints of .
Knowing that the -length of every geodesic is a constant times its -length (and vice-versa) does not immediately imply that is given a constant by . This is because if is a sequence of paths which converge uniformly to , then it is not necessarily true that converges to . For this and other reasons, we will argue in a somewhat different manner than we have indicated above, though our arguments will still be based on the bi-Lipschitz equivalence of metrics and approximate independence statements for the local behavior of a geodesic at different times. We will explain the general strategy in Section 1.5 in more detail.
1.3 Conjectured random planar map connection
As noted above, the -LQG metric should describe the large scale behavior of the graph metric for random planar maps. Since our -LQG metric is in some sense canonical, it is natural to make the following conjecture.
Examples of planar map models to which Conjecture 1.7 should apply include random planar maps weighted by the number of spanning trees (), the Ising model partition function (), the number of bipolar orientations (; [KMSW15]), or the Fortuin-Kasteleyn model partition function (; [She16b]). Another class of models is the so-called mated-CRT maps, which are defined for all ; see [DMS14, GHS17, GMS17].
For , Conjecture 1.7 has already been proven for many different uniform-type random planar maps. The reason for this is that we know that our -LQG metric is equivalent to the metric of [MS15b, MS16b, MS16c] (Corollary 1.4); which in turn is equivalent to a Brownian surface, such as the Brownian map, for certain special -LQG surfaces [MS16b, Corollary 1.5]; which in turn is the scaling limit of uniform random planar maps of various types [Le 13, Mie13].
Conjecture 1.7 has not been proven for any random planar map model for . However, we already have a relationship between the continuum LQG metric and graph distances in random planar maps at the level of exponents for all . Indeed, the quantity appearing in (1.5) describes several exponents associated with random planar maps, such as the ball volume exponent [GHS17, DG18] and the displacement exponent for simple random walk on the map [GM17, GH18]. It is not hard to see (and will be proven in [GP19a]) that is the Hausdorff dimension of .
Conjecture 1.7 can be made somewhat more precise by specifying exactly what type of -LQG surface should arise in the scaling limit. For random planar maps with the topology of the sphere (resp. disk, plane, half-plane) this surface should be the quantum sphere (resp. quantum disk, -quantum cone, -quantum wedge). See [DMS14] for precise definitions of these quantum surfaces. Equivalent definitions of the quantum sphere and quantum disk, respectively, can be found in [DKRV16, HRV18] (see [AHS17] for a proof of the equivalence in the sphere case). Some planar map models have been proven to converge to these quantum surfaces, for general , with respect to topologies which do not encode the metric structure explicitly. Examples of such topologies include convergence in the so-called peanosphere sense [She16b, DMS14] and convergence of the counting measure on vertices to the -LQG measure when the planar map is embedded appropriately into the plane [GMS17].
1.4 Weak LQG metrics and a stronger uniqueness statement
We will prove Theorem 1.1 and 1.2 simultaneously by establishing a uniqueness statement for metrics under a weaker list of axioms, which are satisfied for both the strong LQG metrics considered in Section 1.2 and for subsequential limits of LFPP (as is shown in [DDDF19, DFG19]).
Let be the space of distributions as in Section 1.2. A weak -LQG metric is a measurable function from to the space of continuous metrics on such that the following is true whenever is a whole-plane GFF plus a continuous function.
Length space. Almost surely, is a length space, i.e., the -distance between any two points of is the infimum of the -lengths of -continuous paths (equivalently, Euclidean continuous paths) between the two points.
Locality. Let be a deterministic open set. The internal metric is a.s. determined by .
Weyl scaling. If we define as in (1.8), then a.s. for every continuous function .
Translation invariance. For each fixed deterministic , a.s. .
Tightness across scales. Suppose is a whole-plane GFF and for and let be the average of over the circle . For each , there is a deterministic constant such that the set of laws of the metrics for is tight (w.r.t. the local uniform topology). Furthermore, the closure of this set of laws w.r.t. the Prokhorov topology is contained in the set of laws on continuous metrics on (i.e., every subsequential limit of the laws of the metrics is supported on metrics which induce the Euclidean topology on ). Finally, there exists such that for each ,
Axioms I through III for a weak LQG metric are identical to the corresponding axioms for a strong LQG metric. Axiom IV for a weak LQG metric is equivalent to Axiom IV for a strong LQG metric with . Axiom V for a weak -LQG metric is a substitute for the exact scale invariance property given by Axiom IV for a strong LQG metric. This axiom implies the tightness of various functionals of . For example, if is open and is compact, then the laws of
By the scale invariance of the law of the whole-plane GFF, modulo additive constant, Axiom IV for a strong LQG metric immediately implies Axiom V for a weak -LQG metric with , for as in (1.4). Indeed, using Axiom IV and then Axiom III for a strong -LQG metric shows that
Hence every strong -LQG metric is a weak -LQG metric.
It is shown in [DFG19, Theorem 1.2] that every subsequential limit in probability of the LFPP metrics of (1.6) is of the form where is a weak -LQG metric. Consequently, the following theorem contains both Theorem 1.1 and Theorem 1.2.
Theorem 1.8 (Strong uniqueness of weak LQG metrics).
Let . Every weak -LQG metric is a strong -LQG metric. In particular, by Theorem 1.2, such a metric exists for each and if and are two weak -LQG metrics, then there is a deterministic constant such that if is a whole-plane GFF plus a continuous function, then a.s. .
It turns out that all of our main results are easy consequences of the following statement, which superficially seems to be weaker that Theorem 1.8.
Theorem 1.9 (Weak uniqueness of weak LQG metrics).
Let and let and be two weak -LQG metrics which have the same values of in Axiom V. There is a deterministic constant such that if is a whole-plane GFF plus a continuous function, then a.s. .
Every weak -LQG metric is a strong -LQG metric.
Suppose that is a weak -LQG metric. For , we define
We claim that is a weak -LQG metric with the same scaling constants as . It is easily verified that satisfies Axioms I through IV in the definition of a weak -LQG metric. To check Axiom V, we compute for :
In the case when is a whole-plane GFF, the random variable is centered Gaussian with variance [DS11, Section 3.1]. By (1.12), is bounded above by a constant depending only on (not on ). Axiom V for applied with in place of and in place of therefore implies that the laws of the metrics are tight in the case when is a whole-plane GFF, and that every subsequential limit of the laws of these metrics is supported on metrics (not pseudometrics).
Hence we can apply Theorem 1.9 with to get that for each , there is a deterministic constant such that whenever is a whole-plane GFF plus a continuous function, a.s. . We now argue that is a power of .
For , we have , which implies that a.s. . Therefore,
It is also easy to see that depends continuously on . Indeed, by Axiom III and since , we have . By the continuity of and , it follows that in law as . This gives the continuity of at . Using (1.17) then gives the desired continuity in general.
The relation (1.17) and the continuity of (actually, just Lebesgue measurability is enough) imply that for some . Equivalently, for , a.s.
By Lemma 1.10, every weak -LQG metric is a strong -LQG metric. By (1.15), every strong LQG metric satisfies the axioms in the definition of a weak -LQG metric with . We can therefore apply Theorem 1.9 to get that there is at most one strong LQG metric. This completes the proof of the uniqueness parts of Theorems 1.2 and 1.8.
As for existence, we recall that [DFG19, Theorem 1.2] (building on [DDDF19]) shows that for every sequence of ’s tending to zero, there is a weak -LQG metric and a subsequence along which the re-scaled LFPP metrics converge in probability to , whenever is a whole-plane GFF plus a bounded continuous function. By the uniqueness part of Theorem 1.8, is in fact a strong -LQG metric and any two different subsequential limiting metrics differ by a deterministic multiplicative constant factor. Recall that is the median -distance between the left and right boundaries of the unit square in the case when is a whole-plane GFF normalized so that . Hence for any subsequential limiting metric the median -distance between the left and right boundaries of the unit square is 1. Therefore, the multiplicative constant factor is 1, so the subsequential limit of in probability is unique. This gives Theorem 1.1 and the existence parts of Theorems 1.2 and 1.8. ∎
Finally, we note that our results give non-trivial information about the approximating LFPP metrics from (1.6). Indeed, let be the scaling constants from Theorem 1.1. It is shown in [DG18, Theorem 1.5] that . Using Theorem 1.1, we obtain the following stronger form of this relation.
The function is regularly varying with exponent , i.e., for every one has .
We expect, but do not prove here, that in fact Theorem 1.1 holds with .
Proof of Corollary 1.11.
It is shown in [DFG19, Lemma 2.14] that for any sequence of ’s tending to zero along which the re-scaled LFPP metrics converge in law, also converges (the limit is , with as in Axiom V for the limiting weak -LQG metric). By Theorem 1.1, converges in probability as , so in fact converges, not just subsequentially. This means that is regularly varying with some exponent . Since , we must have . ∎
As explained above, to prove our main results it remains only to prove Theorem 1.9. We emphasize that unlike many results in the theory of LQG, this paper does not build on a large amount of external input. Rather, we will only use some results from the papers [DDDF19, GM19c, DFG19, GM19a], which can be taken as black boxes. All of the externally proven results which we will use are reviewed in Section 2.
Throughout this outline and the rest of the paper, we will use (without comment) the following two basic facts about -geodesics when is a weak -LQG metric and is a whole-plane GFF.
In the remainder of this section we give a very rough idea of the proof of Theorem 1.9. There are a number of technicalities involved, which we will gloss over in order to make the central ideas as transparent as possible. Consequently, some of the statements in this subsection are not exactly accurate without additional caveats. More detailed (and more precise) outlines can be found at the beginnings of the individual sections and subsections.
Main idea of the proof. Suppose and are two weak -LQG metrics as in Theorem 1.9 and let be a whole-plane GFF. As explained in Proposition 2.2, it follows from a general theorem for local metrics of the Gaussian free field [GM19c, Theorem 1.6] that and are bi-Lipschitz equivalent, i.e.,
It is easily seen that and are a.s. equal to deterministic constants (Lemma 3.1). We identify and with these constants (which amounts to re-defining and on an event of probability zero). To prove Theorem 1.9 we will show that .
The basic idea of the proof of this fact is as follows. Suppose by way of contradiction that . Then for any there a.s. exist distinct points such that . In Section 3 (see outline below), using translation invariance of the GFF, modulo additive constant, and the local independence properties of the GFF, we will deduce from this that the following is true. There exists , depending only on the laws of and , such that for each there are many small values of (how small depends on ) for which
where is the Euclidean ball of radius centered at 0. By interchanging the roles of and , we can similarly find , depending only on the laws of and , such that for each , there are many small values of (how small depends on ) for which
See Section 3 for precise statements. The reason why the bounds only hold for “many” choices of , instead of for all , is that we only have tightness across scales (Axiom V), not exact scale invariance. We will use (1.22) to deduce a contradiction to (1.23).
Consider a -geodesic between two fixed points . Using (1.22) and a local independence argument for different segments of (which is explained in the outlines of Sections 4 and 5 below), one can show that it holds with superpolynomially high probability as (i.e., except on an event of probability decaying faster than any positive power of ), at a rate which is uniform over the choice of and , that the following is true. There are times such that and . By the definition (1.21) of , the -distance from to is at most and the -distance from to is at most . Combining these facts shows that with superpolynomially high probability as ,
We now let be as in (1.23) and fix a large constant . For any , we can take a union bound to get that with probability tending to 1 as , at a rate which is uniform in , the bound (1.24) holds simultaneously for all . Now consider an arbitrary pair of points with . Let be the points closest to and , respectively. By the bi-Hölder continuity of and w.r.t. the Euclidean metric [DFG19, Theorem 1.7], if we choose sufficiently large, in a manner depending only on the Hölder exponents (i.e., only on ), then and are much smaller than . From this, we infer that with probability tending to 1 as , at a rate which is uniform in , the bound (1.24) holds simultaneously for all with . If is chosen sufficiently small so that this probability is at least , we get a contradiction to (1.23) with .
The above argument is “morally” correct but is far from a complete proof for three main reasons, which are addressed in Sections 3, 4, and 5, respectively. These three sections are mostly independent from one another: only the main theorem/proposition statements at the beginning of each section are used in later sections.
Section 3: bounds for ratios of distances at many scales. The purpose of Section 3 is to prove (more quantitative versions of) the bounds (1.22) and (1.23) stated above. Since we are only working with a weak -LQG metric, not a strong -LQG metric, we do not have exact scale invariance, just tightness across scales (Axiom V). Consequently, if , then we cannot necessarily say that pairs of points for which exist with uniformly positive probability over different Euclidean scales. That is, it could in principle be that for every small fixed , the probability that there exists with and is very small for some values of . However, we can say that such pairs of points exist with uniformly positive probability for a suitably “dense” set of scales via an argument which proceeds (very roughly) as follows.
Let be small and suppose by way of contradiction that there is a sequence such that is bounded above and below by deterministic constants and the following is true. For each , it holds with probability at least that for every pair of points for which . Using the translation invariance of the metric (Axiom IV) and the local independence properties of the GFF (in particular, Lemma 2.6 below), we see that if are sufficiently small (how small depends only on the laws of and , not on or ), then the following is true. We can cover any fixed compact subset of by Euclidean balls of the form with the property that for every pair of points and . By considering the times when a -geodesic between two fixed points of crosses an annulus for as above, we get that a.s. for a constant . This contradicts the definition (1.21) of .
Hence the set of “bad” scales for which points with and are unlikely to exist cannot be too large, which means that the complementary set of “good” scales for which such points exist with probability at least has to be reasonably dense. This leads to (1.22). The bound (1.23) follows by interchanging the roles of and .
Section 4: independence along an LQG geodesic. Once we know that there are many pairs of points with , we want to use some sort of local independence to say that a -geodesic is extremely likely to get close to at least one such pair of points (i.e., we need the -distance from the to each of and to be much smaller than ). However, -geodesics are highly non-local functionals of the field and do not satisfy any reasonable Markov property. So, techniques for obtaining local independence which may be familiar from the theory of SLE/GFF couplings [SS13, Dub09, MS16d, MS16e, MS16a, MS17, She16a, DMS14] do not apply in our setting.
Instead we need to develop a new set of techniques to obtain local independence at different points of -geodesics. See Figure 1 for an illustration. In fact, we will prove a general theorem (Theorem 4.1) which roughly speaking says the following. Suppose we are given events for and with the following properties. The event is determined by and the part of the -geodesic from to which is contained in . Moreover, for each , the conditional probability of given and the event is a.s. bounded below by a deterministic constant. Then when is small it is very likely that for nearly every choice of , the event occurs for at least one ball hit by .
We will eventually apply this theorem with given by, roughly speaking, the event that gets close to a pair of points with and . This together with the triangle inequality and the bi-Hölder continuity of and w.r.t. the Euclidean metric (to transfer from to a lower bound for