Liouville quantum gravity (LQG) and the Brownian map (TBM) are two distinct models of measure-endowed random surfaces. LQG is defined in terms of a real parameter , and it has long been believed that when , the LQG sphere should be equivalent (in some sense) to TBM. However, the LQG sphere comes equipped with a conformal structure, and TBM comes equipped with a metric space structure, and endowing either one with the other’s structure has been an open problem for some time.
This paper is the first in a three-part series that unifies LQG and TBM by endowing each object with the other’s structure and showing that the resulting laws agree. The present work uses a form of the quantum Loewner evolution (QLE) to construct a metric on a dense subset of a -LQG sphere and to establish certain facts about the law of this metric, which are in agreement with similar facts known for TBM. The subsequent papers will show that this metric extends uniquely and continuously to the entire -LQG surface and that the resulting measure-endowed metric space is TBM.
Liouville quantum gravity and the Brownian map I:\\ The metric \authorJason Miller and Scott Sheffield
- 1 Introduction
- 2 Preliminaries
- 3 Eden model and percolation interface
- 4 Infinite measures on quantum spheres
- 5 Meeting in the middle
- 6 Quantum natural time
- 7 Symmetry
- 8 Metric construction
We have benefited from conversations about this work with many people, a partial list of whom includes Omer Angel, Itai Benjamini, Nicolas Curien, Hugo Duminil-Copin, Amir Dembo, Bertrand Duplantier, Ewain Gwynne, Nina Holden, Jean-François Le Gall, Gregory Miermont, Rémi Rhodes, Steffen Rohde, Oded Schramm, Stanislav Smirnov, Xin Sun, Vincent Vargas, Menglu Wang, Samuel Watson, Wendelin Werner, David Wilson, and Hao Wu.
We would also like to thank the Isaac Newton Institute (INI) for Mathematical Sciences, Cambridge, for its support and hospitality during the program on Random Geometry where part of this work was completed. J.M.’s work was also partially supported by DMS-1204894 and S.S.’s work was also partially supported by DMS-1209044, a fellowship from the Simons Foundation, and EPSRC grants EP/L018896/1 and EP/I03372X/1.
The Liouville quantum gravity (LQG) sphere [DS11, She15, DMS14, MS15b, DKRV14, AHS15] and the Brownian map (TBM) [MM06, LGP08, LG13, Mie13, LG14, MS15b] can both be understood as measures on the space of measure-endowed, sphere-homeomorphic surfaces. The definition of LQG involves a real parameter , and it has long been believed that when , the LQG sphere should be equivalent to TBM. However, the LQG sphere comes equipped with a conformal structure and TBM comes equipped with a metric space structure, and it is far from obvious how to endow either of these objects with the other’s structure.
This paper is the first in a three part series (also including [MS16a, MS16b]) that will provide a robust unification of -LQG and TBM. Over the course of these three papers, we will show the following:
An instance of the -LQG sphere a.s. comes with a canonical metric space structure, and the resulting measure-endowed metric space agrees in law with TBM.
Given an instance of TBM, the -LQG sphere that generates it is a.s. uniquely determined. This implies that an instance of TBM a.s. comes with a canonical (up to Möbius transformation) embedding in the Euclidean sphere. In other words:
An instance of TBM a.s. comes with a canonical conformal structure and the resulting measure-endowed conformal sphere agrees in law with the -LQG sphere.
The canonical (up to Möbius transformation) embedding of TBM (with its intrinsic metric) into the Euclidean sphere (with the Euclidean metric) is a.s. Hölder continuous with Hölder continuous inverse. This in particular implies that a.s. all geodesics in TBM are sent to Hölder continuous curves in the Euclidean sphere.
In [MS16a, MS16b], we will also extend these results to infinite volume surfaces (the so-called Brownian plane [CL12] and the -LQG quantum cone [She15, DMS14]) and to surfaces with boundary (the Brownian disk and its LQG analog).
Before we begin to explain how these results will be proved, let us describe one reason one might expect them to be true. Both TBM and the -LQG sphere are known to be scaling limits of the uniformly random planar map with edges, albeit w.r.t. different topologies.111TBM is the scaling limit w.r.t. the Gromov-Hausdorff topology on metric spaces [LGP08, LG13, Mie13, LG14]. The -LQG sphere (decorated by ) is the scaling limit of the uniformly random planar map (decorated by critical percolation) w.r.t. the so-called peanosphere topology, as well as a stronger topology that encodes loop lengths and intersection patterns (see [She11, DMS14], the forthcoming works [GS15a, GS15b, GM16], and the brief outline in [MS15a]). One should therefore be able to use a compactness argument to show that the uniformly random planar map has a scaling limit (at least subsequentially) in the product of these topologies, and that this scaling limit is a coupling of TBM and the -LQG sphere. It is not obvious that, in this coupling, the instance of TBM and the instance of the -LQG sphere a.s. uniquely determine one another. But it seems reasonable to guess that this would be the case. And it is at least conceivable that one could prove this through a sophisticated analysis of the planar maps themselves (e.g., by showing that pairs of random planar maps are highly likely to be close in one topology if and only if they are close in the other topology).
Another reason to guess that an LQG sphere should have a canonical metric structure, and that TBM should have a canonical conformal structure, is that it is rather easy to formulate reasonable sounding conjectures about how a metric on an LQG sphere might be obtained as limit of approximate metrics, or how a conformal structure on TBM might be obtained as a limit of approximate conformal structures. For example, the peanosphere construction of [DMS14] gives a space-filling curve on the LQG sphere; one might divide space into regions traversed by length- increments of time, declare two such regions adjacent if they intersect, and conjecture that the corresponding graph distance (suitably rescaled) converges to a continuum distance as . Similarly, an instance of TBM comes with a natural space-filling curve; one can use this to define a graph structure as above, embed the graph in the Euclidean sphere using circle packing (or some other method thought to respect conformal structure), and conjecture that as these embeddings converge to a canonical (up to Möbius transformation) embedding of TBM in the Euclidean sphere. In both of these cases, the approximating graph can be constructed in a simple way (in terms of Brownian motion or the Brownian snake) and could in principle be studied directly.
The current series of papers will approach the problem from a completely different direction, which we believe to be easier and arguably more enlightening than the approaches suggested above. Instead of using approximations of the sort described above, we will use a combination of the quantum Loewner evolution (QLE) ideas introduced in [MS13b], TBM analysis that appears in [MS15a], and the -LQG sphere analysis that appears in [MS15b]. There are approximations involved in defining the relevant form of QLE, but they seem to respect the natural symmetries of the problem in a way that the approximation schemes discussed above do not. In particular, our approach will allow us to take full advantage of an exact relationship between the LQG disks “cut out” by and those cut out by a metric exploration.
In order to explain our approach, let us introduce some notation. If is a metric space, like TBM, and then we let denote the radius ball centered at . If the space is homeomorphic to and comes with a distinguished “target” point , then we let denote the filled metric ball of radius , i.e., the set of all points that are disconnected from by . Note that if , then the complement of contains and is homeomorphic to the unit disk .
The starting point of our approach is to let and be points on an LQG-sphere and to define a certain “growth process” growing from to . We assume that and are “quantum typical,” i.e., that given the LQG-sphere itself, the points and are independent samples from the LQG measure on that sphere. The growth process is an increasing family of closed subsets of the LQG-sphere, indexed by a time parameter , which we denote by . Ultimately, the set will represent the filled metric ball corresponding to an appropriately defined metric on the LQG-sphere (when is taken to be the distinguished target point). However, we will get to this correspondence somewhat indirectly. Namely, we will first define as a random growth process (for quantum typical points and ) and only show a posteriori that there is a metric for which the thus defined are a.s. the filled metric balls.
As presented in [MS13b], the idea behind this growth process (whose discrete analog we briefly review and motivate in Section 3) is that one should be able to “reshuffle” the decorated quantum sphere in a particular way in order to obtain a growth process on a LQG surface that hits points in order of their distance from the origin. This process is a variant of the process originally constructed in [MS13b] by starting with an process and then “resampling” the tip location at small capacity time increments to obtain a type of first passage percolation on top of a -LQG surface. The form of that we use here differs from that given in [MS13b] in that we will resample the tip at “quantum natural time” increments as defined in [DMS14] (i.e., time steps which are intrinsic to the surface rather than to its specific choice of embedding). We expect that these two constructions are in fact equivalent, but we will not establish that fact here.
As discussed in [MS13b], the growth process can in some sense be understood as a continuum analog of the Eden model. The idea explained there is that in some small-increment limiting sense, the (random) Eden model growth should correspond to (deterministic) metric growth. In fact, a version of this statement for random planar maps has recently been verified in [CL14a], which shows that on a random planar map, the random metric associated with an Eden model (or first passage percolation) instance closely approximates graph distance.
Once we have defined the growth process for quantum typical points and , we will define the quantity to be the amount of time it takes for a growth process to evolve from to . This is a good candidate to be a distance function, at least for those and for which it is defined. However, while our initial construction of will produce the joint law of the doubly marked LQG surface and the growth process , it will not be obvious from this construction that almost surely. In fact, it will not even be obvious that the growth process is a.s. determined by the LQG sphere and the pair, so we will a priori have to treat as a random variable whose value might depend on more than the LQG-surface and the pair.
The bulk of the current paper is dedicated to showing that if one first samples a -LQG sphere, and then samples as i.i.d. samples from its LQG measure, and then samples conditionally independent growth processes from each to each , then it is almost surely the case that the defined from these growth processes is a metric, and that this metric is determined by the LQG sphere and the points, as stated in Theorem 1.1 below.
Both the -LQG sphere and TBM have some natural variants that differ in how one handles the issues of total area and special marked points; these variants are explained for example in [MS15a, MS15b]. On both sides, there is a natural unit area sphere measure in which the total area measure is a.s. one. On both sides, one can represent a sphere of arbitrary positive and finite area by a pair , where is a unit area sphere and is a positive real number. The pair represents the unit area sphere except with area scaled by a factor of and distance (where defined) scaled by a factor of . On both sides it turns out to be natural to define infinite measures on quantum spheres such that the “total area” marginal has the form for some . In particular, on both sides, one can define a natural “grand canonical” quantum sphere measure on spheres with marked points (see the notation in Section 1.3). Sampling from this infinite measure amounts to
first sampling a unit area sphere ,
then sampling marked points i.i.d. from the measure on ,
then independently selecting from the infinite measure and rescaling the sphere’s area by a factor of (and distance by a factor of ).
Theorem 1.1, stated below, applies to all of these variants. Recall that in the context of an infinite measure, almost surely means outside a set of measure zero.
Suppose that is an instance of the -quantum sphere, as defined in [DMS14, MS15b] (either the unit area version or the “grand canonical” version involving one or more distinguished points). Let be a sequence of points chosen independently from the quantum area measure on . Then it is a.s. the case that for each pair , the quantity is uniquely determined by and the points and . Moreover, it is a.s. the case that
for all distinct and , and
satisfies the (strict) triangle inequality for all distinct , , and .
The fact that the triangle inequality in Theorem 1.1 is a.s. strict implies that if the metric can be extended to a geodesic metric on the entire LQG-sphere (something we will establish in the subsequent paper [MS16a]) then in this metric it is a.s. the case that none of the points on the countable sequence lies on a geodesic between two other points in the sequence. This is unsurprising given that, in TBM, the measure of a geodesic between two randomly chosen points is almost surely zero. (This is well known and essentially immediate from the definition of TBM; see [MS15a] for some discussion of this point.)
The construction of the metric in Theorem 1.1 is “local” in the sense that it only requires that the field near any given point is absolutely continuous with respect to a -LQG sphere. In particular, Theorem 1.1 yields a construction of the metric on a countable, dense subset of any -LQG surface chosen i.i.d. from the quantum measure. Moreover, the results of the later papers [MS16a, MS16b] also apply in this generality, which allows one to define geodesic metrics on other -LQG surfaces, such as the torus constructed in [DRV15].
The proof of Theorem 1.1 is inspired by a closely related argument used in a paper by the second author, Sam Watson, and Hao Wu (still in preparation) to define a metric on the set of loops in a process. To briefly sketch how the proof goes, suppose that we choose a -LQG sphere with marked points and , and then choose a growth process from to and a conditionally independent growth process from to . We also let be chosen uniformly in independently of everything else. Let be the joint distribution of . Since the natural measure on -LQG spheres is an infinite measure, so is . However, we can make sense of conditioned on as a probability measure. Given , we have that and are well defined as random variables denoting the respective time durations of and . As discussed above, we interpret (resp. ) as a measure of the distance from to (resp. to ). Write for the weighted measure . In light of the uniqueness of Radon-Nikodym derivatives, in order to show that almost surely, it will suffice to show that .
The main input into the proof of this is Lemma 1.2, stated below (which will later be restated slightly more precisely and proved as Theorem 7.1). Suppose we sample from , then let — so that is uniform in , and then define . Both and are understood as growth processes truncated at random times, as illustrated in Figure 1.1. We also let and . Under , we have that is uniform in .
Using the definitions above, the -induced law of is equal to the -induced law of .
The proof of Lemma 1.2 is in some sense the heart of the paper. It is established in Section 7, using tools developed over several previous sections, which in turn rely on the detailed understanding of processes on -LQG spheres developed in [DMS14, MS15b] as well as in [MS12a, MS12b, MS12c, MS13a]. We note that the intuition behind this symmetry was also sketched at the end of [MS15a] in the context of TBM.
To derive from Lemma 1.2, it will suffice to show the following:
The conditional law of , given is the same as the conditional law of , given .
Intuitively, sampling from either conditional law should amount to just continuing the evolution of and on , beyond their given stopping times, independently of each other. However, it will take some work to make this intuition precise. The remainder of this subsection provides a short overview of the argument.
Because of the way that was constructed, one can show quite easily that the conditional law of given the five-tuple is a function of the four-tuple . (The existence of regular conditional probabilities follows from the fact that the Gaussian free field and associated growth processes can be defined as random variables in a standard Borel space; see further discussion for example in [SS13, MS13b].) The proof of Lemma 1.3 will be essentially done once we establish the following claim: the conditional law of given the five-tuple is described by the same function applied to the four-tuple . Indeed, a symmetric argument implies an analogous statement about the conditional law of under and given the corresponding five-tuple. Lemma 1.3 will follow readily from this symmetric pair of statements and the a priori conditional independence of and given .
The claim stated just above may appear to be obvious, but there is still some subtlety arising from the fact that and are not defined in symmetric ways a priori, and in fact is a complicated stopping time for (which depends both on and on the additional randomness encoded in and ), and we have not proved anything like a strong Markov property that would hold for arbitrary stopping times. To deal with this rather technical point, we will approximate both and by stopping times that a.s. take only countably many values, noting that the required Markov property is easy to derive for such stopping times. To prove that the conditional laws computed using these approximations converge to the appropriate limits, we will check that for any bounded function the process is a.s. continuous (as a function of ) at the times and , respectively. Since these processes are martingales, they a.s. have only countably many discontinuities; it will thus suffice to prove that when is given and (resp. ) is chosen uniformly from (resp. ) the probability that either or assumes any fixed value is zero. This is obviously true for , and it follows for because the map (a random function that depends on ) is non-increasing by definition, and the symmetry property implies that a.s. first hits at exactly time , which implies that there a.s. cannot be a positive interval of values on which is constant.
We will see a posteriori that , which we will prove by using the fact that and the symmetry of Lemma 1.2, which implies that both and are -conditionally uniform on , once is given. We will also use this fact to derive the triangle inequality. Note that if is a third point and we are working on the event that , then and must intersect each other, at least at the point . In fact, it will not be hard to see that a.s. for some the processes and still intersect. This implies that if then . Plugging in , we obtain the strict triangle inequality.
1.2 Further observations and sequel overview
In the course of establishing Theorem 1.1, it will also become clear (almost immediately from the definitions) that the growth process from to and the growth process from to almost surely agree up until some random time at which and are first separated from each other, after which the two processes evolve independently. Thus one can describe the full collection of growth processes from to all the other points in terms of a single “branching” growth process with countably many “branch times” (i.e., times at which some and are separated for the first time).
It will also become clear from our construction that when exploring from a marked point to a marked point , one can make sense of the length of , and that as a process indexed by this evolves as the time-reversal of an excursion of a continuous state branching process (CSBP), with the jumps in this process corresponding to branch times.222We will see in [MS16a] that if one defines a quantum-time on an infinite volume LQG surface, namely a quantum cone, then the evolution of the boundary length is a process that matches the one described by Krikun (discrete) [Kri05] and Curien and Le Gall [CL14b] for the Brownian plane [CL12]. We will review the definition of a CSBP in Section 2.1. Letting vary over all of the points , one obtains a branching version of a time-reversed CSBP excursion, and it will become clear that the law of this branching process agrees with the analogous law for TBM, as explained in [MS15a].
All of this suggests that we are well on our way to establishing the equivalence of the -LQG sphere and TBM. As further evidence in this direction, note that it was established in [MS15a] that the time-reversed branching process (together with a countable set of real numbers indicating where along the boundary each “pinch point” occurs) contains all of the information necessary to reconstruct an instance of the entire Brownian map. That is, given a complete understanding of the exploration process rooted at a single point, one can a.s. reconstruct the distances between all pairs of points. This suggests (though we will not make this precise until the subsequent paper [MS16a]) that, given the information described in the QLE branching process provided in this paper, one should also be able to recover an entire Brownian map instance.
In order to finish the project, the program in [MS16a] will be to
Derive some continuity estimates and use them to show that a.s. extends uniquely to a metric defined on the entire LQG sphere, and to establish Hölder continuity for the identify map from the sphere endowed with the Euclidean metric to the sphere endowed with the random metric, and then
This will imply that the metric space described by has the law of TBM and, moreover, that the instance of TBM is a.s. determined by the underlying -LQG sphere. The program in [MS16b] will be to prove that in the coupling between TBM and the -LQG sphere, the former a.s. determines the latter, i.e., to show that an instance of TBM a.s. has a canonical embedding into the sphere. Thus we will have that the -LQG sphere and TBM are equivalent in the sense that an instance of one a.s. determines the other. The ideas used in [MS16b] will be related to the arguments used in [DMS14] to show that an instance of the peanosphere a.s. has a canonical embedding.
1.3 Prequel overview
As noted in Section 1.1, both TBM and the -LQG sphere have natural infinite volume variants that in some sense correspond to grand canonical ensembles decorated by some fixed number of marked points. In this paper, because we deal frequently with exploration processes from one marked point to another, we will be particularly interested in the natural infinite measures on doubly marked spheres. We recall that
In [MS15a] this natural measure on doubly marked Brownian map spheres with two marked points is denoted (and more generally refers to the measure with marked points).
In [MS15b] the natural measure on doubly marked -LQG spheres is denoted by .
As noted in Section 1.1, in both cases, the law of the overall area is given (up to a multiplicative constant) by . In both cases, the conditional law of the surface given is that of a sample from a probability measure on unit area surfaces (with the measure rescaled by a factor of , and distance rescaled by — though of course distance is not a priori defined on the LQG side). We remark that in much of the literature on TBM the unit area measure is the primary focus of attention (and it is denoted by in [MS15a]).
The paper [MS15b] explains how to explore a doubly marked surface sampled from with an curve drawn from to . The paper [MS15a] explains how to explore a doubly marked surface sampled from by exploring the so-called “metric net,” which consists of the set of points that lie on the outer boundary of for some . (We are abusing notation slightly here in that represents a quantum surface in the first case and a metric space in the second, and these are a priori different types of objects.) In both cases, the exploration/growth procedure “cuts out” a countable collection of disks, each of which comes with a well defined boundary length. Also in both cases, the process that encodes the boundary length corresponds (up to time change) to the set of jumps in the time-reversal of a -stable Lévy excursion with only positive jumps. Moreover, in both cases, the boundary length of each disk “cut out” is encoded by the length of the corresponding jump in the time-reversed -stable Lévy excursion. Finally, in both cases, the disks can be understood as conditionally independent measure-endowed random surfaces, given their boundary lengths.
The intuitive reason for the similarities between these two types of explorations is explained in the QLE paper [MS13b], and briefly reviewed in Section 3. The basic idea is that in the discrete models involving triangulations, the conditional law of the unexplored region (the component containing ) does not depend on the rule one uses to decide which triangle to explore next; if one is exploring via the Eden model, one picks a random location on the boundary to explore, and if one is exploring a percolation interface, one explores along a given path. The law of the set of disks cut out by the exploration is the same in both cases.
The law of a “cut out” disk, given that its boundary length is , is referred to as in [MS15a]. If one explores up to some stopping time before encountering , then the conditional law of the unexplored region containing is that of a marked Brownian disk with boundary length (here is the marked point), and is referred to as in [MS15a]. It is not hard to describe how these two measures are related. If one forgets the marked point , then both and describe probability measures on the space of quantum disks; and from this perspective, the Radon-Nikodym derivative of w.r.t. is given (up to multiplicative constant) by the total surface area. Given the quantum disk sampled from , the conditional law of the marked point is that of a sample from the quantum measure on the surface.
Precisely analogous statements are given in [MS15b] for the exploration of a sample from .333These results are in turn consequences of the fact, derived by the authors and Duplantier in an infinite volume setting in [DMS14], that one can weld together two so-called Lévy trees of -LQG disks to produce a new -LQG surface decorated by an independent curve that represents the interface between the two trees. The following objects are shown in [MS15b] well defined, and are analogous to objects produced by the measures and in [MS15a]:
The -LQG disk with boundary length . This is a random quantum surface whose law is the conditional law of a surface cut out by the exploration, given only its boundary length (and not its embedding in the larger surface).
The marked -LQG disk with boundary length . This is a random quantum surface whose law is obtained by weighting the unmarked law by total area, and letting the conditional law of given the surface be that of a uniformly random sample from the area measure (normalized to be a probability measure). It represents the conditional law of the unexplored quantum component containing .
Consider a doubly marked -LQG sphere decorated by an independent whole plane path from its first marked point to its second marked point . We consider to be parameterized by its quantum natural time. Fix an and let denote the outer boundary of the closed set , i.e., the boundary of the -containing component of the complement of . Then the conditional law of the -containing region (given that its boundary length is ) is that of a marked -LQG disk with boundary length . In particular, since this law is rotationally invariant, the overall law of the surface is unchanged by the following random operation: “cut” along , rotate the disk cut out by a uniformly random number in , and then weld this disk back to the beaded quantum surface (again matching up quantum boundary lengths).
It is natural to allow to range over integer multiples of a constant . As illustrated in Figure 1.2, we let denote the “necklace” described by the union , which we interpret as a beaded quantum surface (see [DMS14]) attached to a “string” of some well defined length. Applying the above resampling for each integer multiple of corresponds to “reshuffling” these necklaces in the manner depicted in Figure 3.4.
Fix and apply the random rotation described in Proposition 1.4 for each that is an integer multiple of . Taking any subsequential limit as , we obtain a coupling of a -quantum sphere with a growth process on that sphere, such that the law of the ordered set of disks cut out by that process is the same as in the case.
The growth process obtained this way is what we will call the quantum natural time (as opposed to the capacity time process described in [MS13b], which we expect but do not prove to be equivalent to the quantum time version). As already noted in Section 1.1, we will make extensive use of quantum natural time in this paper. When we use the term without a qualifier, we will mean the quantum natural time variant.
Let us highlight one subtle point about this paper. Although we a priori construct only subsequential limits for the growth process using the procedure described in Proposition 1.5, we ultimately show that the metric defined on a countable sequence of i.i.d. points chosen from the quantum measure does not depend on the particular choice of subsequence. Once we know this metric we know, for each and each and in , which points from the set lie in the set . Since is closed, we would expect it to be given by precisely the closure of this set of points, which would imply that the growth process described in Proposition 1.5 is a.s. defined as a true (non-subsequential) limit. This would follow immediately if we knew, say, that was a.s. the closure of its interior. However, we will not prove in this paper that this is the case. That is, we will not rule out the possibility that the boundary of contains extra “tentacles” that possess zero quantum area and somehow fail to intersect any of the values. Ruling out this type of behavior will be part of the program in [MS16a], where we establish a number of continuity estimates for and . Upon showing this, we will be able to remove the word “subsequential” from the statement of Proposition 1.5.
The remainder of this article is structured as follows. In Section 2 we review preliminary facts about continuous state branching processes, quantum surfaces, and conformal removability. In Section 3, we recall some of the discrete constructions on random planar triangulations that appeared in [MS13b], which we use to explain and motivate our continuum growth processes. In particular, we will recall that on these triangulated surfaces random metric explorations are in some sense “reshufflings” of percolation explorations, and in Section 4 we construct quantum-time using an analogous reshuffling of . In Section 5 we establish a certain symmetry property for continuum percolation explorations () on -LQG surfaces (a precursor to the main symmetry result we require). Then in Section 6 we will give the construction of the quantum natural time variant of . In Section 7, we establish the main symmetry result we require, and in Section 8 we will finish the proof of Theorem 1.1.
2.1 Continuous state branching processes
We will now review some of the basic properties of continuous state branching processes (CSBPs) and their relationship to Lévy processes. CSBPs will arise in this article because they describe the time-evolution of the quantum boundary length of the boundary of a . We refer the reader to [Ber96] for an introduction to Lévy processes and to [LG99, Kyp06] for an introduction to CSBPs.
A CSBP with branching mechanism (or -CSBP for short) is a Markov process on whose transition kernels are characterized by the property that
where , , is the non-negative solution to the differential equation
be the extinction time for . Then we have that [Kyp06, Corollary 10.9]
A -CSBP can be constructed from a Lévy process with only positive jumps and vice-versa [Lam67] (see also [Kyp06, Theorem 10.2]). Namely, suppose that is a Lévy process with Laplace exponent . That is, if then we have that
Then the time-changed process is a -CSBP. Conversely, if is a -CSBP and we let
then is a Lévy process with Laplace exponent .
We will be interested in the particular case that for . For this choice, we note that
2.2 Quantum surfaces
Suppose that is an instance of (a form of) the Gaussian free field (GFF) on a planar domain and is fixed. Then the -LQG surface associated with is described by the measure which is formally given by where denotes Lebesgue measure on . Since the GFF does not take values at points (it is a random variable which takes values in the space of distributions), it takes some care to make this definition precise. One way of doing so is to let, for each and such that , be the average of on (see [DS11, Section 3] for more on the circle average process). The process is jointly continuous in and one can define to be the weak limit as along negative powers of of [DS11]; the normalization factor is necessary for the limit to be non-trivial. We will often write for the measure . In the case that has free boundary conditions, one can also construct the natural boundary length measure in a similar manner.
The regularization procedure used to construct leads to the following change of coordinates formula [DS11, Proposition 2.1]. Let
Suppose that are planar domains and is a conformal map. If is (a form of) a GFF on and
for all measurable sets . This allows us to define an equivalence relation on pairs by declaring and to be equivalent if and are related as in (2.10). An equivalence class of such a is then referred to as a quantum surface.
More generally, suppose that are planar domains, for are given points, and is a distribution on . Then we say that the marked quantum surfaces are equivalent if there exists a conformal transformation , for each , and , are related as in (2.10).
In this work, we will be primarily interested in two types of quantum surfaces, namely quantum disks and spheres. We will remind the reader of the particular construction of a quantum sphere that we will be interested in for this work in Section 4.1. We also refer the reader to [MS15b] and [DMS14] for a careful definition of a quantum disk as well as several equivalent constructions of a quantum sphere.
2.3 Conformal removability
An LQG surface can be obtained by endowing a topological surface with both a good measure and a conformal structure in a random way. (And we can imagine that these two structures are added in either order.) Given two topological disks with boundary (each endowed with a good area measure in the interior, and a good length measure on the boundary) it is a simple matter to produce a new good-measure-endowed topological surface by taking a quotient that involves gluing (all or part of) the boundaries to each other in a boundary length preserving way.
The problem of conformally welding two surfaces is the problem of obtaining a conformal structure on the combined surface, given the conformal structure on the individual surfaces. (See, e.g., [Bis07] for further discussion and references.) To make sense of this idea, we will draw from the theory of removable sets, as explained below.
A compact subset of a domain is called (conformally) removable if every homeomorphism from into that is conformal on is also conformal on all of . A Jordan domain is said to be a Hölder domain if any conformal transformation from to is Hölder continuous all of the way up to . It was shown by Jones and Smirnov [JS00] that if is the boundary of a Hölder domain, then is removable; it is also noted there that if a compact set is removable as a subset of , then it is removable in any domain containing , including all of . Thus, at least for compact sets , one can speak of removability without specifying a particular domain .
The following proposition illustrates the importance of removability in the setting of quantum surfaces (see also [DMS14, Section 1.5]):
Suppose that is a quantum surface, is compact such that for disjoint. Suppose that is another quantum surface, is compact such that for disjoint. Assume that is equivalent to as a quantum surface for and that, furthermore, there exists conformal transformations for which extend to a homeomorphism . If is conformally removable, then and are equivalent as quantum surfaces.
This follows immediately from the definition of conformal removability. ∎
One example of a setting in which Proposition 2.1 applies is when the quantum surface is given by a so-called quantum wedge and is the range of an curve for [She15, DMS14]. A quantum wedge naturally comes with two marked points and , which are also the seed and the target point of the curve. In this case, the conformal maps (which are defined on the left and right components of ) are chosen so that the quantum length of the image of a segment of as measured from the left and right sides matches up. With this choice, the extend to a homeomorphism of the whole domain and it shown in [RS05] that the range of is almost surely conformally removable, so Proposition 2.1 applies.
When we apply Proposition 2.1 in the current work, we will not always be welding quantum surfaces according to quantum boundary length as in [She15, DMS14] because it will not always be obviously true that we will be in a setting in which and have an intrinsically defined quantum boundary length. This may a priori be the case for the process we construct here because we will not rule out the possibility that the process contains “spikes” (i.e., we will not show in this work that it is the case that the range of a is equal to the closure of its interior; this will be a consequence of the results of [MS16a]).
3 Eden model and percolation interface
In this section we briefly recall a few constructions from [MS13b, Section 2], together with some figures included there. Figure 3.1 shows a triangulation of the sphere with two distinguished edges and , and the caption describes a mechanism for choosing a random path in the dual graph of the triangulation, consisting of distinct triangles , that goes from to . It will be useful to imagine that we begin with a single -gon and then grow the path dynamically, exploring new territory as we go. At any given step, we keep track of the total number edges on the boundary of the already-explored region and the number of vertices remaining to be seen in the component of the unexplored region that contains the target edge. The caption of Figure 3.2 explains one step of the exploration process. The exploration process induces a Markov chain on the set of pairs with and . In this chain, the coordinate is almost surely non-increasing, and the coordinate can only increase by when the coordinate decreases by .
Now consider the version of the Eden model in which new triangles are only added to the unexplored region containing the target edge, as illustrated Figure 3.3. In both Figure 3.1 and Figure 3.3, each time an exploration step separates the unexplored region into two pieces (each containing at least one triangle) we refer to the one that does not contain the target as a bubble. The exploration process described in Figure 3.1 created two bubbles (the two small white components), and the exploration process described in Figure 3.3 created one (colored blue). We can interpret the bubble as a triangulation of a polygon, rooted at a boundary edge (the edge it shares with the triangle that was observed when the bubble was created).
The specific growth pattern in Figure 3.3 is very different from the one depicted in Figure 3.1. However, the analysis used in Figure 3.2 applies equally well to both scenarios. The only difference between the two is that in Figure 3.3 one re-randomizes the seed edge (choosing it uniformly from all possible values) after each step.
In either of these models, we can define to be the boundary of the target-containing unexplored region after steps. If is the corresponding Markov chain, then the length of is for each . Let denote the union of the edges and vertices in , the edges and vertices in and the triangle and bubble (if applicable) added at step , as in Figure 3.4. We refer to each as a necklace since it typically contains a cycle of edges together with a cluster of one or more triangles hanging off of it. The analysis used in Figure 3.2 (and discussed above) immediately implies the following:
Consider a random rooted triangulation of the sphere with a fixed number of vertices together with two distinguished edges chosen uniformly from the set of possible edges. If we start at one edge and explore using the Eden model as in Figure 3.3, or if we explore using the percolation interface of Figure 3.1, we will find that the following are the same:
The law of the Markov chain (which terminates when the target 2-gon is reached).
The law of the total number of triangles observed before the target is reached.
The law of the sequence of necklaces.
Indeed, one way to construct an instance of the Eden model process is to start with an instance of the percolation interface exploration process and then randomly rotate the necklaces in the manner illustrated in Figure 3.4.
4 Infinite measures on quantum spheres
4.1 Lévy excursion description of doubly-marked quantum spheres
The purpose of this section is to review the results established in [MS15b] which are relevant for this article. First, we let be the measure which is defined as follows. Suppose that is a -stable Lévy process with only upward jumps and let be its running infimum. Then we let be the Itô excursion measure associated with the excursions that makes from . The law of the duration of such an excursion follows a power law. Indeed, following [Ber96], the process of sampling can be described as follows (see [Ber96, Section VIII.4]):
Pick a lifetime from the measure on where denotes Lebesgue measure and is a constant. Here, (where ) is the so-called positivity parameter of the process [Ber96, Section VIII.1].
Given , pick an excursion of length from the normalized excursion measure associated with a -stable Lévy process with only positive jumps and then rescale it spatially and in time so that it has length .
As explained in [MS15b], we can construct a doubly-marked quantum sphere decorated by a non-crossing path connecting and from such an excursion as follows. For each upward jump of , we sample a conditionally independent quantum disk whose boundary length is equal to the size of the jump. We assume that each of the quantum disks has a marked boundary point sampled uniformly from its boundary measure together with a uniformly chosen orientation of its boundary. We assume that the marked points and orientations are chosen conditionally independently given the realizations of the quantum disks. Let denote the collection of marked and oriented quantum disks sampled in this way. Then the pair together uniquely determines a doubly-marked surface which is homeomorphic to the sphere together with a non-crossing path which connects and . In particular, the time-reversal of describes the evolution of the quantum boundary length of the complementary component of in which contains and the jumps of describe the boundary lengths of the quantum disks that cuts off from . For each , we let be the component of which contains . The time-parameterization of so that the quantum boundary length of the component of which contains is equal to is the so-called quantum natural time introduced in [DMS14].
One of the main results of [MS15b] is that a doubly-marked surface/path pair produced from conditioned to have quantum mass equal to (although is infinite, this conditioning yields a probability measure) has the law of the unit area quantum sphere constructed in [DMS14], the points conditional on are chosen uniformly at random from the quantum measure, and the conditional law of given is that of a whole-plane process connecting and . This holds more generally when we condition the surface to have quantum mass equal to for any fixed except in this setting is a quantum sphere of mass rather than . (As noted earlier, a sample from the law of such a surface can be produced by starting with a unit area quantum sphere and then scaling its associated mass measure by the factor .)
This relationship between and the law of a unit area quantum sphere decorated with an independent whole-plane process implies that possesses certain symmetries. These symmetries will be important later on so we will pause for a moment to point them out.
If we condition on , then the points and are both chosen independently from the quantum area measure on .
If we condition on , , and , then is whole-plane from to .
The amount of quantum natural time elapsed for to travel from to is equal to (the time corresponding to the Lévy excursion).
This also implies that is invariant under the operation of swapping and and then reversing the time of [MS13a] (with the quantum natural time parameterization). To see the symmetry of the quantum natural time parameterization under time-reversal, we have from the previous observations that the law of the ordered collection of bubbles cut off by from its target point is invariant under the operation of swapping and and reversing the time of . The claim thus follows because the quantum natural time parameterization can be constructed by fixing , counting the number of bubbles cut off by with quantum boundary length in , and then normalizing by a constant times the factor . That this is the correct normalization follows since the Lévy measure for a -stable Lévy process is given by a constant times where denotes Lebesgue measure on . See, for example, [MS15b, Section 6.2] for additional discussion of this point as well as Remark 6.4 below in the context of the construction of .
We also emphasize that under , we have that:
is distributed uniformly from the quantum boundary measure associated with the quantum surface parameterized by (see [MS15b, Proposition 6.4]).
The components of , viewed as quantum surfaces, are conditionally independent given their boundary lengths. Those components which do not contain are quantum disks given their boundary lengths. The component which does contain has the law of a quantum disk weighted by its total quantum area.
4.2 Weighted measures
Throughout, we will work with the following two measures which are defined with as the starting point. Namely, with equal to the length of the associated Lévy excursion and the quantum boundary length of the complementary component of containing we write
where denotes Lebesgue measure. We note that the marginal of on and is given by , i.e. by weighting by the length of the Lévy excursion. (The additional subscript “” is to indicate that is a weighted measure.) It will be convenient throughout to think of as a measure on triples , , where is a uniformly random point chosen from the total length of the Lévy excursion.
The marginal of on and is given by where
As we will see later, after the “reshuffling” this will have the interpretation of taking and then weighting it by the amount of quantum distance from to . (The additional subscript “” is to indicate that is the “distance weighted” measure.)
We finish this section by recording the following proposition which relates the conditional law of and given and to .
Given , the conditional distribution of is the same as the conditional distribution of when we condition on the event that the length of the Lévy excursion is at least .
For both and , given and , the conditional distribution of is the same as the conditional distribution of when we condition on the event that the length of the Lévy excursion is at least and the given value of .
We will explain the argument in the case that ; the same argument gives part (ii).
If we fix the value of , the conditional distribution of the Lévy excursion in the definition of is given by where is the measure on -stable Lévy excursions which arises by scaling spatially and in time so that the excursion length is equal to . This representation clearly implies (i). A similar argument gives (ii). ∎
4.3 Continuum scaling exponents
We now determine the distribution of and under .
There exists constants such that
Proof of Proposition 4.2.
For an excursion sampled from we write . By scaling and the explicit form of described above, it is not difficult to see by making the change of variables that there exists a constant such that
For each , we let and let
By combining this with (2.6), it therefore follows that there exists a constant such that
Sending implies the result. ∎
5 Meeting in the middle
In this section, we shall assume that we are working in the setting described in Section 4.2. The main result is the following theorem which proves a certain symmetry statement for the measure . This result will later be used in Section 7 to prove an analogous symmetry result for which, in turn, is one of the main inputs in the proof of Theorem 1.1. See Figure 5.1 for an illustration of the result.