A universal law for Voronoï cell volumes in infinitely large maps
Abstract.
We discuss the volume of Voronoï cells defined by two marked vertices picked randomly at a fixed given mutual distance in random planar quadrangulations. We consider the regime where the mutual distance is kept finite while the total volume of the quadrangulation tends to infinity. In this regime, exactly one of the Voronoï cells keeps a finite volume, which scales as for large . We analyze the universal probability distribution of this, properly rescaled, finite volume and present an explicit formula for its Laplace transform.
1. Introduction
In a recent paper [5], we analyzed the volume distribution of Voronoï cells for some families of random bipointed planar maps. Recall that a planar map is a connected graph embedded in the sphere: it is bipointed if it has two marked distinct vertices. These marked vertices allow us to partition the map into two Voronoï cells, where each cell corresponds, so to say, to the part of the map closer to one marked vertex than to the other. The volume of, say the second Voronoï cell (that centered around the second marked vertex) is then a finite fraction of the total volume of the map, with , while the first cell clearly spans the complementary fraction . The main result proven in [5] is that, for several families of random bipointed maps with a fixed total volume, and in the limit where this volume becomes infinitely large, the law for the fraction of the total volume spanned by the second Voronoï cell is uniform in the interval , a property conjectured by Chapuy in [4] among other more general conjectures. Here it is important to stress that the above result holds when the two marked vertices are chosen uniformly at random in the map. In particular, their mutual distance is left arbitrary^{1}^{1}1For one family on maps considered in [5], it was assumed for convenience that the mutual distance be even, but lifting this constraint has no influence on the obtained result..
This paper deals on the contrary with Voronoï cells within random bipointed maps where the two marked vertices are picked randomly at a fixed given mutual distance. Considering again the limit of maps with an infinitely large volume and keeping the (fixed) mutual distance between the marked vertices finite, we find that only one of the Voronoï cells becomes infinitely large while the volume of the other remains finite. In particular, the fraction of the total volume spanned by this latter cell tends to while that of the infinite cell tends to . In other words, having imposed a fixed finite mutual distance between the marked vertices drastically modifies the law for the fraction which is now concentrated at if it is precisely the second Voronoï cell which remains finite or at if this second cell becomes infinite.
In this regime of fixed mutual distance, a good measure of the Voronoï cell extent is now provided by the volume of that of the two Voronoï cells which remains finite. The main goal of this paper is to compute the law for this finite volume, in particular in a universal regime where the mutual distance, although kept finite, is large.
The paper is organized as follows: we first introduce in Section 2 the family of bipointed maps that will shall study (i.e. bipointed quadrangulations), define the volumes of the associated Voronoï cells and introduce some generating function with some control on these volumes (Section 2.1). We then discuss the scaling function which captures the properties of this generating function in some particular scaling regime (Section 2.2), and whose knowledge is the key of the subsequent calculations. Section 3 is devoted to our analysis of Voronoï cell volumes in the regime of interest in this paper, namely when the maps become infinitely large and the mutual distance between the marked vertices remains finite. We first analyze (Section 3.1) the law for the fraction of the total volume of the map spanned by the second Voronoï cell and show, as announced above, that it is evenly concentrated at or . We then analyze (Section 3.2) map configurations for which the volume of the second Voronoï cell remains finite and show how to obtain, from the simple knowledge of the scaling function introduced above, the law for this (properly rescaled) volume when the mutual distance becomes large. This leads to an explicit universal expression (Section 3.3) for the probability distribution of the finite Voronoï cell volume (in practice for its Laplace transform), whose properties are discussed in details. Section 4 proposes an instructive comparison of our result with that, much simpler, obtained for Voronoï cells within bipointed random trees. Section 5 discusses the case of asymmetric Voronoï cells where some explicit bias in the evaluation of distances is introduced. Our conclusions are gathered in Section 6. A few technical details, as well as explicit but heavy intermediate expressions, are given in various appendices.
2. Voronoï cells in bipointed maps
2.1. A generating function for bipointed maps with a control on their Voronoï cell volumes
The objects under study in this paper are bipointed planar quadrangulations, namely planar maps whose all faces have degree , and with two marked distinct vertices. We moreover demand that these vertices, distinguished as and , be at some even graph distance , namely
for some fixed given integer . Given and , the corresponding two Voronoï cells are obtained via some splitting of the map into two domains which, so to say, regroup vertices which are closer to one marked vertex than to the other. As discussed in details in [5], a canonical way to perform this splitting consists in applying the wellknow Miermont bijection [7] which transforms a bipointed planar quadrangulation into a socalled planar isolabelled twoface map (il.2.f.m), namely a planar map with exactly two faces, distinguished as and and with vertices labelled by positive integers satisfying:

labels on adjacent vertices differ by or ;

the minimum label for the set of vertices incident to is ;

the minimum label for the set of vertices incident to is .
As recalled in [5], the Miermont bijection provides a onetoone correspondence between bipointed planar quadrangulations and planar il.2.f.m, the labels of the vertices corresponding precisely to their distance to the closest marked vertex in the quadrangulation. More interestingly, by drawing the original quadrangulation on top of its image, the two faces and define de facto two domains in the quadrangulation which are perfect realizations of the desired two Voronoï cells as, by construction, each of these domains regroups vertices closer to one marked vertex. Since faces of the quadrangulation are, under the Miermont bijection, in correspondence with edges of the il.2.f.m, the volume ( number of faces) of a given cell in the quadrangulation is measured by half the number of edge sides incident to the corresponding face in the il.2.f.m. Note that this volume is in general some halfinteger since a number of faces of the quadrangulation may be shared by the two cells (see [5] for details). To be precise, an il.2.f.m is made of a simple closed loop separating its two faces and ^{2}^{2}2This loop is simply formed by the cyclic sequence of edges incident to both faces. together with a number of subtrees attached to vertices along , possibly on each side of the loop. If we call and the total number of edges for subtrees in the face and respectively, and the length ( number of edges) of the loop , the volumes and of the Voronoï cells are respectively
for a total volume
Finally, the requirement that translates into the following fourth label constraint:

the minimum label for the set of vertices incident to is .
Having defined Voronoï cells, we may control their volume by considering the generating function of bipointed planar quadrangulations where , with a weight
From the Miermont bijection and the associated canonical construction of Voronoï cells, is also the generating function of il.2.f.m satisfying the extra requirement with a weight
As such, may, via some appropriate decomposition of the i.l.2.f.m, be written as (see [5])
(1) 
(here is the finite difference operator ), where is some generating function for appropriate chains of labelled trees (which correspond to appropriate open sequences of edges with subtrees attached on either side of the incident vertices). Without entering into details, it is enough for the scope of this paper to know that the generating function is entirely determined^{3}^{3}3This relation fully determines for all order by order in and , i.e. is fully determined order by order in . by the relation (obtained by a simple splitting of the chains)
(2) 
for , where the quantity (as well as its analog ) is a well known generating function for appropriate labelled trees. It is given explicitly by
(3) 
where is taken in the range and parametrizes (in the range for a proper convergence of the generating function). For , the solution of (2) can be made explicit and reads
(4) 
Unfortunately, no such explicit expression is known for when and the relation (1) might thus appear of no practical use at a first glance. As discussed in [5], this is not quite true as we may recourse to appropriate scaling limits of all the above generating functions to extract explicit statistics on Voronoï cell volumes in a limit where the maps become (infinitely) large. Let us now discuss this point.
2.2. The associated scaling function
The limit of large quadrangulations (i.e. with a large number of faces) is captured by the singularity of whenever or tends toward its critical value . As we shall see, in all cases of interest, this singularity may be analyzed by setting
(5) 
and letting tend to . In this limit, we have for instance the following expansion for the quantity parametrizing in (3):
so that, for (i.e. ), we easily get from the exact expression (4) of the expansion
(6) 
Since is regular when , the most singular part of this generating function is given by
and we thus deduce that the number of bipointed planar quadrangulations with faces and with their two marked vertices at distance behaves at large as
(7) 
When itself becomes large, this number behaves as
(8) 
Note that this later estimate assumes that becomes first arbitrarily large with a value of remaining finite, and only then is set to be large. This order of limits corresponds to what is usually called the local limit. In particular, and do not scale with each other. Now it is interesting to note that getting this last result (8) does not require the full knowledge of and may be obtained upon using instead some simpler scaling function which captures the behavior of in a particular scaling regime. Consider indeed the generating function in a regime where as above by letting in (5), but where we let simultaneously and become large upon setting
with and kept finite. In this scaling regime, we have the expansion
where the function is given explicitly from (4) by
This in turn implies the expansion
which yields
where the scaling function associated with reads explicitly
(9) 
A crucial remark is that we recognize in this latter small expansion of the large leading behavior^{4}^{4}4In particular, we have the large expansion: . of the coefficients in the expansion (6) for in the local limit. For instance, the large behavior of the singular term (proportional to ) in (6) is given by
For (in which case we have the direct identification ), the left hand side is precisely the coefficient in (7), so that the result (8) may thus be read off directly on the expression of the scaling function via
(10) 
without recourse to the explicit knowledge of the full generating function .
The origin of this “scaling correspondence”, which connects the local limit at large to the scaling limit at small is explained in details in the next section. This correspondence is in fact a general property and can be applied in the situation where . It therefore allows us to access the large limit of the large asymptotics of (again sending first) from the simple knowledge of the scaling function associated with . As of now, let us already fix our notations for scaling functions when and are arbitrary: parametrizing and as in (5) above, we have when the expansion
with a scaling function which, from (2) expanded at lowest nontrivial order in , is solution of the nonlinear partial differential equation
(11) 
Here is the first nontrivial term in the small expansion of , namely, from its explicit expression (3),
(12) 
As for , we may now use (1) to relate the associated scaling function to , namely
(13) 
Scaling functions are in general much simpler than the associated full generating functions. In particular, although we have no formula for for arbitrary and , an explicit expression for is known for arbitrary and , as first obtained in [5] upon solving (11) with appropriate boundary conditions. We may thus recourse to this result to get an explicit expression for the scaling function itself via (13). The corresponding formula is quite heavy and its form is not quite illuminating. Still, we display it in Appendix A for completeness (the reader may refer to this expression to check the various limits and expansions of displayed hereafter in the paper).
As opposed to , the scaling function is thus known exactly and we will now show in details how to use the scaling correspondence to deduce from its small expansion the large limit of the large asymptotics of and control the volume of, say, the second Voronoï cell in large quadrangulations, by some appropriate choice of .
3. Infinitely large maps with two vertices at finite distance
This section is devoted to estimating the law for the volumes spanned by the Voronoï cells in bipointed quadrangulations whose total volume ( number of faces) tends to infinity. Calling and the two Voronoï cell volumes, we have so it is enough to control one of two volumes, say . Here the distance between the two marked vertices is kept finite (possibly large) when .
3.1. The law for the proportion of the total volume spanned by one Voronoï cell
For large and finite , the first natural way to measure is to express it in units of , i.e. consider the proportion
of the total volume spanned by the second Voronoï cell. We have of course and the large asymptotic probability law for may be obtained from via
since .
From the scaling correspondence, the large asymptotics of is, at large , encoded in the small expansion of the scaling function for some appropriate . The right hand side of the above equality may thus be computed explicitly at large from the knowledge of . This computation, together with the precise correspondence between the large local limit and the small scaling limit, is discussed in details in Appendix B. We decided however not to develop the calculation here since the resulting law is in fact trivial. As might have been guessed by the reader, we indeed find
(14) 
or equivalently
This result simply states that, for and finite large, only one of the Voronoï cells has a volume of order with, by symmetry,
(15) 
The main purpose of this paper, discussed in the following sections, is precisely to characterize the volume of the Voronoï cell which is an . As we shall see, the volume of this Voronoï cell remains actually finite and scales as when becomes large.
3.2. Infinitely large maps with a finite Voronoï cell
This section and the next one present our main result, namely the law for the (properly rescaled) volume of the Voronoï cell which is not of order when . More precisely, we will concentrate here on map configurations for which the total volume tends to infinity but the volume is kept finite. We will then verify a posteriori that the number of these configurations represents of the total number of bipointed maps whenever is large. This will de facto prove that the configurations for which in (15) are in fact, with probability , configurations for which is finite.
Let us denote by
the number of planar bipointed quadrangulations with fixed given values of and . In the limit with a fixed finite , this number may be estimated from the leading singularity of when for a fixed value of (see [6] for a detailed argument of a fully similar estimate in the context of hull volumes). We have indeed
where is the coefficient of the leading singularity of when at fixed , hence is obtained from the expansion^{5}^{5}5The precise form of this expansion is dictated by the similar explicit expansion (6) for . In particular, the absence of singular term is imposed by the fact that such a term, if present, would imply that be of order at large while this quantity is clearly bounded by which, as we have seen, is of order only.
Upon normalizing by the total number of bipointed maps with fixed and , whose asymptotic behavior is given by (7), we deduce the limiting probability that the second Voronoï cell has volume :
This probability for arbitrary finite may be encoded in the generating function
(16) 
where is a weight per unit volume. Recall that may take half integer values so that the sum on the left hand side above actually runs over all (positive) halfintegers.
Let us now discuss the scaling correspondence in details^{6}^{6}6We discuss here the general case where is fixed while . Our arguments could be repeated verbatim to the case to explain the scaling correspondence in this case, as observed directly from the explicit expressions of and .. Its origin is best understood by considering the all order expansion of for , namely
(with as discussed in the footnote 5). We may indeed, via the identification for , relate the scaling function to this expansion upon writing
Since depends only on the product , the quantity , which depends a priori on , and , is actually a function of the two variables and only, or equivalently of the two variables and . We deduce from the very existence of the scaling function above that^{7}^{7}7The fact that all the , , are not zero is verified a posteriori by the fact that has a small expansion involving non vanishing coefficients for all , .
(17) 
for (while ) with, moreover, the direct identification
with . This latter identity allows us in turn to identify the functions via
(18) 
where the last term was obtained by setting in the middle term since, being independent of , the middle term should be too for consistency. We may now come back to our estimate of (16) when is large. To obtain a non trivial law at large , we must measure in units of , i.e. consider the probability distribution for the rescaled volume defined by
This law is indeed captured by choosing , in which case the second argument, , of the numerator in (16) behaves as
Taking and in the above estimate (17) and using the identification (18), we may now write
leading eventually, using (10) and (9), to
(19) 
Since we have at our disposal an explicit expression for , this equation will give us a direct access to the desired law for .
3.3. Explicit expressions and plots
As a first, rather trivial, check of our expression (19), let us estimate the probability that remains finite in infinitely large planar bipointed quadrangulations. This probability is obtained by summing over all allowed finite values of , i.e. by setting in (16), i.e. in (19). It therefore takes the large value
For , the explicit expression for simplifies into
where the () and () are polynomials of degree and respectively in the variable , given by
We have in particular the small expansion
(20) 
which leads to a probability that be finite equal to
We thus see that configurations for which the second Voronoï cell remains finite whenever represent at large precisely of all the configurations. This is fully consistent with our result (15) provided that the configurations for which we found are actually configurations for which remains finite. Otherwise stated, configurations for which both and would diverge at large are negligible at (large) finite .
Beyond this first result at , we can consider, for any , the expectation value of for bipointed quadrangulations with and finite , conditioned to have their second Voronoï cell finite^{8}^{8}8Alternatively, we may lift this conditioning and interpret as the expectation value of where is the rescaled volume of the smallest Voronoï cell. . It is given by
and has a large limiting value
From the explicit expression of displayed in Appendix A, we deduce after some quite heavy computation the following expression for :
(21) 
where , and () are polynomials in of degree at most (with coefficients linear in for the last three polynomials), given explicitly by
The function