Volumes in the Uniform Infinite Planar Triangulation: from skeletons to generating functions
VVolumes in the Uniform Infinite Planar Triangulation:from skeletons to generating functions L AURENT M ÉNARD ∗ Modal’X, Université Paris Ouest and LiX, École Polytechnique
Abstract
We develop a method to compute the generating function of the number of vertices insidecertain regions of the Uniform Infinite Planar Triangulation (UIPT). The computations aremostly combinatorial in flavor and the main tool is the decomposition of the UIPT into layers,called the skeleton decomposition, introduced by Krikun [20]. In particular, we get explicitformulas for the generating functions of the number of vertices inside hulls (or completedmetric balls) centered around the root, and the number of vertices inside geodesic slices ofthese hulls. We also recover known results about the scaling limit of the volume of hullspreviously obtained by Curien and Le Gall by studying the peeling process of the UIPT in[17].
The probabilistic study of large random planar maps takes its roots in theoretical physics, whereplanar maps are considered as approximations of universal two dimensional random geometriesin Liouville quantum gravity theory (see for instance the book [4]). In the past decade, a lot ofwork has been devoted to make rigorous sense of this idea with the construction and study ofthe so-called Brownian map. The surveys [22, 26] will give the interested reader a nice overviewof the field as well as an up to date list of references.Since they are instrumental in every proof of convergence to the Brownian map, the mostsuccessful tools to study random planar maps are undoubtedly the various bijections betweencertain classes of maps and decorated trees. The search for such bijections was initiated by Coriand Vauquelin [14] and perfected by Schaeffer [29]. Since then, a lot of bijections in the samespirit have been discovered, (see in particular the one by Bouttier, Di Francesco and Guitter [11]).These bijections are particularly well suited to study metric properties of large random maps(see the seminal work of Chassaing and Schaeffer [13]), and they have lead to the remarkableproofs of convergence in the Gromov-Hausdorff topology of wide families of random maps tothe Brownian map by Le Gall [23] and Miermont [25] independently, paving the way to otherresults of convergence [1, 2, 10].Another very powerful tool to study random maps is the so called peeling process – informallya Markovian exploration procedure – introduced by Watabiki [30] and used immediately byWatabiki and Ambjørn to derive heuristics for the Hausdorff dimension of random maps in[3]. Probabilists started to show interest in this procedure a bit later, starting by Angel [5], whoformalized it in the setting of the Uniform Infinite Planar Triangulation (UIPT). Since then, this ∗ [email protected] a r X i v : . [ m a t h . P R ] A p r rocess has received growing attention and proved valuable to study not only the geometry ofrandom maps [5, 9, 12, 17], but also random walks [8], percolation [5, 6, 24, 28], and even, tosome extent, conformal aspects [15].In this work we will use another tool, introduced by Krikun [20], to study the UIPT, calledthe skeleton decomposition. Before we present this tool, let us recall that a planar map is aproper embedding of a connected multi-graph in the two dimensional sphere, considered up toorientation preserving homeomorphisms. The maps we consider will always be rooted (theyhave a distinguished oriented edge), and we will focus on rooted triangulations of type I in theterminology of Angel and Schramm [7], meaning that loops and multiple edges are allowedand that every face of the map is a triangle. The UIPT is the infinite random lattice defined asthe local limit of uniformly distributed rooted planar triangulations with n faces as n → ∞ (seeAngel and Schramm [7]). We will denote the UIPT by T ∞ and, if M is a (finite) planar map, wewill denote its number of vertices by | M | .For every integer r ≥
1, the ball B r ( T ∞ ) is the submap of T ∞ composed of all its faces havingat least one vertex at distance stricly less that r from the origin of the root edge. Since the UIPTis almost surely one ended, of all the connected component of T ∞ \ B r ( T ∞ ) , only one is infiniteand the hull B • r ( T ∞ ) is the complement in T ∞ of this unique infinite connected component (seeFigure 1 for an illustration). The layers of the UIPT are the sets B • r ( T ∞ ) \ B • r − ( T ∞ ) for r ≥
1. Theskeleton decomposition of the UIPT roughly states that the geometry of the layers of the UIPT isin one-to-one correspondance with a critical branching process and a collection of independentBoltzmann (or free) triangulations with a boundary (see Figure 2). We will give a detailedpresentation of this decomposition in Section 2. ∞ r T ∞ B r ( T ∞ ) B • r ( T ∞ ) ∂B • r ( T ∞ ) Figure 1:
Illustration of the ball of radius r in the UIPT and the corresponding hull. This decomposition was used by Krikun in [20] to study the length of the boundary of thehulls B • r ( T ∞ ) of the UIPT and in [ ] for similar considerations on the Uniform Infinite PlanarQuadrangulation. Since then, this decomposition has not received much attention with thenotable exception of the recent work by Curien and Le Gall [16], where it is used to study localmodifications of the graph distance in the UIPT.We will use the skeleton decomposition of the UIPT to get exact expressions for the generatingfunctions of the number of vertices inside certain regions of hulls, starting with the hulls2hemselves. Theorem 1.
For any s ∈ [
0, 1 ] and any nonnegative integer r one has E (cid:104) s | B • r ( T ∞ ) | (cid:105) = cosh (cid:18) sinh − (cid:18)(cid:113) ( − t ) t (cid:19) + r cosh − (cid:18)(cid:113) − tt (cid:19)(cid:19)(cid:18) cosh (cid:18) sinh − (cid:18)(cid:113) ( − t ) t (cid:19) + r cosh − (cid:18)(cid:113) − tt (cid:19)(cid:19) + (cid:19) where t is the unique solution in [
0, 1 ] of the equation s = t ( − t ) . An easy consequence of this Theorem is the scaling limitlim R → ∞ E (cid:104) e − λ | B •(cid:98) xR (cid:99) ( T ∞ ) | / R (cid:105) already obtained in [17] via the peeling process. Indeed, put s = e − λ / R and r = (cid:98) xR (cid:99) for some λ , x > R . Then t = − √ λ /3 R + o ( R − ) and cosh − (cid:32)(cid:114) − tt (cid:33) ∼ (cid:114) − tt − ∼ ( λ ) R giving lim R → ∞ E (cid:104) e − λ | B •(cid:98) xR (cid:99) ( T ∞ ) | / R (cid:105) = cosh (cid:0) ( λ ) x (cid:1)(cid:16) cosh (( λ ) x ) + (cid:17) in accordance with [17, 18] for type I triangulations.We also get an explicit expression for the generating function of the volume of hulls condi-tionally on their perimeter (see Proposition 2 for a precise statement). This allows to recover thefollowing scaling limit, already appearing in [18], Theorem 1.4, as the Laplace transform of thevolume of hulls of the Brownian plane conditionally on the perimeter. Corollary 1.
Fix x , (cid:96) > , then, for any λ > , one has lim R → ∞ E (cid:104) e − λ | B •(cid:98) xR (cid:99) ( T ∞ ) | / R (cid:12)(cid:12)(cid:12) | ∂ B •(cid:98) xR (cid:99) ( T ∞ ) | = (cid:98) (cid:96) R (cid:99) (cid:105) = x ( λ ) cosh (cid:16) ( λ ) x (cid:17) sinh (cid:16) ( λ ) x (cid:17) exp (cid:18) − (cid:96) (cid:18) ( λ ) (cid:18) coth (cid:16) ( λ ) x (cid:17) − (cid:19) − x (cid:19)(cid:19) .Our approach also allows us to compute the exact generating function of the difference ofvolume between hulls of the UIPT (see Proposition 2), and then recover one of the main resultsof [17], namely the scaling limit of the volumes of Hulls to a stochastic process. This convergenceholds jointly with the scaling limit of the perimeter of the hulls and we need to introduce somenotation taken from [17] to state it. 3et ( X t ) t ≥ be the Feller Markov process with values in R + whose semigroup is characterizedby E (cid:104) e − λ X t (cid:12)(cid:12)(cid:12) X = x (cid:105) = exp (cid:18) − x (cid:16) λ − + t /2 (cid:17) − (cid:19) for every x , t ≥ λ >
0. The process X is a continuous time branching process withbranching mechanism given by u (cid:55)→ u . As explain in [18], one can construct a stochasticprocess ( L t ) t ≥ with càdlàg paths such that the time-reversed process ( L ( − t ) − ) t ≤ is distributedas X "started from + ∞ at time − ∞ " and conditioned to hit 0 at time 0. We also let ( ξ i ) i ≥ be asequence of independent real valued random variables with density1 √ π x e − x { x > } and assume that this sequence is independent of the process L . Finally we set M t = ∑ s i ≤ t ξ i ( ∆ L s i ) ,where ( s i ) i ≥ is a measurable enumeration of the jumps of L . We recover the following result,first proved in [17] by studying the peeling process of the UIPT: Theorem 2 ([17], scaling limit of the hull process) . We have the following convergence in distributionin the sense of Skorokhod: (cid:16) R − | ∂ B •(cid:98) xR (cid:99) ( T ∞ ) | , R − | B •(cid:98) xR (cid:99) ( T ∞ ) | (cid:17) x ≥ ( d ) −−−→ R → ∞ (cid:0) · L x , 4 · · M x (cid:1) x ≥ .As for Theorem 1, our proof is based on the skeleton decomposition of random triangulationsand explicit computations of generating functions. The convergence of perimeters towards theprocess L was already established by Krikun [20] using this decomposition and we prove thejoint convergence of the second component.Finally, we study the volume of geodesic slices of the UIPT, defined by analogy with geodesicslices of the Brownian map (see Miller and Sheffield [27]). Fix r >
0, and orient ∂ B • r ( T ∞ ) in such a way that the root edge of T ∞ lies on its right hand side. Now pick two vertices v , v (cid:48) ∈ ∂ B • r ( T ∞ ) , the geodesic slice S ( r , v , v (cid:48) ) is the submap of B • r ( T ∞ ) bounded by the twoleftmost geodesics (see Section 5 for a precise definition) started respectively at v and v (cid:48) to theroot, and by the oriented arc from v to v (cid:48) along ∂ B • r ( T ∞ ) (See Figure 4 for an illustration). Noticethat B • r ( T ∞ ) = S ( r , v , v (cid:48) ) ∪ S ( r , v (cid:48) , v ) . We will also denote by v ∧ v (cid:48) the vertex where the twoleftmost geodesics started at v and v (cid:48) coalesce.For technical reasons, it will be easier to study the volume of geodesic slices minus thenumber of vertices on one of the two geodesics bounding it (for S ( r , v , v (cid:48) ) , we are talking aboutexcluding a number of vertices between 2 and r + r anyway. Theorem 3.
Fix n , r , q and q , . . . , q n some non negative integers such that q + · · · + q n = q. Condi-tionally on the event {| ∂ B • r ( T ∞ ) | = q } , let v be a vertex of ∂ B • r ( T ∞ ) chosen uniformly at random and et v , . . . , v n be placed in that order on the oriented cycle ∂ B • r ( T ∞ ) such that the oriented arc from v j tov j + along ∂ B • r ( T ∞ ) has length q j for every j (we set v n + = v ). Then, for s , . . . , s n ∈ [
0, 1 ] , one has E (cid:34) n ∏ j = s | S ( r , v j , v j + ) |− d ( v j , v j ∧ v j + ) − j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ∂ B • r ( T ∞ ) | = q (cid:35) = n ∏ j = t j ϕ { r } t j ( ) ϕ { r } ( ) q j × n ∑ k = q k q t k ϕ { r } (cid:48) t k ( ) ϕ { r } (cid:48) ( ) ϕ { r } ( ) ϕ { r } t k ( ) where, for every j ∈ {
1, . . . , n } , t j is the unique solution in [
0, 1 ] of the equation s j = t j ( − t j ) andthe functions ϕ { r } t and ϕ { r } are computed explicitly in Lemma 3. Equivalently, Theorem 3 states that, for each k , the root vertex of T ∞ belongs to the slice S ( r , v k , v k + ) with probability q k / q and that its volume has generating function t k ϕ { r } t k ( ) ϕ { r } ( ) q k − · ϕ { r } (cid:48) t k ( ) ϕ { r } (cid:48) ( ) ,and that conditionally on this event, the volumes of the other slices are independent and havegenerating functions given by t j ϕ { r } t j ( ) ϕ { r } ( ) q j for every j (cid:54) = k . It is also worth noticing that the generating function of the volume of theslice containing the root vertex is exactly the same as the hull of T ∞ conditionally on the event {| ∂ B • r ( T ∞ ) | = q k } , suggesting that this slice has the same law as a hull once the two geodesicboundaries are glued.Since geodesic slices do not form a growing family as the radius of the hulls grows, it isless natural to look for a scaling limit of their volume as a stochastic processes as in Theorem 2.However, it is still quite straightforward to derive asymptotics from Theorem 3 and obtain Corollary 2.
Fix n > an integer and (cid:96) , x > some real numbers. Fix also (cid:96) , . . . , (cid:96) n some nonnegative reals such that (cid:96) + · · · + (cid:96) n = (cid:96) . For every integer R > , conditionally on the event {| ∂ B •(cid:98) xR (cid:99) ( T ∞ ) | = (cid:98) (cid:96) R (cid:99)} , let v be a vertex of ∂ B •(cid:98) xR (cid:99) ( T ∞ ) chosen uniformly at random and letv , . . . , v n be placed in that order on the oriented cycle ∂ B •(cid:98) xR (cid:99) ( T ∞ ) such that the oriented arc from v j tov j + along ∂ B •(cid:98) xR (cid:99) ( T ∞ ) has length ∼ (cid:96) j R as R → ∞ . Then, for λ , . . . , λ n > , one has lim R → ∞ E (cid:34) n ∏ j = e − λ j | S ( r , v j , v j + ) | / R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ∂ B •(cid:98) xR (cid:99) ( T ∞ ) | = (cid:98) (cid:96) R (cid:99) (cid:35) = (cid:32) n ∑ i = (cid:96) i (cid:96) x ( λ i ) cosh (cid:0) ( λ i ) x (cid:1) sinh (( λ i ) x ) (cid:33) × exp (cid:32) − n ∑ i = (cid:96) i (cid:18) ( λ ) (cid:18) coth (cid:16) ( λ ) x (cid:17) − (cid:19) − x (cid:19)(cid:33) .5s for Corolloray 1, this can be interpreted in terms of the Brownian plane: each slice hasprobability (cid:96) i / (cid:96) to contain the root, in which case its volume has the same law as the volume ofthe hull of the Brownian plane condionally on the perimeter being (cid:96) i . In addition, conditionallyon this event, the volume of the other slices are independent and their Laplace transform isgiven by exp (cid:18) − (cid:96) j (cid:18) ( λ ) (cid:18) coth (cid:16) ( λ ) x (cid:17) − (cid:19) − x (cid:19)(cid:19) for every j (cid:54) = i .The paper is organized as follows. In Section 2 we recall some results about the generatingfunctions of triangulations counted by boundary length and inner vertices and we describethe decomposition of the UIPT into layers. In Section 3 we present our method and use it toprove Theorem 1 and Corollary 1. Section 4 studies the difference of volume between hullsand contains the proof of Theorem 2. Finally, Section 5 studies geodesic slices and contains theproofs of Theorem 3 and Corollary 2. Acknowledgments.
The author would like to thank Julien Bureaux for pointing out the linkwith hyperbolic functions in Lemma 3, yielding a simpler proof and a much nicer formula. Theauthor acknowledges support form the ANR grant "GRaphes et Arbres ALéatoires" (ANR-14-CE25-0014) and from CNRS.
As already mentioned in the introduction, the triangulations we consider in this work are typeI triangulations in the terminology of Angel and Schramm [7] – loops and multiple edges areallowed – and will always be rooted even when not mentioned explicitely. More precisely, wedeal with triangulations with simple boundary, that is rooted planar maps (the root of a map isa distinguished oriented edge and the root vertex of a rooted map is the origin of its root edge)such that every face is a triangle except for the face incident to the right hand side of the rootedge which can be any simple polygon. If the length of the boundary face is p , we will speak oftriangulations of the p -gon.One of the advantages of dealing with type I triangulations for our purpose is that trian-gulations of the sphere can be thought of as triangulations of the 1-gon as already mentionedin [16]. To see that, split the root edge of any triangulation into a double edge and then add aloop inside the region bounded by the new double edge and re-root the triangulation at thisloop oriented clockwise (so that the interior of the loop lies on its right hand side). Note thatthis construction also works if the root is itself a loop. This tranformation is a bijection betweentriangulations of the sphere and triangulations of the 1-gon.The enumeration of triangulations of the p − gon is now well known and can be found forexample in [16, 20]. Let T n , p be the set of triangulations of the p − gon with n inner vertices ( i.e. vertices that do not belong to the boundary face) and define the bivariate generating series T ( x , y ) = ∑ p ≥ ∑ n ≥ (cid:12)(cid:12) T n , p (cid:12)(cid:12) x n y p − . (1)6utte’s equation reads, for y > T ( x , y ) = y + x · T ( x , y ) − T ( x , 0 ) y + T ( x , y ) . (2)This equation can be solved using the quadratic method and the solution is explicit in terms ofthe unique solution of the equation x = h ( x ) ( − h ( x )) (3)such that h ( ) =
0. This function h seen a Taylor series has non negative coefficients and itsradius of convergence is ρ : = √ α : = h ( ρ ) =
112 . (5)The solution of equation (2) is then well defined on [ ρ ] × [ α ] and given by T ( x , 0 ) = h + x − h x (6)for x ∈ [ ρ ] and T ( x , y ) = y − x y + (cid:112) ( y − x ) − y + xyT ( x , 0 ) y = y − x y + (cid:112) x + y − y + yh − yh y (7)for ( x , y ) ∈ [ ρ ] × [ α ] . These expressions are compatible when taking the limit x → y →
0. Notice also that T ( ρ , α ) is finite: T ( ρ , α ) = − √
36 .The formulas (6) and (7) allow to compute explicitely the number of triangulations of the p -gon with a given number of inner vertices. However we will not need the exact formulas,only the following asymptotic expression: (cid:12)(cid:12) T n , p (cid:12)(cid:12) ∼ n → ∞ C ( p ) ρ − n n − for every p ≥ C ( p ) = p − p ( p ) !4 √ π ( p ! ) .In the following we will always denote by T p ( x ) = [ y p − ] T ( x , y ) the generating series oftriangulations with boundary length p counted by inner vertices.7 .2 Skeleton decomposition of finite triangulations We present here the skeleton decomposition of triangulations as first defined by Krikun [20] fortype II triangulations and later by Curien and Le Gall [16] for type I triangulations. First, weneed to define balls and hulls for finite triangulations.Let T be a triangulation of the sphere seen as a triangulation of the 1-gon. For every integer r >
0, the ball B r ( T ) of radius r centered at the root vertex of T is the planar map obtained bytaking the union of the faces of T that have at least one vertex at distance less than or equal to r − T . Now let v be a distinguished vertex of T and fix r > v and the root vertex of T is strictly larger than r . In that case, the vertex v belongs to the complement of the ball B r ( T ) and we define the r − hull B • r ( T , v ) of the pointedmap ( T , v ) as the union of B r ( T ) and all the connected components of the complement in T of B r ( T ) except the one that contains v .Define the boundary ∂ B • r ( T , v ) of B • r ( T , v ) as the set of vertices of B • r ( T , v ) having at leastone neighbour in the complement of B • r ( T , v ) , with the edges joining any pair of such vertices.An important observation is that ∂ B • r ( T , v ) is a simple cycle of T and that its vertices are allat distance exactly r from the root vertex of T . The planar map B • r ( T , v ) is therefore almost atriangulation with a simple boundary, the difference being that it is rooted at the orginal rootedge of T instead of an edge of the boundary face. It is a special case of a triangulation of thecylinder defined in [16]: Definition.
Let r ≥ triangulation of the cylinder of height r is a rooted planarmap such that all faces are triangles except for two distinguished faces verifying:1. The boundaries of the two distiguished faces form two disjoint simple cycles.2. The boundary of one of the two distinguished faces contains the root edge, and this faceis on the right hand side of the root edge. We call this face the root face and the otherdistiguished face the exterior face.3. Every vertex of the exterior face is at graph distance exactly r from the boundary of theroot face, and edges of the boundary of the exterior face also belong to a triangle whosethird vertex is at distance r − r , p , q ≥
1, a triangulation of the ( r , p , q ) -cylinder is a triangulation of thecylinder of height r such that its root face has degree p and its exterior face has degree q .With that terminology, the planar maps ∆ such that ∆ = B • r ( T , v ) for some integer r andsome pointed triangulation of the sphere ( T , v ) are the triangulations of the ( r , 1, q ) − cylinderfor some integer q ≥
1. Triangulations of the cylinder will also allow us to describe the geometryof triangulations between hulls. More precisely, if ( T , v ) is a pointed triangulation of the sphereand r > r > v is at distance strictly larger than r from the rootvertex of T , we define the layer between heights r and r of ( T , v ) by L • r , r ( T , v ) = (cid:0) B • r ( T , v ) \ B • r ( T , v ) (cid:1) ∪ ∂ B • r ( T , v ) .The planar maps ∆ such that ∆ = L • r , r ( T , v ) for some integers r > r > ( T , v ) are the triangulations of the ( r , p , q ) − cylinder for some integers p , q ≥ r , p , q > ∆ a triangulation of the ( r , p , q ) − cylinder. The skeleton decomposition of ∆ consists of an ordered forest of q rooted plane trees with maximal height r and a collection oftriangulations with a boundary indexed by the vertices of the forest of height stricly less than r .Borrowing from Krikun [20] and Curien and Le Gall [16], we define the growing sequence ofhulls of ∆ as follows: for 1 ≤ j ≤ r −
1, the ball B j ( ∆ ) is the union of all faces of ∆ having a vertexat distance stricly smaller than j from the root face, and the hull B • j ( ∆ ) consists of B j ( ∆ ) and allthe connected components of its complement in ∆ except the one containing the exterior face.By convention B • r ( ∆ ) = ∆ . For every j , the hull B • j ( ∆ ) is a triangulation of the ( j , p , q (cid:48) ) -cylindlerfor some non negative integer q (cid:48) , and we denote its exterior boundary by ∂ j ∆ . By convention ∂ ∆ is the boundary of the root face of ∆ . In addition, every cycle ∂ j ∆ is oriented so that B • j ( ∆ ) is always on the right hand side of ∂ j ∆ .Now let N ( ∆ ) be the collection of all edges of ∆ that belong to one of the cycles ∂ i ∆ for some0 ≤ i ≤ r . This set is a discrete version of the metric net of the Brownian map introduced byMiller and Sheffield [27]. In order to define a genealogy on N ( ∆ ) , notice that, for 1 ≤ i ≤ r ,every edge of ∂ i ∆ belongs to exactly one face of ∆ whose third vertex belongs to ∂ i − ∆ (it isthe face on its right hand side). Such faces are usually called down triangles of height i . Now,for any 1 ≤ i ≤ r , we say that an edge e ∈ ∂ i ∆ is the parent of an edge e (cid:48) ∈ ∂ i − ∆ if the firstvertex belonging to a down triangle of height i encountered when turning around the orientedcycle ∂ i − ∆ and starting at the end vertex of the oriented edge e (cid:48) belongs to the down triangleassociated to e . See Figure 2 for an illustration.These relations define a forest F of q rooted trees, its vertices being in one-to-one correspon-dence with the edges of N ( ∆ ) , that inherit from the planar structure of ∆ , making them planarrooted trees. In addition we can order canonicaly the trees, starting from the one containing theroot edge of ∆ and following the orientation of ∂ r ∆ . Notice also that every tree of the forest hasheight smaller than or equal to r and that the whole forest has exactly p vertices at height r . τ τ τ τ τ q vM v Figure 2:
Skeleton decomposition of a triangulation of the cylinder. The distinguishedvertex corresponding to the root edge of the triangulation is the red one on the bottomleft. Left: construction of the forest. Right: triangulation with a boundary filling a slot.
To completely describe ∆ , in addition to the forest F that gives the full structure of N ( ∆ ) andthe associated down triangles, we need to specify the structure of the submaps of ∆ lying in theinterstices, or slots, bounded by its down triangles. More precisely, to each edge e ∈ ∂ i ∆ where9 ≤ i ≤ r , we associate a slot bounded by its children and the two edges joining the startingvertex of e to ∂ i − ∆ (if e has no child, these two edges may or may not be glued into a singleedge). This slot is rooted at its unique boundary edge belonging to the down triangle associatedto e , the orientation chosen so that the interior of the slot is on the left hand side of the root. SeeFigure 2 for an illustration. With these conventions, the slot associated to an edge e is filled witha well defined triangulation of the ( c e + ) -gon, where c e is the number of children of e in theforest F . The triangulation of the ( r , p , q ) -cylinder ∆ is then fully characterized by the forest F and the collection of triangulations with a boundary associated to the vertices of F of heightstricly less than r .To summerize, let us say that a pointed forest is ( r , p , q ) -admissible if1. It is composed of an ordered sequence of q rooted plane trees of height lesser than or equalto r .2. It has exaclty p vertices at height r ,3. the distinguished vertex has height r and belongs to the first tree.We denote by F ( r , p , q ) the set of all ( r , p , q ) -admissible forests, and for any F ∈ F ( r , p , q ) wedenote by F (cid:63) for the set of vertices of F at height stricly smaller than r .The skeleton decomposition presented above is a bijection between triangulations of the ( r , p , q ) -cylinders and pairs consisting of a ( r , p , q ) -admissible forest F and a collection ( M v ) v ∈ F (cid:63) ,where, for each v ∈ F (cid:63) and denoting by c v the number of children of v in F , M v is a triangulationof the ( c v + ) -gon. We say that the forest associated to a triangulation of a cylinder ∆ is itsskeleton and denote it by Skel ( ∆ ) .As metioned earlier, this decomposition allows to canonically root the layers of a triangula-tion by rooting each layer at the ancestor of the root edge of the triangulation in its skeleton. Thanks to the spatial Markov property (see [7], Theorem 5.1), the skeleton decomposition isparticularly well suited to study the UIPT. Indeed, for any intergers r , q ≥ ( r , 1, q ) -admissible forest F , this property states that conditionally on the event { Skel ( B • r ( T ∞ )) = F } , thetriangulations filling the slots associated to the down triangles constitute a family of independentBoltzmann triangulations (cid:16) T ( c v + ) (cid:17) v ∈ F (cid:63) where, for any integer p ≥
1, the law of the Boltzmanntriangulation of the p − gon is given by P (cid:16) T ( p ) = T (cid:17) = ρ n T p ( ρ ) for any triangulation of the p − gon T with n inner vertices.From this, a lot of information on the skeleton decomposition of the UIPT can be dug, suchas the following Lemma that will be instrumental for our purpose. Lemma 1 ([16, 20]) . Fix r , p , q > and let ∆ be a triangulation of the ( r , p , q ) -cylinder. The skeleton of ∆ is a ( r , p , q ) − admissible forest F ∈ F ( r , p , q ) . For each v ∈ F (cid:63) , we denote the triangulation filling the lot associated to v by M v and the number of its inner vertices by n v . Then, for any r (cid:48) ≥ , P (cid:0) L • r (cid:48) , r (cid:48) + r ( T ∞ ) = ∆ (cid:12)(cid:12) | ∂ B • r (cid:48) ( T ∞ ) | = p (cid:1) = α q C ( q ) α p C ( p ) ∏ v ∈ F (cid:63) α c ( v ) − ρ n v + , = α q C ( q ) α p C ( p ) ∏ v ∈ F (cid:63) θ ( c v ) ∏ v ∈ F (cid:63) ρ n v T c v + ( ρ ) , where θ is the critical offspring distribution whose generating function ϕ is given by ϕ ( u ) = ∞ ∑ i = θ ( i ) u i = ρα u ( T ( ρ , α u ) − T ( ρ , 0 )) = − (cid:18) + √ − u (cid:19) − , u ∈ [
0, 1 ] .Lemma 1 is not hard to establish (see [16, 20] for the proof), and its main interest is thatit allows do do exact computations by interpreting the product over vertices of the forest asthe probability of some events for a branching process associated to ϕ . As we will do similarcomputations in various situations, let us give an example taken from [20] for the sake of clarity,and because it will be needed later. Say we want to compute P ( | ∂ B • r ( T ∞ ) | = q ) for some q > ∂ B • ( T ∞ ) is the root edge of T ∞ , which we recall is a loop, it has length 1 and the formulaof Lemma 1 directly gives: P ( | ∂ B • r ( T ∞ ) | = q ) = α q C ( q ) α C ( ) ∑ F ∈F ( r ,1, q ) ∏ v ∈ F (cid:63) θ ( c v ) = α q C ( q ) α C ( ) q ∑ F ∈F (cid:48) ( r ,1, q ) ∏ v ∈ F (cid:63) θ ( c v ) ,where F (cid:48) ( r , 1, q ) is the set of all ordered forests of rooted plane trees with height lesser than orequal to r , the whole forest having a single vertex at height r . Thus F (cid:48) ( r , 1, q ) is just the set offorests in F ( r , 1, q ) up to a circular permutation, explaining the factor 1/ q . But now the quantity ∑ F ∈F (cid:48) ( r ,1, q ) ∏ v ∈ F (cid:63) θ ( c v ) is exactly the probability that a Galton-Waton branching process with offspring distributiongiven by ϕ started with q particles has a single particle at generation r . Therefore, we have P ( | ∂ B • r ( T ∞ ) | = q ) = α q C ( q ) α C ( ) q [ u ]( ϕ { r } ( u )) q where ϕ { r } ( u ) = ϕ ◦ · · · ◦ ϕ (cid:124) (cid:123)(cid:122) (cid:125) r times ( u ) and [ u ] f ( u ) is the coeffiscient in u of the Taylor series at 0 of thefunction f . The iterates ϕ { r } can be computed explicitely (see Lemma 3 with t = P ( | ∂ B • r ( T ∞ ) | = q ) = α q C ( q ) α C ( ) (cid:18) − ( r + ) (cid:19) q − ( r + ) . (8) In this section, we will focus on the generating function of the number of vertices in the hulls ofthe UIPT. To that aim, we start with the following result:11 emma 2.
For any integer r > , s ∈ [
0, 1 ] and t ∈ [
0, 1 ] , one has E (cid:104) s | B • r ( T ∞ ) | (cid:105) = s ∑ q ≥ ( α t ) q C ( q ) α tC ( ) ∑ F ∈F ( r ,1, q ) ∏ v ∈ F (cid:63) ρ s · ( α t ) c ( v ) − · T c ( v )+ ( ρ s ) . Remark.
This Lemma and Proposition 1 are in fact a consequences of Proposition 2, but we stillprovide independent proofs because they provide a nice framework to introduce the functions ϕ t in (9) that are central to this work. Proof of Lemma 2.
Fix r , q > ∆ a triangulation of the ( r , 1, q ) -cylinder having F ∈ F ( r , 1, q ) as skeleton. Recall that for every v ∈ F (cid:63) we denote by c v the number of children of v in F and by n v the number of inner vertices of the triangulation of the ( c v + ) -gon filling the slot associatedto v . With these notations we have | ∆ | − = ∑ v ∈ F (cid:63) ( n v + ) ,the − r in F . Lemma 1 then gives s | ∆ | P ( B • R ( T ∞ ) = ∆ ) = s α q C ( q ) α C ( ) ∏ v ∈ F (cid:63) α c ( v ) − ( s ρ ) n v + and summing over every triangulation of the ( r , 1, q ) -cylinder having F as skeleton we obtain ∑ ∆ : Skel ( ∆ )= F s | ∆ | P ( B • r ( T ∞ ) = ∆ ) = s α q C ( q ) α C ( ) ∏ v ∈ F (cid:63) α c ( v ) − ∑ n v ≥ |T n v , c v + | · ( s ρ ) n v + = s α q C ( q ) α C ( ) ∏ v ∈ F (cid:63) ρ s · α c ( v ) − · T c ( v )+ ( ρ s ) .Since for any t ∈ [
0, 1 ] and any forest F ∈ F ( r , 1, q ) one has ∏ v ∈ F (cid:63) t c ( v ) − = t − q ,we can write ∑ ∆ : Skel ( ∆ )= F s | ∆ | P ( B • R ( T ∞ ) = ∆ ) = s ( α t ) q C ( q ) α tC ( ) ∏ v ∈ F (cid:63) ρ s · ( α t ) c ( v ) − · T c ( v )+ ( ρ s ) and the result follows by summing over q ≥ ( r , 1, q ) -admissible forest.As was done for Lemma 1, we want to interpret the numbers (cid:0) ρ s · ( α t ) i − · T i + ( ρ s ) (cid:1) i ≥ appearing in Lemma 2 as an offspring probability distribution. For ( s , t ) ∈ [
0, 1 ] , the generatingfunction of these numbers is defined, for every u ∈ [
0, 1 ] , by Φ s , t ( u ) = ∑ i ≥ ρ s · ( α t ) i − · T i + ( ρ s ) u i = ρ s ( α t ) u ( T ( ρ s , α tu ) − T ( ρ s , 0 )) .12he functions Φ s , t are clearly non negative and increasing, thus we just have to pick ( s , t ) suchthat Φ s , t ( ) =
1. Using formulas (6) and (7), simple computations yield Φ s , t ( ) = t · α − h ( ρ s ) − t · α − h ( ρ s ) − s + s (cid:112) s + t − t + t · α − h ( ρ s ) − t · α − h ( ρ s ) t .To solve this equation, we first notice that, from equation (3) satisfied by h , we have s = α − h ( ρ s ) (cid:16) − α − h ( ρ s ) (cid:17) .This suggests to consider t ( s ) ∈ [
0, 1 ] such that h ( ρ s ) = α t ( s ) ,or equivalently with equation (3), s = t ( s ) ( − t ( s )) .This parametrization yields Φ s , t ( s ) ( ) = t − t − s + s √ s − t + t t = ( s , t ) ∈ [
0, 1 ] such that s = t √ − t . For suchpairs we define, for every u ∈ [
0, 1 ] , ϕ t ( u ) : = Φ s , t ( s ) ( u )= ut − ut − s + s √ s + t u − t u + t u − t ut u which is the generating function of a probability distribution. Simple computations give thefollowing alternative expression: ϕ t ( u ) = − (cid:32) √ − u (cid:114) − tt + (cid:115) + − u (cid:18) − tt (cid:19)(cid:33) − . (9)This expression is not unlike the expression of ϕ given in Lemma 1, and ϕ = ϕ which is nosurprise.The next result gives an expression of the generating function of the volume of hulls of theUIPT in terms of iterates of the functions ϕ t . We will see in its proof that it takes advantage ofthe branching process associated to ϕ t . Proposition 1.
Fix r > and a pair ( s , t ) ∈ [
0, 1 ] such that s = t √ − t, then E (cid:104) s | B • r ( T ∞ ) | (cid:105) = s (cid:16) − t ϕ { r } t ( ) (cid:17) − ϕ { r } (cid:48) t ( ) where ϕ { r } t ( u ) = ϕ t ◦ · · · ◦ ϕ t (cid:124) (cid:123)(cid:122) (cid:125) r times ( u ) . roof. First, we interpret the sum over forests in F ( r , 1, q ) appearing in the statement of Lemma1 as the probability of an event for a branching process with offspring distribution given by ϕ t .To do that we first write ∑ F ∈F ( r ,1, q ) ∏ v ∈ F (cid:63) ρ s · ( α t ) c ( v ) − · T c ( v )+ ( ρ s ) = ∑ F ∈F ( r ,1, q ) ∏ v ∈ F (cid:63) [ u c v ] ϕ t ( u )= q ∑ F ∈F (cid:48) ( r ,1, q ) ∏ v ∈ F (cid:63) [ u c v ] ϕ t ( u ) (10)where F (cid:48) ( r , 1, q ) is the set of all ordered forests of q rooted plane trees of height lesser than orequal to r and having exactly one vertex at height r . The forests in F (cid:48) ( r , 1, q ) are obtained fromthe forests in F ( r , 1, q ) by a circular permutation of the order of their trees, so that the vertex atheight r does not necessarily belong to the first tree, explaning the factor q . But now, the righthand side of (10) without this factor q is exactly the probability that a Galton-Watson branchingprocess with offspring distribution given by ϕ t started with q particles has exaclty one particleat generation r . This probability is [ u ] (cid:16) ϕ { r } t ( u ) (cid:17) q and thus E (cid:104) s | B • r ( T ∞ ) | (cid:105) = s ∑ q ≥ ( α t ) q C ( q ) α tC ( ) q [ u ] (cid:16) ϕ { r } t ( u ) (cid:17) q , = s [ u ] α t ∑ q ≥ (cid:18) qq (cid:19) (cid:16) α t ϕ { r } t ( u ) (cid:17) q , = s [ u ] (cid:16) − · α t ϕ { r } t ( u ) (cid:17) − − α t , = s [ u ] t (cid:18)(cid:16) − t ϕ { r } t ( u ) (cid:17) − − (cid:19) , = s (cid:16) − t ϕ { r } t ( ) (cid:17) − [ u ] ϕ { r } t ( u ) ,giving the result. Before proving Theorem 1, let us first compute explicitely the iterates ϕ { r } t appearing in Proposi-tion 1: Lemma 3.
Fix t ∈ [
0, 1 [ and r ∈ N , then, for every u ∈ [
0, 1 ] , ϕ { r } t ( u ) = − − tt (cid:18) sinh (cid:18) sinh − (cid:16)(cid:113) ( − t ) t ( − u ) (cid:17) + r cosh − (cid:18)(cid:113) − tt (cid:19)(cid:19)(cid:19) and ϕ { r } ( u ) = − (cid:16) √ − u + r (cid:17) .14 roof. Fix t , u ∈ [
0, 1 ] and, for every n ∈ N , denote v n = (cid:113) − ϕ { n } t ( u ) .From the expression of ϕ t given in equation (9) we deduce that the sequence ( v n ) n ≥ satisfies (cid:40) v = √ − u v n + = av n + (cid:112) + ( a − ) v n (11)with a = (cid:114) − tt ≥ a =
1, the sequence ( v n ) has arithmetic progression and the result is trivial. Therefore wesuppose t <
1, and thus a >
1. Define w n > ( w n ) = (cid:112) a − v n ;then the recursion relation (11) satisfied by ( v n ) n ≥ becomessinh ( w n + ) = a sinh ( w n ) + (cid:112) a − ( w n )= sinh (cid:16) w n + cosh − ( a ) (cid:17) .This shows that the sequence ( w n ) has arithmetic progression and we have, for every n ≥ w n = sinh − (cid:16)(cid:112) a − v (cid:17) + n cosh − ( a ) and the result follows easily. Remark.
The sequence ( v n ) defined by (11) satisfies the following second order linear recursion v n + = av n − v n − that can be derived directly from (11) by noticing that v n + − av n v n + + v n = = v n + − av n ( v n + − v n − ) − v n − = ( v n + − v n − )( v n + − av n + v n − ) .This gives an alternate derivation of v n where hyperbolic functions do not appear directly. Proof of Theorem 1.
Theorem 1 is now a direct consequence of Proposition 1 and Lemma 3. Indeedwe have from Lemma 3 ϕ { r } t ( ) = − − tt (cid:18) sinh (cid:18) sinh − (cid:18)(cid:113) − tt (cid:19) + r cosh − (cid:18)(cid:113) − tt (cid:19)(cid:19)(cid:19) [ u ] ϕ { r } t ( u ) = (cid:18) − tt (cid:19) (cid:18) − tt (cid:19) − (cid:32) sinh (cid:32) sinh − (cid:32)(cid:114) − tt (cid:33) + r cosh − (cid:32)(cid:114) − tt (cid:33)(cid:33)(cid:33) − × cosh (cid:32) sinh − (cid:32)(cid:114) − tt (cid:33) + r cosh − (cid:32)(cid:114) − tt (cid:33)(cid:33) .The result then follows from Proposition 1.The proof of Corollary 1 relies on Proposition 2 proved in the next Section but we give ithere since it is more in the spirit of this section. Proof of Corollary 1.
Proposition 2 gives with r (cid:48) = p = E (cid:104) s | B • r ( T ∞ ) | (cid:12)(cid:12)(cid:12) | ∂ B • r ( T ∞ ) | = q (cid:105) = st q − (cid:16) ϕ { r } t ( ) (cid:17) q − ϕ { r }(cid:48) t ( ) (cid:16) ϕ { r } ( ) (cid:17) q − ϕ { r }(cid:48) ( ) .Putting s = e − λ / R , r = (cid:98) xR (cid:99) and q = (cid:98) (cid:96) R (cid:99) we have the following asymptotics: t q = − √ λ /3 R + O (cid:18) R (cid:19) , ϕ (cid:98) xR (cid:99) t ( ) = − √ λ R (cid:16) sinh (cid:16) ( λ ) x (cid:17)(cid:17) − + O (cid:18) R (cid:19) , ϕ (cid:98) xR (cid:99) ( ) = − ( (cid:98) xR (cid:99) ) , ϕ (cid:98) xR (cid:99) (cid:48) t ( ) ∼ (cid:32) √ λ R (cid:33) cosh (cid:0) ( λ ) x (cid:1) ( sinh (( λ ) x )) , ϕ (cid:98) xR (cid:99) (cid:48) ( ) ∼ ( xR ) ,and the result follows easily. In order to prove Theorem 2, we first compute the generating function of the volume of layersof the UIPT:
Proposition 2.
Let r , r (cid:48) , p , q be nonegative integers and ( s , t ) ∈ [
0, 1 ] such that s = t √ − t, then E (cid:104) s | L • r (cid:48) , r (cid:48) + r ( T ∞ ) | (cid:12)(cid:12)(cid:12) | ∂ B • r (cid:48) + r ( T ∞ ) | = q , | ∂ B • r (cid:48) ( T ∞ ) | = p (cid:105) = s p t q − p [ u p ] (cid:16) ϕ { r } t ( u ) (cid:17) q [ u p ] (cid:16) ϕ { r } ( u ) (cid:17) q .16 roof. The proof of this Proposition is very much in the spirit of the proofs of Lemma 2 andProposition 1. Indeed, let ∆ be a triangulation of the ( r , p , q ) − cylinder having F ∈ F ( r , p , q ) asskeleton. We have | ∆ | − p = ∑ v ∈ F (cid:63) ( n v + ) ,giving, with Lemma 1 and summing over every triangulation having F as skeleton, ∑ ∆ : Skel ( ∆ )= F s | ∆ | P (cid:0) L • r (cid:48) , r (cid:48) + r ( T ∞ ) = ∆ (cid:12)(cid:12) | ∂ B • r (cid:48) ( T ∞ ) | = p (cid:1) = s p α q C ( q ) α p C ( p ) ∏ v ∈ F (cid:63) α c ( v ) − ∑ n v ≥ |T n v , c v + | ( s ρ ) n v + = s p α q C ( q ) α p C ( p ) ∏ v ∈ F (cid:63) ρ s · α c ( v ) − · T c ( v )+ ( ρ s )= s p ( α t ) q C ( q )( α t ) p C ( p ) ∏ v ∈ F (cid:63) [ u c v ] ϕ t ( u ) .Summing over every ( r , p , q ) -admissible forest then gives E (cid:20) s | L • r (cid:48) , r (cid:48) + r ( T ∞ ) | (cid:8) | ∂ B • r (cid:48) + r ( T ∞ ) | = q (cid:9)(cid:12)(cid:12)(cid:12)(cid:12) | ∂ B • r (cid:48) ( T ∞ ) | = p (cid:21) = s p ( α t ) q C ( q )( α t ) p C ( p ) ∑ F ∈F ( r , p , q ) ∏ v ∈ F (cid:63) [ u c v ] ϕ t ( u ) .Now, if F (cid:48) ( r , p , q ) denotes the set of all ( r , p , q ) -admissible forests up to a cyclic permutation ofthe order of the trees, each tree in F ( r , p , q ) corresponds to exactly q trees of F (cid:48) ( r , p , q ) , therefore E (cid:20) s | L • r (cid:48) , r (cid:48) + r ( T ∞ ) | (cid:8) | ∂ B • r (cid:48) + r ( T ∞ ) | = q (cid:9)(cid:12)(cid:12)(cid:12)(cid:12) | ∂ B • r (cid:48) ( T ∞ ) | = p (cid:21) = s p ( α t ) q C ( q )( α t ) p C ( p ) q ∑ F ∈F (cid:48) ( r , p , q ) ∏ v ∈ F (cid:63) [ u c v ] ϕ t ( u ) .The trees in F (cid:48) ( r , p , q ) have a distinguished vertex at height r , and if F (cid:48)(cid:48) ( r , p , q ) denotes the setof all rooted forests of height lesser than or equal to r , with q trees, and having a total number p of vertices at height r , each forest of F (cid:48)(cid:48) ( r , p , q ) corresponds to exactly p forests in F (cid:48) ( r , p , q ) ,thus E (cid:20) s | L • r (cid:48) , r (cid:48) + r ( T ∞ ) | (cid:8) | ∂ B • r (cid:48) + r ( T ∞ ) | = q (cid:9)(cid:12)(cid:12)(cid:12)(cid:12) | ∂ B • r (cid:48) ( T ∞ ) | = p (cid:21) = s p ( α t ) q C ( q )( α t ) p C ( p ) pq ∑ F ∈F (cid:48)(cid:48) ( r , p , q ) ∏ v ∈ F (cid:63) [ u c v ] ϕ t ( u ) .But now the sum ∑ F ∈F (cid:48)(cid:48) ( r , p , q ) ∏ v ∈ F (cid:63) [ u c v ] ϕ t ( u ) is the probability that a Galton-Watson process with offspring distribution given by ϕ t startedwith q particles has p particles at generation r . This yields E (cid:20) s | L • r (cid:48) , r (cid:48) + r ( T ∞ ) | (cid:8) | ∂ B • r (cid:48) + r ( T ∞ ) | = q (cid:9)(cid:12)(cid:12)(cid:12)(cid:12) | ∂ B • r (cid:48) ( T ∞ ) | = p (cid:21) = s p ( α t ) q C ( q )( α t ) p C ( p ) pq [ u p ] (cid:16) ϕ { r } t ( u ) (cid:17) q .Using the same reasoning, we can easily get P (cid:0)(cid:8) | ∂ B • r (cid:48) + r ( T ∞ ) | = q (cid:9)(cid:12)(cid:12) | ∂ B • r (cid:48) ( T ∞ ) | = p (cid:1) = α q C ( q ) α p C ( p ) pq [ u p ] (cid:16) ϕ { r } ( u ) (cid:17) q and the result follows. 17s we will see in the proof of Theorem 2, the jumps of the process of hull perimeters willinduce jumps for the process of hull volumes. This motivates the following technical result,which is a consequence of Proposition 2, and will be used in the proof of Therem 2. Corollary 3.
Fix an integer r > and (cid:96) > δ > . Let ( p n , q n ) n ≥ be non negative integers such thatn − p n → (cid:96) and n − q n → (cid:96) − δ as n → ∞ . Then, conditionally on the events (cid:8) | ∂ B • r + ( T ∞ ) | = q n (cid:9) ∩ (cid:8) | ∂ B • r ( T ∞ ) | = p n (cid:9) , the following convergence in distribution holdsn − | B • r + ( T ∞ ) \ B • r ( T ∞ ) | ( d ) −−−→ n → ∞ δ · ξ where ξ is a random variable with density √ π x e − x { x > } .Proof. Proposition 2 gives, for any ( s , t ) ∈ [
0, 1 ] with s = t √ − t , E (cid:104) s | B • r + ( T ∞ ) \ B • r ( T ∞ ) | (cid:12)(cid:12)(cid:12) | ∂ B • r + ( T ∞ ) | = q n , | ∂ B • r ( T ∞ ) | = p n (cid:105) = t q n − p n [ u p n ] (cid:16) ϕ { r } t ( u ) (cid:17) q n [ u p n ] (cid:16) ϕ { r } ( u ) (cid:17) q n .We can study the asymptotic behavior of the quantity [ u p n ] ( ϕ t ( u )) q n with standart analytictechniques: [ u p n ] ( ϕ t ( u )) q n = i π (cid:73) γ ϕ t ( z ) q n z p n + dz where γ is a small enough contour enclosing the origin. The function ϕ t being analytic in C \ [ + ∞ [ , it is possible to deform the contour γ into a Henkel-type contour γ n withoutchanging the value of the integral (the modulus of the integrand decreases exponentially fast for | z | large). For n ≥
1, we can take γ n to be the reunion on the semi infinite line − i / n + [ + ∞ [ ,oriented from right to left, the semi circle 1 + n e i ] π /2,3 π /2 [ oriented clockwise, and the semiinfinite line + i / n + [ + ∞ [ oriented from left to right (see Figure 3 for an illustration). Thechange of variable z → + z / p n then gives [ u p n ] ( ϕ t ( u )) q n = i π (cid:73) γ pn ϕ t ( z ) q n z p n + dz = i π p n (cid:73) H ϕ t ( + z / p n ) q n ( + z / p n ) p n + dz where H is the Henkel contour, that is the reunion of the semi infinite line − i + [ + ∞ [ , orientedfrom right to left, the semi circle e i ] π /2,3 π /2 [ oriented clockwise, and the semi infinite line i +[ + ∞ [ oriented from left to right (see Figure 3 for an illustration). /n /n /n γ n H Figure 3:
The contours H and γ n . z ∈ C \ [ + ∞ [ : ϕ t ( z ) = − t − t ( − z ) (cid:32) + (cid:114) − z t − t (cid:33) − .If s n = e − λ / n , then t n = − √ λ /3 n + O ( n − ) , thus for z ∈ H : ϕ t n ( + z / p n ) = + zp n (cid:32) − √ λ n + O ( n − ) (cid:33) · + (cid:115) − zp n + √ λ n + O ( n − ) − , = + zp n + z (cid:16) (cid:96) √ λ − z (cid:17) p n + O ( p − n ) .Then we have, for z ∈ H , ϕ t n ( + z / p n ) q n = e z ( − δ / (cid:96) ) (cid:18) + ( − δ / (cid:96) ) z (cid:16) (cid:96) √ λ − z (cid:17) p − n + O ( p − n ) (cid:19) ,giving ϕ t n ( + z / p n ) q n ( + z / p n ) p n + = e − z δ / (cid:96) (cid:18) + ( − δ / (cid:96) ) z (cid:16) (cid:96) √ λ − z (cid:17) p − n + O ( p − n ) (cid:19) .Since for any α ∈ R i π (cid:73) H ( − z ) α e − z dz = Γ ( − α ) ,we get, at least on a formal level, [ u p n ] ( ϕ t ( u )) q n = ( − δ / (cid:96) ) i π p n (cid:73) H e − z δ / (cid:96) z ( (cid:96) √ λ − z ) dz + O ( p n − ) , = ( − δ / (cid:96) ) i π p n δ / (cid:96) e − δ √ λ (cid:18) (cid:96) δ (cid:19) (cid:73) H e − z (cid:16) δ √ λ + z (cid:17) ( − z ) dz + O ( p − n ) , = ( − δ / (cid:96) ) p n (cid:18) (cid:96) δ (cid:19) e − δ √ λ (cid:32) δ √ λ Γ ( − ) + − Γ ( − ) (cid:33) + O ( p − n ) , = ( − δ / (cid:96) ) p n (cid:18) (cid:96) δ (cid:19) e − δ √ λ (cid:32) − δ √ λ Γ ( − ) + − Γ ( − ) (cid:33) + O ( p − n ) . (12)The justification of the formal argument used to derive (12) is quite standart in analytic combi-natorics. For example it is identical to the one done in the proof of Theorem VI.1 of [19].This asymptotic expansion yields [ u p n ] ( ϕ t n ( u )) q n [ u p n ] ( ϕ ( u )) q n −−−→ n → ∞ e − δ √ λ (cid:18) δ √ λ + (cid:19) ,and E (cid:20) exp (cid:18) − λ n | B • r + ( T ∞ ) \ B • r ( T ∞ ) | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) | ∂ B • r + ( T ∞ ) | = q n , | ∂ B • r ( T ∞ ) | = p n (cid:21) = t q n − p n n [ u p n ] ( ϕ t n ( u )) q n [ u p n ] ( ϕ ( u )) q n −−−→ n → ∞ e − δ √ λ (cid:18) δ √ λ + (cid:19) ,19nally giving the result since the Laplace transform of ξ is given by E (cid:104) e − λξ (cid:105) = ( + √ λ ) e − λ for every λ > Proof of Theorem 2.
With the help of Corollary 3, the proof of this result is similar to the proof ofTheorem 1 in [17]. First, we can restrict the time interval to [
0, 1 ] and verify that (cid:16) R − | ∂ B •(cid:98) xR (cid:99) ( T ∞ ) | , R − | B •(cid:98) xR (cid:99) ( T ∞ ) | (cid:17) x ∈ [ ] ( d ) −−−→ R → ∞ (cid:0) · L x , 4 · · M x (cid:1) x ∈ [ ] .The convergence of the first component (cid:16) R − | ∂ B •(cid:98) xR (cid:99) ( T ∞ ) | (cid:17) x ∈ [ ] ( d ) −−−→ R → ∞ (cid:0) · L x (cid:1) x ∈ [ ] . (13)is already proved in [20] via the skeleton decomposition and in [17] via the peeling process.Therefore, we will study the second component given the first one.For every r ≥ V r : = | B • r ( T ∞ ) | = + r ∑ i = U i where, for every i ≥ U i = | B • i ( T ∞ ) \ B • i − ( T ∞ ) | .Fix ε > R >
0, Corollary 3 suggests to introduce, for r ∈ {
1, . . . , R } , V > ε r = r ∑ i = U i { P i < P i − − ε R } , V ≤ ε r = r ∑ i = U i { P i ≥ P i − − ε R } ,where P i = | ∂ B • i ( T ∞ ) | for every i ≥ R − V ≤ ε R is small uniformly in R when ε is small. We will proceedwith a first moment argument, and a first step is to give a bound on the expectation of U i conditionnaly on the event { P i − = p } , for i ≥
1. Fix p , q ≥ F be a ( p , q ) -admissibleforest. Recall that the spatial Markov property of the UIPT states that, conditionnaly on theevent { Skel ( L • i − i ( T ∞ )) = F } , the layer L • i − i ( T ∞ ) is composed of its down triangles and acollection of indenpendent Boltzmann triangulations (cid:16) T ( c v + ) (cid:17) v ∈ F (cid:63) . There exists a universalconstant C > p ≥
1, one has E (cid:104) | T ( p ) | (cid:105) ≤ C p (see for example [17], Proposition 8) and therefore E (cid:2) U i (cid:12)(cid:12) Skel ( L • i − i ( T ∞ )) = F (cid:3) ≤ C ∑ v ∈ F (cid:63) ( c v + ) = C ∑ v ∈ F (cid:63) c v + C q + C p .20sing Lemma 1, we get, for every i ≥ p ≥ E (cid:104) U i { P i = q } (cid:12)(cid:12)(cid:12) P i − = p (cid:105) = ∑ F ∈F ( p , q ) E (cid:2) U i (cid:12)(cid:12) Skel ( L • i − i ( T ∞ )) = F (cid:3) · α q C ( q ) α p C ( p ) ∏ v ∈ F (cid:63) [ u c v ] ϕ ≤ α q C ( q ) α p C ( p ) pq (cid:32) ( C q + C p ) [ u p ] ϕ q + C ∑ n + ··· n q = p q ∑ i = n i q ∏ i = [ u n i ] ϕ (cid:33) .The sum on the right hand side of the last equation is exactly E (cid:104) ( N + · · · + N q ) { N + ··· + N q = p } (cid:105) = q E (cid:104) N { N + ··· + N q = p } (cid:105) ,where N , . . . , N q are independent random variables distributed according to ϕ . We have E (cid:104) N { N + ··· + N q = p } (cid:105) = [ u p − ] (cid:16) ϕ (cid:48)(cid:48) ϕ q − (cid:17) + pq [ u p ] ϕ q for every p ≥
2, yiedling E (cid:104) U i { P i = q } (cid:12)(cid:12)(cid:12) P i − = p (cid:105) ≤ α q C ( q ) α p C ( p ) pq (cid:16) ( C q + C p ) [ u p ] ϕ q + C · q [ u p − ] (cid:16) ϕ (cid:48)(cid:48) ϕ q − (cid:17)(cid:17) . (14)Using ∑ q ≥ q α q C ( q ) u q = C (cid:48) · (cid:16) ( − u ) − − (cid:17) for some C (cid:48) > E [ U i | P i − = p ] ≤ p α p C ( p ) (cid:18) C [ u p ] (cid:18) ϕ ( − ϕ ) (cid:19) + C · p [ u p ] ( − ϕ ) − + C [ u p − ] (cid:18) ϕ (cid:48)(cid:48) ( − ϕ ) (cid:19)(cid:19) for every p ≥
1, where C , C , C > − ϕ ∼ − u and ϕ (cid:48)(cid:48) ∼ ( − u ) − as u →
1, it is easy to see that E [ U i | P i − = p ] ≤ p α p C ( p ) C p (15)for some constant C >
0. Therefore, if p ≤ ε R and ε is small enough, using the fact that p α p C ( p ) = O ( p ) , we have E [ U i | P i − = p ] ≤ C p ≤ C (cid:48) R ε (16)where C , C (cid:48) > K >
1, equations (15) and (8) give E (cid:104) U i { P i − ≥ K R } (cid:105) ≤ C ∑ p ≥ K R p (cid:18) − i (cid:19) p − i , ≤ C (cid:48) i ∑ p ≥ K R p i e − p / i .21he function u (cid:55)→ u e − u being decreasing for u >
2, the last inequality transforms to E (cid:104) U i { P i − ≥ K R } (cid:105) ≤ C (cid:48) i (cid:90) u ≥ K R u i e − u / i du , ≤ C (cid:48) i (cid:90) u ≥ K R i u e − u du .Since we only consider i ∈ {
1, . . . , R } , we have E (cid:104) U i { P i − ≥ K R } (cid:105) ≤ C (cid:48) i (cid:90) u ≥ K u e − u du . ≤ C R K e − K ≤ C (cid:48) R ε (17)for K = ε − and ε small enough.Finally, if p = (cid:98) (cid:96) R (cid:99) for some (cid:96) ∈ [ ε , K ] , we have using (14): E (cid:104) U i { P i ≥ p − ε R } (cid:12)(cid:12)(cid:12) P i − = p (cid:105) ≤ C p [ u p ] ϕ p + −(cid:98) ε R (cid:99) ( − ϕ ) + C p [ u p ] ϕ p −(cid:98) ε R (cid:99) ( − ϕ ) + C p [ u p − ] ϕ p −(cid:98) ε R (cid:99) ϕ (cid:48)(cid:48) ( − ϕ ) (18)where we also used the fact that 1 p α p C ( p ) ∼ p → ∞ Cp .The same methods of singularity analysis than the ones used in the proof of Corollary 2 give, as R → ∞ , (cid:104) u (cid:98) (cid:96) R (cid:99) (cid:105) ϕ (cid:98) (cid:96) R (cid:99) + −(cid:98) ε R (cid:99) ( − ϕ ) ∼ i π (cid:96) R (cid:73) H e − z ε (cid:96) (cid:18) − z (cid:96) R (cid:19) − dz = R ε Γ ( ) , (cid:104) u (cid:98) (cid:96) R (cid:99) (cid:105) ϕ (cid:98) (cid:96) R (cid:99)−(cid:98) ε R (cid:99) ( − ϕ ) ∼ i π (cid:96) R (cid:73) H e − z ε (cid:96) (cid:18) − z (cid:96) R (cid:19) − dz = Γ ( ) R ε , (cid:104) u (cid:98) (cid:96) R (cid:99)− (cid:105) ϕ (cid:98) (cid:96) R (cid:99)−(cid:98) ε R (cid:99) ϕ (cid:48)(cid:48) ( − ϕ ) ∼ i π (cid:96) R (cid:73) H e − z ε (cid:96) (cid:18) − z (cid:96) R (cid:19) − dz = R ε Γ ( ) .These last three asymptotic behaviors and (18) finally give, for any p ∈ [ ε R , K R ] , E (cid:104) U i { P i ≥ p − ε R } (cid:12)(cid:12)(cid:12) P i − = p (cid:105) ≤ C (cid:48) ε K R + C (cid:48) ε − K R + C (cid:48) ε K R . (19)Combining (16), (17) and (19) give, for every R , R − E (cid:104) V ≤ ε R (cid:105) ≤ C ε ,and thus, for every δ >
0, we havesup R ≥ P (cid:32) sup x ∈ [ ] (cid:12)(cid:12)(cid:12) R − V (cid:98) xR (cid:99) − R − V ≤ ε (cid:98) xR (cid:99) (cid:12)(cid:12)(cid:12) > δ (cid:33) −−→ ε →
0. (20)22e now turn to V > ε r and use the reasoning of the proof of Theorem 1 of [17] (we give thefull reasoning for the sake of completness). Denote by x , x , . . . the jump times of L before time1. For every r ≥
1, let (cid:96) ( r ) , . . . , (cid:96) ( r ) r be the integers i ∈ {
1, . . . , r } listed in increasing order of thequantities P i − P i − (and the usual order of N for indices such that P i − P i − is equal to a givenvalue). It follows from the convergence (13) that, for every integer K ≥ (cid:16) R − (cid:96) ( R ) , . . . , R − (cid:96) ( R ) K , R − (cid:16) P (cid:96) ( R ) − P (cid:96) ( R ) − (cid:17) , . . . , R − (cid:16) P (cid:96) ( R ) K − P (cid:96) ( R ) K − (cid:17)(cid:17) ( d ) −−−→ R → ∞ (cid:0) x , . . . , x K , 3 · ∆ L x , . . . , 3 · ∆ L x K (cid:1) , (21)and this convergence holds jointly with the convergence (13). In addition, using Corollary 3, wealso get U (cid:96) ( R ) (cid:16) P (cid:96) ( R ) − P (cid:96) ( R ) − (cid:17) , · · · , U (cid:96) ( R ) K (cid:16) P (cid:96) ( R ) K − P (cid:96) ( R ) K − (cid:17) ( d ) −−−→ R → ∞ (cid:0) · · ξ , . . . , 4 · · ξ K (cid:1) , (22)jointly with the convergences (13) and (21), where the random variables ξ i are independentcopies of the random variable ξ of Corollary 3, and independent of the process L .Chosing K sufficiently large such that the probability of | ∆ L x K | < ε / ( · ) is close to 1, wecan combine (21) and (22) to obtain the joint convergence (cid:16) R − P (cid:98) Rx (cid:99) , R − V > ε (cid:98) Rx (cid:99) (cid:17) x ∈ [ ] ( d ) −−−→ R → ∞ (cid:0) · L x , 4 · · M ε x (cid:1) x ∈ [ ] ,where the process ( M ε x ) x ∈ [ ] is defined by M ε x = ∑ i ≥ { x i ≤ x , | ∆ L xi | > ε / ( · ) } ξ i ( ∆ L x i ) .It is easy to verify that, for every δ > P (cid:32) sup x ∈ [ ] |M x − M ε x | > δ (cid:33) −−→ ε → Fix r > v ∈ ∂ B • r ( T ∞ ) . There are several geodesic paths from v to the root vertex and wewill distinguish a canonical one, called the leftmost geodesic. Informally, it is constructed fromthe following local rule: at each step, take the leftmost available neighbour that takes you closerto the root. More precisely, the vertex v ∈ ∂ B • r ( T ∞ ) is connected to several vertices of ∂ B • r − ( T ∞ ) and we can enumerate them in clockwise order, starting from the first one after the edge of ∂ B • r ( T ∞ ) whose initial vertex is v . The first step of the leftmost geodesic from v to the root vertexis the last edge appearing in this enumeration and the path is constructed by induction. Notice23hat the first step of the leftmost geodesic is an edge of the down triangle associated to the edgeof ∂ B • r ( T ∞ ) on the left hand side of v (see Figure 4 for an illustration).Now pick v , v (cid:48) ∈ ∂ B • r ( T ∞ ) , the two leftmost geodesics started respectively at v and v (cid:48) willcoalesce at a vertex denoted by v ∧ v (cid:48) . The geodesic slice S ( r , v , v (cid:48) ) is the submap of B • r ( T ∞ ) bounded by these two paths and the part of ∂ B • r ( T ∞ ) going from v to v (cid:48) (recall that ∂ B • r ( T ∞ ) is oriented so that B • r ( T ∞ ) lies on its right hand side). As a consequence of the definition ofleftmost geodesics, the slice S ( r , v , v (cid:48) ) is completely described by the trees of the skeleton of B • r ( T ∞ ) whose root lies, following the orientation of ∂ B • r ( T ∞ ) , between v and v (cid:48) . Indeed, it iscomposed of the down triangles and the slots associated to the vertices of these trees. Figure 4contains an illustration of this fact. v v v ∧ v Figure 4:
In red, two leftmost geodesic paths to the root started respectively at v and v (cid:48) , up to their coalescence point v ∧ v (cid:48) . The geodesic slice S ( r , v , v (cid:48) ) is is the part ofthe map lying inside the two red paths and below the path joining v and v (cid:48) . Proof of Theorem 3.
Since, for any v , v (cid:48) ∈ ∂ B • r ( T ∞ ) , the slice S ( r , v , v (cid:48) ) corresponds to the trees ofSkel ( B • r ( T ∞ )) whose root lie between v and v (cid:48) , we need to identify these trees. Indeed, the firsttree of Skel ( B • r ( T ∞ )) plays a special role (it is the only one of height r ) and the geometry of theslice is not the same whether this tree is rooted between v and v (cid:48) or not. Equivalently, this meansthat the slice containing the root vertex of T ∞ will play a special role.We denote by F = ( τ , . . . , τ q ) the skeleton of B • r ( T ∞ ) . Recall that it is an ordered forest,and more precisely a ( r , 1, q ) -admissible forest. The vertex v is the vertex on the left-hand sideof the root of τ i for some i between 1 and q , and the part of the skeleton describing the slice S ( r , v j , v j + ) is the ordered forest F i , j = ( τ i + q + ··· + q j − , . . . , τ i + q + ··· + q j − ) ,24here τ q + k = τ k for every k ∈ {
1, . . . , q } (we also always set v n + = v ). The vertex v beingchosen uniformly, this happens with probability q for every i ∈ {
1, . . . q } .Now, fix ∆ a triangulation of the ( r , 1, q ) − cylinder with skeleton F = ( τ , . . . , τ q ) . For v , v (cid:48) ∈ ∂ ∆ , we denote by ∆ ( v , v (cid:48) ) the geodesic slice define by the arc from v to v (cid:48) and the twoleftmost geodesic started respectively ar v and v (cid:48) . If v chosen uniformly and then ( v , . . . , v n ) aresuch that the length of the arc for v j to v j + along ∂ ∆ has length q j , then, for any s , . . . , s n ∈ [
0, 1 ] E (cid:34) n ∏ j = s | ∆ ( v j , v j + ) |− d ( v j , v j ∧ v j + ) − j (cid:35) = q q ∑ i = n ∏ j = ∏ v ∈ F (cid:63) i , j s n v + j ,where the expectation takes into account only the randomness of v (the map ∆ is deterministichere). This is where it is easier to consider | ∆ ( v j , v j + ) | − d ( v j , v j ∧ v j + ) − | ∆ ( v j , v j + ) | . Indeed, in the previous formula, the terms s n v + count the number of innervertices in blocs as well as the top vertex of each block. This means that every vertex of theleftmost geodesic on the right hand side of the slice is not counted explaining the deduction of d ( v j , v j ∧ v j + ) + d ( v j , v j ∧ v j + ) ). This is not much harder todo, but it leads to a much more complicated formula and does not have a lot of benefits.Lemma 1 gives E (cid:34) n ∏ j = s | S ( r , v j , v j + ) |− d ( v j , v j ∧ v j + ) − j { B • r ( T ∞ )= ∆ } (cid:35) = α q C ( q ) α C ( ) q q ∑ i = n ∏ j = ∏ v ∈ F (cid:63) i , j α c ( v ) − ( ρ s j ) n v + , = α q C ( q ) α C ( ) q q ∑ i = n ∏ j = t q j − { τ ∈ Fi , j } j ∏ v ∈ F (cid:63) i , j (cid:0) α t j (cid:1) c v − ( ρ s j ) n v + .Summing over every triangulation ∆ having F as skeleton then gives E (cid:34) n ∏ j = s | S ( r , v j , v j + ) |− d ( v j , v j ∧ v j + ) − j { Skel ( B • r ( T ∞ ))= F } (cid:35) = α q C ( q ) α C ( ) q q ∑ i = n ∏ j = t q j − { τ ∈ Fi , j } j ∏ v ∈ F (cid:63) i , j [ u c v ] ϕ t j ( u ) .Finally summing over every admissible forest yields E (cid:34) n ∏ j = s | S ( r , v j , v j + ) |− d ( v j , v j ∧ v j + ) − j {| ∂ B • r ( T ∞ ) | = q } (cid:35) = α q C ( q ) α C ( ) q ∑ F ∈F ( r ,1, q ) q ∑ i = n ∏ j = t q j − { τ ∈ Fi , j } j ∏ v ∈ F (cid:63) i , j [ u c v ] ϕ t j ( u ) , = α q C ( q ) α C ( ) q ∑ F ∈F (cid:48) ( r ,1, q ) n ∏ j = t q j − { h ( F j )= r } j ∏ v ∈ F (cid:63) j [ u c v ] ϕ t j ( u ) ,where h ( · ) denotes the maximal height of a forest. If, for k ≥
1, we denote by F rk the set of all25rdered forests of k trees with maximal height stricly less than r , we get E (cid:34) n ∏ j = s | S ( r , v j , v j + ) |− d ( v j , v j ∧ v j + ) − j {| ∂ B • r ( T ∞ ) | = q } (cid:35) = α q C ( q ) α C ( ) q n ∑ k = t q k − k ∑ F k ∈F (cid:48) ( r ,1, q k ) ∏ v ∈ F (cid:63) k [ u c v ] ϕ t k ( u ) × ∏ j (cid:54) = k t q j j ∑ F j ∈F rqj ∏ v ∈ F j [ u c v ] ϕ t j ( u ) , = α q C ( q ) α C ( ) q (cid:32) n ∏ j = (cid:16) t j ϕ { r } t j ( ) (cid:17) q j (cid:33) × n ∑ k = t k [ u ] (cid:16) ϕ { r } t k ( u ) (cid:17) q k (cid:16) ϕ { r } t k ( ) (cid:17) q k , = α q C ( q ) α C ( ) q (cid:32) n ∏ j = (cid:16) t j ϕ { r } t j ( ) (cid:17) q j (cid:33) × n ∑ k = q k t k ϕ { r } (cid:48) t k ( ) ϕ { r } t k ( ) .Finally we have E (cid:34) n ∏ j = s | S ( r , v j , v j + ) |− d ( v j , v j ∧ v j + ) − j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ∂ B • r ( T ∞ ) | = q (cid:35) = n ∏ j = t j ϕ { r } t j ( ) ϕ { r } ( ) q j × n ∑ k = q k q t k ϕ { r } (cid:48) t k ( ) ϕ { r } (cid:48) ( ) ϕ { r } ( ) ϕ { r } t k ( ) giving the result. Proof of Corollary 2.
This is a direct consequence of Theorem 3 using the same asymptotics as inthe proof of Corollary 1.
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