A polynomial upper bound for the mixing time of edge rotations on planar maps
aa r X i v : . [ m a t h . P R ] J a n A polynomial upper bound for the mixing time ofedge rotations on planar maps Alessandra Caraceni ♦ Abstract
We consider a natural local dynamic on the set of all rooted planar maps with n edges thatis in some sense analogous to “edge flip” Markov chains, which have been considered beforeon a variety of combinatorial structures (triangulations of the n -gon and quadrangulations of thesphere, among others). We provide the first polynomial upper bound for the mixing time of this“edge rotation” chain on planar maps: we show that the spectral gap of the edge rotation chain isbounded below by an appropriate constant times n − / . In doing so, we provide a partially newproof of the fact that the same bound applies to the spectral gap of edge flips on quadrangulationsas defined in [8], which makes it possible to generalise the result of [8] to a variant of the edge flipchain related to edge rotations via Tutte’s bijection. This work is concerned with estimating the mixing time of a particular Markov chain on the setof all possible (rooted) planar maps with n edges.Many di ff erent Markov chains with a geometric flavour have been considered on a variety ofinteresting state spaces given by the sets of all possible planar combinatorial structures of a certaintype and size – e.g. plane trees, binary trees, triangulations of the n -gon, lattice triangulations,quadrangulations of the sphere, etc.A natural family of Markov chains which have sparked a lot of interest, both because of theirdeceptive simplicity and their potential applications (e.g. to systematic biology [2]), is that of“edge flip” chains. The archetypal example of an edge flip chain is Aldous’ so-called triangulationwalk [1], defined on the state space of all possible triangulations of the n -gon (i.e. of maximalconfigurations of non-crossing diagonals). Its transitions are edge flips in the following sense:given a triangulation of the n -gon, a single step of the chain consists in choosing a diagonaluniformly at random and, with probability 1 /
2, replacing it with the other diagonal of the uniquequadrilateral formed by the two triangles adjacent to it (see Figure 1a).Giving a sharp estimate for the mixing time of the triangulation walk as a function of n isa notoriously di ffi cult open problem. The lower bound of Ω ( n / ) shown by Molloy, Reed andSteiger [11], which is in fact Aldous’ original conjecture for the actual growth rate of the relaxationtime, is still quite distant from the best upper bound to date, which is the O ( n log n ) obtained byMcShine and Tetali [10]. ♦ Department of Statistics, University of Oxford, UK.
[email protected] .The author would like to acknowledge that part of this work was done while they were supported by the EPSRC grantEP / N004566 / ut triangulations of the n -gon are not the only structures that are well-suited to supporting anedge flip chain, though they provide perhaps the simplest possible example; edge flip dynamicshave been considered for example on lattice triangulations [6, 7, 13] and rectangular dissections [5,4]. Recently, Alexandre Stau ff er and the author proved a polynomial upper bound for the mixingtime of edge flips on quadrangulations of the sphere [8].Some very natural classes of combinatorial objects able to support edge flip chains are specificsets of so-called planar maps , where by planar map we mean a connected, locally finite planar(multi)graph endowed with a cellular embedding in the two-dimensional sphere, considered upto orientation-preserving homeomorphisms of the sphere itself. For example, one might considertriangulations of the sphere with n edges – that is, planar maps whose faces have degree 3 – ratherthan triangulations of the n -gon. An edge flip would then consist in choosing an edge uniformly atrandom and, with probability 1 /
2, replacing it with the other diagonal of the quadrilateral formedby the two faces adjacent to it – or, if the edge is adjacent to only one face, leaving it unchanged(see Figure 1b). This chain has been considered by Budzinski in [3], where he shows a lowerbound of Ω ( n / ) for the mixing time.Analogous chains can be defined on the set of p -angulations of the sphere with n edges for any p >
3: one chooses an edge uniformly at random and, if it is adjacent to two distinct faces, erasesit to obtain a (2 p − f , and then draws an edge joining the i -th corner of f , where i is chosen uniformly at random in { , , . . . , p − } (and corners are labelled, say, clockwise), tocorner i + p − p − p -angular faces within f . Some care must betaken (and some non-canonical choices made) in dealing with edges that are adjacent to a singleface on both sides.An especially attractive case is p =
4, namely, that of quadrangulations (see Figure 1c). In thiscase, an edge separating two faces, if flipped, will be replaced by one of three edges cutting thehexagon created in its absence “in half”, chosen uniformly at random. In particular, it remainsunchanged with probability 1 /
3. It is therefore natural to define a flip for a quadrangulation edgeadjacent to the same face on both sides as leaving the edge unchanged with probability 1 / /
3, replacing it with an edge joining its degree 1 endpoint to the unique vertex ofthe face which was not an endpoint of the original edge (see Figure 7, and more generally Section 4for a detailed description of the dynamics).The case of quadrangulations of the sphere is interesting for multiple reasons. One is that it isstill very simple and preserves a strong similarity to the case of edge flips on triangulations of thesphere and of the n -gon. Another is the fact that quadrangulations in particular come with a veryhandy toolset, including Schae ff er-type bijections with labelled plane trees [12]: they fall withinthe scope of so-called Catalan structures, that is, combinatorial structures whose enumeration isclosely related to Catalan numbers (e.g. plane trees, triangulations of the n -gon, binary trees etc.);as a consequence, opportunities arise for a number of possible Markov chain comparisons.One such comparison, made with a “leaf translation” Markov chain on labelled plane trees, iswhat made it possible to show the main result of [8], namely an upper bound of order n / forthe relaxation time of the edge flip Markov chain on quadrangulations of the sphere.It should now be mentioned that, in order to have the Schae ff er bijection with labelled planetrees and to have Catalan numbers emerge when enumerating quadrangulations, one considers pointed , rooted quadrangulations of the sphere – that is, quadrangulations endowed with a distin-guished vertex and a distinguished oriented edge. Redefining the dynamics to take the pointingand rooting into account poses no di ffi culties; the choice made in [8] is that of performing edge flipsexactly as described, preserving the pointing and the orientation of the root edge when flipped(Figure 7). It seems quite reasonable that pointing and rooting should not be truly relevant, andindeed the pointing can be quickly dealt with and does not appear in the results of [8]. As for the a) (b) (c) Figure 1: An edge flip performed on a triangulation of the octagon (a); an edge flip on arooted triangulation of the sphere – drawn so that the infinite face lies to the right of theroot edge (b); an edge flip on a quadrangulation of the sphere, where both possible newalternative edges are drawn, dashed, in di ff erent colours (c). rooting, however, it is worth noting that its role is more central. While for example it is naturalto conjecture that the upper bound of O ( n / ) for the relaxation time proved in [8] should alsohold for the mixing time of – say – a Markov chain that censors flips of the root edge, or thatexcludes the root edge from the set of “flippable” edges, this fact is not easy to show; moreover,the proof in the aforementioned paper relies heavily on some ad hoc geometric constructions thatbuild upon the Schae ff er bijection, and root edge flips feature prominently in its canonical paths,so that adapting the proof is utterly non-trivial.On the other hand, the argument in [8] does have the potential for generalisation, and onemay very well wish to apply variants of it to other edge flip Markov chains and to other classes ofplanar maps.We have mentioned how one could consider edge flips on p -angulations for p ,
4; one otheravenue for generalisation would be to consider, rather than edge flips on p -angulations, dynamicson the set of all planar maps with – say – a fixed number of edges, with no restrictions on facedegrees. This is exactly what we propose to do in this paper. We shall consider a natural dynamicon planar maps that, in analogy to edge flips, involves the local manipulation of a single randomedge at each step. What we will introduce is a Markov chain which we will call the edge rotation chain on (rooted) planar maps with n edges. A single step consists essentially in choosing anoriented edge uniformly at random and sliding its “tip” one step to the left, or one step to theright, or leaving everything unchanged (each with probability 1 / rooted planar maps, we can take advantageof how general rooted planar maps with a fixed number of edges can themselves be thought ofas Catalan structures. Indeed, thanks to Tutte’s bijection [14] we shall directly relate the edgerotation chain to a version of the edge flip Markov chain on rooted quadrangulations where thequadrangulation root edge is not included in the set of “flippable” edges.We will then proceed to give an upper bound that will apply to both the mixing time of the n edges; in eachcase, an oriented edge is shown, together with the two (or one, in the last picture) facesadjacent to it. One can rotate it clockwise or counterclocwise, which in some cases maycreate a loop (second picture). If the edge is itself a loop enclosing a face of degree 1 (thirdpicture) one of the two edge rotations causes it to “detach itself” from the boundary of itsexternal face and create a new degree 1 vertex. If the edge is oriented towards an endpointof degree 1 (fourth picture) then rotating it in either direction eliminates that endpoint infavour of a loop. A complete presentation (not in terms of the rotated oriented edge butof the corner that the tip “rotates through”) is given in Section 3. edge rotation chain and that of the variant edge flip chain on rooted quadrangulations. Our mainresult is the following: Theorem 1.
Let ν n and µ n be the spectral gaps of the (non-root-flipping) edge flip Markov chain ˜ F n onthe set Q n of quadrangulations with n faces and of the edge rotation Markov chain R n on the set M n ofrooted planar maps with n edges, respectively. We have ν n = µ n , and there are positive constants C , C (independent of n) such that C n − / ≥ ν n ≥ C n − / for all n. Consequently, the mixing time of both chains is O ( n / ) . The proof will combine part of the approach of [8] with some new ideas, which render it almostcompletely independent of Schae ff er’s bijection: we shall construct probabilistic canonical pathson the set of rooted quadrangulations rather than the set of plane trees, thus making the approachmore readily generalisable.Section 2 introduces relevant objects – maps, quadrangulations – and contains a brief descrip-tion of Tutte’s bijection, which will be used in Section 4 to relate the edge rotation chain presentedin Section 3 to an edge flip chain.The rest of the paper will develop the necessary tools to prove Theorem 1. The argument isbased on an algorithm to grow quadrangulations uniformly at random by “adding faces” one ata time (Section 5) and a construction of probabilistic canonical paths (Section 6) which is truly thecore of this paper. Section 7 concludes the proof. Definition 2.1. A planar map is a connected, locally finite planar (multi)graph endowed witha cellular embedding in the two-dimensional sphere, considered up to orientation-preservinghomeomorphisms of the sphere itself.Of course, planar maps inherit terminology and features from graphs – we shall speak of their vertices and edges – but with their built-in planar embedding comes the added perk of having c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c Figure 3: A rooted planar map drawn in the plane in such a way that the infinite facecontains the root corner. A face of degree 9 is shaded and the labels c , . . . , c placed alongits clockwise contour. well defined faces (i.e. the connected components of the complement of the image of vertices andedges via the cellular embedding, see Figure 3). It will often prove useful to endow an edge withan orientation (each edge has two possible orientations). Given an oriented edge ~ e in a map m whose endpoints are a vertex e − (the tail) and a vertex e + , we shall informally say that the corner corresponding to ~ e is a suitably small neighbourhood of the vertex e − intersected with the facelying directly to the right of ~ e .We will speak of corners as “belonging to” faces (the corner corresponding to ~ e belongs tothe face lying directly to the right of ~ e ) and also to vertices (the corner corresponding to ~ e is acorner of vertex e − ). Corners of a single vertex and corners of a single face have two naturalcyclic orderings: clockwise and counterclockwise . Given a face f of a map m , we shall call the cyclicsequence ( c i ) deg fi = of all corners of f in clockwise (resp. counterclockwise) order, where the indexis considered modulo deg f , a clockwise (resp. counterclockwise ) contour of f ; the number deg f ofcorners of f is the degree of the face f . When mentioning a contour of the face f without specifyingits direction, we shall be referring to its clockwise contour.A rooted planar map is a pair ( m , c ), where m is a planar map and c is a corner of m ; since – asexplained above – there is a direct correspondence between corners and oriented edges, we mayalso see a rooted planar map as being endowed with a distinguished oriented edge rather than adistinguished corner: we will adopt either point of view, depending of what is most convenient.We shall call the vertex that the root corner c belongs to, i.e. the tail of the root edge, the origin of the rooted map ( m , c ); we shall often denote the origin of a map by ∅ . Note that all maps we willrefer to in this paper will be rooted; we will therefore, for the sake of simplicity, usually denotethem by a single letter and not as a pair: we will write M n for the set of all rooted planar mapswith n edges and will write m ∈ M n to indicate that m is a planar map with n edges and is alsoendowed with a root corner / edge, which will normally be denoted by ρ . Definition 2.2.
A quadrangulation is a planar map all of whose faces have degree 4. We shall ccccccccccccccccc + c + c + c + c + c + c + c + c + c + c + c + c + c + c + c + c + c − c − c − c − c − c − c − c − c − c − c − c − c − c − c − c − c − e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e − e − e − e − e − e − e − e − e − e − e − e − e − e − e − e − e − e + e + e + e + e + e + e + e + e + e + e + e + e + e + e + e + e + ccccccccccccccccce = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e + e + e + e + e + e + e + e + e + e + e + e + e + e + e + e + e + e − e − e − e − e − e − e − e − e − e − e − e − e − e − e − e − e − c = c + = c − c = c + = c − c = c + = c − c = c + = c − c = c + = c − c = c + = c − c = c + = c − c = c + = c − c = c + = c − c = c + = c − c = c + = c − c = c + = c − c = c + = c − c = c + = c − c = c + = c − c = c + = c − c = c + = c − e + = e − e + = e − e + = e − e + = e − e + = e − e + = e − e + = e − e + = e − e + = e − e + = e − e + = e − e + = e − e + = e − e + = e − e + = e − e + = e − e + = e − e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = ccccccccccccccccce − = e + e − = e + e − = e + e − = e + e − = e + e − = e + e − = e + e − = e + e − = e + e − = e + e − = e + e − = e + e − = e + e − = e + e − = e + e − = e + e − = e + c − c − c − c − c − c − c − c − c − c − c − c − c − c − c − c − c − c + c + c + c + c + c + c + c + c + c + c + c + c + c + c + c + c + Figure 5: The corners c , c − , c + within the face f c ; the edge e = drawn by the procedure isdashed. Notice that e = is a loop if e − and e + are corners of the same vertex, which happenswhen deg f c = c is 1 (second and fourth image). write Q n for the set of all rooted quadrangulations with n faces.It is a classical result of Tutte [14] that we have | Q n | = | M n | ; and in fact, Tutte himself providesa simple explicit bijection Φ : M n → Q n , which we shall briefly describe here before making use ofit for our purposes.Given a rooted planar map m ∈ M n , build a new rooted planar map as follows: • draw one vertex within each face of m ; • connect each newly drawn vertex to all corners in the face it belongs to (draw new edges insuch a way that they do not cross); • erase all original edges of m ; • there is one edge drawn by this procedure that crosses the original root corner of m ; let thatedge be the new root edge, oriented away from the original root corner.The procedure described above yields a rooted planar map Φ ( m ) which has | V ( m ) | + | F ( m ) | = n + | E ( m ) | = n edges, hence n faces, each of which can be shown to be a quadrangle; inother words, Φ ( m ) ∈ Q n .An inverse procedure can be described just as easily: given a quadrangulation q ∈ Q n , • partition the set of vertices of q into two parts: we shall call real vertices those at even graphdistance from the origin and face vertices those at odd distance (notice that real vertices areonly adjacent to face vertices and vice-versa, so each face has two corners of real vertices andtwo corners of face vertices); • within each face, draw an edge joining its two corners belonging to real vertices; • erase all face vertices and all original edges of q ; • root the newly formed map in the one corner that the original root edge of q was issued from.The map resulting from this procedure, which clearly has n edges, one for each face of q , is noneother than Φ − ( q ). Indeed, we have the following: Theorem 2 (Tutte) . The mapping Φ is a bijection between the set M n of rooted planar maps with n edgesand the set Q n of rooted quadrangulations with n faces; it induces a correspondence between the set of edgesof each map m and the set of faces of Φ ( m ) . M n Let m be a map in M n , let c be a corner of m other than the root corner and let s be an element of { = , + , −} . Construct a map m c , s ∈ M n as follows: the corner c belongs to a face f c of m ; let c − and c + be the corners immediately before andimmediately after c in a clockwise contour of f c ; • let e − and e + be the edges of f c joining c − to c and c to c + respectively; in the case wheredeg f c =
2, in which c − = c + , e − and e + are the two edges forming the boundary of f c , andthey are named in such a way that f c lies to the right of the edge e + , oriented away from c (Figure 5); in the case where deg f c =
1, in which c = c − = c + , we set e − = e + to be the oneloop which constitutes the boundary of f c ; • if deg f c >
1, draw an edge e = joining corner c − to corner c + (in the case where c − = c + andthe case where the vertex of c has degree 1, notice that e = will be a loop); • if c − = c = c + (that is if deg f c =
1) draw a new vertex within the loop e − and join it to thevertex of c by a new edge e = ; • notice now that, whatever the case for deg f c , one new edge e = has been drawn and atriangular face containing c , whose boundary edges are { e = , e − , e + } (which are not necessarilyall distinct), has been created; • finally, erase the edge e s (and any vertices adjacent only to e s ); if s = ± and the root corner ρ of m is in { c − , c + } , set the new root corner to be the part of ρ that did not belong to thetriangular face containing c created by drawing e = ; otherwise, set the new root corner to bethe one that contains the original corner ρ (which is “larger” than the original only if e s is anedge adjacent to ρ ). The new rooted map obtained in this way is m c , s ; it has exactly as manyedges as m and therefore belongs to M n .We shall say that m c , s is obtained from m via an edge rotation ; though rotating edges is notexplicitly mentioned in the construction above, the reason for the name should be clear: exceptfor some rather degenerate cases, the whole construction – when s ∈ {− , + } – essentially consistsof “rotating” the edge e s within the corner c s by “detaching it” from the vertex of c and insteadsetting its other endpoint to be within the next corner in the clockwise / counterclockwise contourof f c , thus e ff ectively turning it into the new edge e = (see Figure 6). Even the case where e − = e + ,which turns an edge with an endpoint of degree 1 into a loop and vice-versa, can be thought of asan edge rotation of sorts.We can naturally identify the edge e s in m with the “rotated edge” e = in m c , s (when s ∈ { + , −} )and thus have a natural identification between edges of m and edges of m c , s . Faces and verticescannot be as readily identified between m and m c , s , because the number of faces and vertices mayincrease or decrease; and, even though the number of corners remains unchanged after an edgerotation, defining a 1-to-1 correspondence is not entirely canonical, although there is one that iscompatible with our choice of the rerooting, in the sense that it allows us to interpret the newchoice of the root corner as “leaving it unchanged”.Corners other than those involving the vertices of c , c − , c + are of course untouched, and will bedenoted by the same symbols in m and m c , s . If s ∈ {− , + } , then we shall identify c − and c + with thetwo (not necessarily distinct) corners joined by the newly drawn edge e = in the face lying directlyto the right of e = , oriented from c − to c + (again, see Figure 6).Notice that, if s = + , then the number of corners around the vertex v + of c + is unchanged; ifthose corners were c + , c , . . . , c k in clockwise order around v + , starting with c + in the map m , wewill identify them with the k + c + in m c , s , having already identified c + , by just keeping the same order.The dynamic on maps we shall be considering throughout this paper is given by a Markovchain R n on M n which is such that, assuming R nk = m for some m ∈ M n , we have R nk + = m c , s , where c and s are independent random variables, s being uniformly distributed in { = , + , −} and c being c f c f c f c f c f c f c f c f c f c f c f c f c f c f c f c f c e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e + e + e + e + e + e + e + e + e + e + e + e + e + e + e + e + e + e − e − e − e − e − e − e − e − e − e − e − e − e − e − e − e − e − c + c + c + c + c + c + c + c + c + c + c + c + c + c + c + c + c + ccccccccccccccccc c − c − c − c − c − c − c − c − c − c − c − c − c − c − c − c − c − f c f c f c f c f c f c f c f c f c f c f c f c f c f c f c f c f c e + e + e + e + e + e + e + e + e + e + e + e + e + e + e + e + e + e − e − e − e − e − e − e − e − e − e − e − e − e − e − e − e − e − m c , = m c , = m c , = m c , = m c , = m c , = m c , = m c , = m c , = m c , = m c , = m c , = m c , = m c , = m c , = m c , = m c , = c + c + c + c + c + c + c + c + c + c + c + c + c + c + c + c + c + ccccccccccccccccc c − c − c − c − c − c − c − c − c − c − c − c − c − c − c − c − c − f c f c f c f c f c f c f c f c f c f c f c f c f c f c f c f c f c e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e − e − e − e − e − e − e − e − e − e − e − e − e − e − e − e − e − m c , + m c , + m c , + m c , + m c , + m c , + m c , + m c , + m c , + m c , + m c , + m c , + m c , + m c , + m c , + m c , + m c , + c + c + c + c + c + c + c + c + c + c + c + c + c + c + c + c + c + c − c − c − c − c − c − c − c − c − c − c − c − c − c − c − c − c − f c f c f c f c f c f c f c f c f c f c f c f c f c f c f c f c f c e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e = e + e + e + e + e + e + e + e + e + e + e + e + e + e + e + e + e + m c , − m c , − m c , − m c , − m c , − m c , − m c , − m c , − m c , − m c , − m c , − m c , − m c , − m c , − m c , − m c , − m c , − c + c + c + c + c + c + c + c + c + c + c + c + c + c + c + c + c + c − c − c − c − c − c − c − c − c − c − c − c − c − c − c − c − c − Figure 6: The three maps of the form m c , s as constructed from m ∈ M n . a corner of m other than its root corner, chosen uniformly at random; in other words, transitionsprobabilities for R n are of the form p R ( m , m ′ ) = n − X c ∈ C ( m ) \ ρ (1 m ′ = m c , + + m ′ = m c , − + m ′ = m ) , (1)where C ( m ) is the set of all corners of m and ρ is its root corner. Lemma 3.1.
The Markov chain R n is reversible, aperiodic and irreducible.Proof. The first two properties are clear by construction (they can be inferred immediately fromexpression (1) for the transition probabilities).As for irreducibility, we shall show that every map m ∈ M n can be turned into the map m madeof n nested loops, rooted in the corner within the central loop, via a sequence of edge rotations.Given m ∈ M n , consider the face f ρ containing the root corner ρ of m . Suppose it has clockwisecontour ρ, c , . . . , c k , with k ≥
1: by considering the edge-rotated map m c , − we can reduce thedegree of f ρ by 1 (by which we mean the the face containing the root corner in m c , − has degree k rather than k + m , via a sequence of edge rotations, to a map whoseroot corner lies within a loop (i.e. such that the root edge is a loop). Now, given any map ˜ m suchthat the root edge is a loop, consider the first corner c in counterclockwise order around the origin,starting with the root corner ρ , such that c lies within a face f c of degree strictly more than 2, whoseclockwise contour we will call c , c , . . . , c k (with k > m c , − decreases the degreeof the face f c by 1: repeating this operation yields a map such that the root edge is a loop and allcorners around the origin lie within faces of degree 1 or 2. Such a map can only be m : if onedraws it on the plane in such a way that the root corner lies within the infinite face, so that the rootedge is an “external” loop, one finds that the finite face adjacent to it is either a degree 1 face – inwhich case the map has only one edge – or has degree 2, in which case its boundary is completedby one “internal” loop; repeating this argument inductively identifies the map as m . (cid:3) eeeeeeeeeeeeeeee q q e , + q e , − v vwwq q e , + = q e , − eeeeeeeeeeeeeeeee Figure 7: Clockwise and counterclockwise flips for a simple and a double edge in aquadrangulation.
We now wish to relate our edge rotation dynamic on planar maps to the Markov chain of edgeflips on quadrangulations as introduced in [8] – or rather, to a slight variant thereof.The edge flip Markov chain F n on the set Q n was introduced in [8] as a chain whose stepsconsist in, given a quadrangulation, selecting one of its edges uniformly at random and thenmaking an independent uniform choice among the following three options: leaving it unchanged,flipping it clockwise or flipping it counterclockwise. The choice of the root edge was allowed andflipping the root edge would preserve its orientation.More formally, given a quadrangulation q ∈ Q n and an edge e of q , we denote by q e , + (resp. q e , − ), the quadrangulation obtained from q by flipping edge e clockwise (resp. counterclockwise ), by which wemean the quadrangulation given by the following procedure: • if e is adjacent to two distinct faces of q , erase e from q (thus obtaining a new face with exactly6 corners) and replace it with the edge obtained by rotating e clockwise (resp. counterclock-wise) by one corner (see Figure 7). • if e is an internal edge within a degenerate face, let v be the vertex of that face that is not an endpoint of e and let w be the endpoint of e having degree 1; erase e and replace it withan edge within the same face having endpoints v , w . If e is the root edge of q , let the newlydrawn edge be the root of q e , + (resp. q e , − ), oriented in the same way as before (with respectto w ).The edge flip Markov chain as originally described has transition probabilities p F ( q , q ′ ) = n X e ∈ E ( q ) (cid:16) q ′ = q e , + + q ′ = q e , − + q = q ′ (cid:17) . In order to directly relate a dynamic on maps to edge flips on quadrangulations, however, oneis led to consider a variant of the chain F n that does not allow flipping the root edge. Indeed,the Tutte bijection Φ assigns very di ff erent roles to quadrangulation vertices at even and odddistance from the origin: since a root flip (at least as described) would change the parity of thedistance to the origin for each vertex in the quadrangulation, the two maps corresponding to thequadrangulation before and after the flip are potentially completely di ff erent from each other.It therefore becomes necessary to redefine or completely eliminate root edge flips from thechain F n ; in particular, we shall from here on consider a new edge flip Markov chain where thechoice of the edge to flip is uniform among all edges other than the root edge . We shall still refer to q e , + q e , − q q e , + = q e , − Figure 8: Clockwise and counterclockwise flips for a simple and a double edge in aquadrangulation. this as the edge flip
Markov chain on Q n and we shall denote it by ˜ F n ; its transition probabilitiesare of the form p ˜ F ( q , q ′ ) = n − X e ∈ E ( q ) \ ρ (cid:16) q ′ = q e , + + q ′ = q e , − + q = q ′ (cid:17) , where ρ is the root edge of q . Proposition 4.1.
Given a quadrangulation q ∈ Q n with root edge ρ , an edge e ∈ E ( q ) \ ρ and s ∈ { + , −} ,we have Φ ( q e , s ) = Φ ( q ) c , s , where Φ is the Tutte bijection from Section 2 and c is the corner of Φ ( q ) thatcorresponds to the edge e of q.Proof. Consider the case where e is not an internal edge within a degenerate face, but rather isadjacent to two distinct faces of q , within each of which a map edge is drawn by the construction Φ . Orient the edge e away from its endpoint at even distance from the origin and let f − be the facelying to its right and f + the one lying to its left. If we take c to be the corner of Φ ( q ) that the edge e is issued from, it should be clear that the map edges e − and e + constructed as a function of c inSection 3 correspond to quadrangulation faces f − and f + respectively.Now consider for example the edge-flipped quadrangulation q e , + ; it is clear that, since e isnot the root edge, the parity of distances from the origin is unchanged. The endpoints of e − aretherefore still “real vertices” which need to be joined by a map edge lying within the quadrangularface next to the flipped edge e : we can draw e − exactly as before. This is in contrast to the edge e + , which would now cross the flipped edge e , and therefore needs to be erased. The edge thatreplaces e + is an edge e = that would form a triangle containing c , along with e − and e + , in theoriginal map (left part of Figure 8).Furthermore, notice that the fact that the root edge is unchanged in q e , s implies that the rootcorner of Φ ( q e , s ) must still be the one that the quadrangulation root edge is issued from. In orderfor this to be true in the non-trivial cases where the root corner of Φ ( q ) is “split” by the addition of e = , the root corner must become the part of the corner lying outside the e − , e + , e = triangle, exactlyas described in Section 3.This shows that Φ ( q e , s ) = Φ ( q ) c , s when e is not an internal edge within a degenerate face and s = + (including the case where e = ends up being a loop, which one can see arise in the situationdepicted in Figure 8 when edges on the boundary of f + and f − are identified); the case of s = − isidentical.Now consider the case where the endpoint u of e at odd distance from the origin has degree 1in q . This corresponds to its degenerate face having two face vertices and one real vertex, and amap loop edge ( e − = e + ) being drawn within it by the construction Φ . In this case, we know that q e , s , for s = ± , is the quadrangulation that replaces e with an edge drawn between u and the othervertex on the external boundary of the degenerate face. It is immediately apparent that Φ ( q e , s ) is, ndeed, the map which replaces the loop e − = e + with an edge e = having a brand new vertex as anendpoint, which is Φ ( q ) c , s . The rooting poses no real issues, since e cannot be the root edge of q and the identification of corners between Φ ( q ) and Φ ( q e , s ) is clear.Finally, the case where the endpoint of e at even distance from the root has degree 1 in q isprecisely the inverse of the one above. (cid:3) We have thus shown that the two Markov chains ˜ F n and R n are isomorphic; in particular, theyhave the same relaxation and mixing time.The main result in [8] consisted in the following bounds for the spectral gap of the Markovchain F n : Theorem (C., Stau ff er) . Let γ n be the spectral gap of the edge flip Markov chain F n on the set Q n of rootedquadrangulations with n faces. There are positive constants C , C independent of n such thatC n − ≤ γ n ≤ C n − . Consequently, the mixing time for F n is O ( n / ) . While the upper bound above for the spectral gap of F n immediately yields a lower bound forthe relaxation time of ˜ F n and therefore R n , an upper bound for the relaxation time of ˜ F n cannottrivially be gleaned from [8]. The rest of this paper will therefore be devoted to analysing thechain ˜ F n to obtain an upper bound which applies to the edge rotation Markov chain. Thoughthe general strategy is not dissimilar to the one employed in [8], some ad hoc constructions andideas will be necessary; as a result, we will have a partially new proof of an upper bound forthe relaxation time of the original chain F n which (mostly) does not rely on the Cori-Vanquelin-Schae ff er correspondence with plane trees, and should therefore be better suited for furthergeneralisations. Consider the following operation which, given a quadrangulation q ∈ Q n (with n >
1) and a corner c of q , yields a quadrangulation coll( q , c ) in Q n − . If the face f of q containing c has 4 distinct verticesand c , c , c , c is a clockwise contour of f , then “collapse” f by identifying the edge joining c to c with the one joining c to c and the edge joining c to c with the one joining c to c as in Figure 9,thus identifying the vertices of corners c and c . If f has fewer than 4 distinct vertices then ithas exactly 3, one of which is adjacent to two corners c , c of f , where c , c , c , c is a clockwisecontour; in this case, whatever c , identify the edge joining c to c with the edge joining c to c and the edge joining c to c with the edge joining c to c , thus also identifying the vertices of c and c . Note that the latter procedure applies to the case of f being a degenerate face (lower partof Figure 9). If the root corner does not belong to f , it is simply preserved; if it belongs to f , thenwe root the quadrangulation coll( q , c ), in such a way that the root edge is the collapsed image ofthe original root edge, oriented as before.We shall say that two quadrangulations q ∈ Q n and q ′ ∈ Q n − di ff er by collapsing a face if there isa corner c of q such that q ′ = coll( q , c ).What we need is a “hierarchy” like the one described for coloured plane trees in Section 5of [8], but for quadrangulations, where the adjacency condition of di ff ering by erasing a leaf isreplaced by the one given by collapsing faces. What we wish to produce is a collection of mappings g n : Q n × Q n − → R ≥ with the following properties:(i) g n ( q , q ′ ) = q ′ cannot be obtained from q by collapsing a face; ffffffffffffffff fffffffffffffffff fffffffffffffffff f Figure 9: Collapsing a face in a quadrangulation. (ii) X q ′ ∈ Q n − g n ( q , q ′ ) = q in Q n ;(iii) X q ∈ Q n g n ( q , q ′ ) = | Q n || Q n − | for all q ′ in Q n − .Such a collection of mappings g n can be built explicitly from the mappings f n given in Section 5of [8]. In order to do this, we need to briefly recall some notation and standard results. Definition 5.1. A labelled tree is a plane tree t (i.e. a rooted planar map with a single face) endowedwith a labelling l : V ( t ) → Z such that • if ∅ is the origin of t , l ( ∅ ) = • for any vertex v ∈ V ( t ) \ {∅} , | l ( v ) − l ( p ( v )) | ∈ { , − , } , where p ( v ) denotes the parent of v .We shall call LT n the set of all labelled trees with n edges, and conventionally set LT = {•} to bethe set containing the graph with a single vertex labelled 0.We shall also write Q • n for the set of all pairs ( q , δ ), where q ∈ Q n and δ ∈ V ( q ), i.e. for the setof all pointed (rooted) quadrangulations of the sphere with n faces. We shall conventionally definethe set Q = {→} as the one containing a rooted planar map with a single edge and two distinctvertices; as a consequence, Q • has two elements. Also by convention, but consistently with ourprevious definition, we shall set coll( q , c ), where c is any corner in a quadrangulation q ∈ Q , to bethe “root-only map” →∈ Q .Labelled trees (and pointed quadrangulations) will be useful thanks to the Cori-Vanquelin-Schae ff er correspondence (see [12]), which is an explicit bijective construction φ : LT n ×{− , } → Q • n transforming tree labels into quadrangulation graph distances: given t ∈ LT n and ε ∈ {− , } , themapping φ naturally induces an identification between vertices of t and vertices of φ ( t , ε ) otherthan the distinguished vertex such that, if l is the labelling of t , we have l ( v ) = d gr ( v , δ ) − d gr ( δ, ∅ ),where v is interpreted as a vertex of t in the left hand side of the equation and as a vertex of φ ( t , ε )in the right hand side, ∅ is the origin of φ ( t , ε ) and δ its distinguished vertex, and d gr is the graphdistance on the vertex set of φ ( t , ε ).In Section 5 of [8] we provided a collection of maps f n : LT n × LT n − → R with the exactproperties stated for g n above, where every instance of Q k is replaced by LT k and (i) is replaced by(i) f n ( t , t ′ ) = t ′ cannot be obtained from t by erasing a leaf.In particular, we showed that such properties hold for the collection of maps constructedrecursively as follows: a a − aa + a − aa − a − a − v and the edge ( v , p ( v )) joining it to itsparent) from a labelled tree t ∈ LT n , whatever the labels, corresponds to collapsing the facebuilt around the edge ( v , p ( v )) in the pointed quadrangulation Φ ( t , ε ). • if ( t , t ′ ) ∈ LT n × LT n − do not di ff er by erasing a leaf, f n ( t , t ′ ) = • if ( t , t ′ ) ∈ LT × LT , set f ( t , t ′ ) = • if ( t , t ′ ) ∈ LT n × LT n − , where n >
1, di ff er by erasing a leaf, consider the subtrees L ( t ) , L ( t ′ )containing the leftmost child of the root vertex and its descendants in t , t ′ respectively; if | L ( t ) | = i and | L ( t ′ ) | = i − i >
0, set f n ( t , t ′ ) = i ( i + n − i − n − n ( n + f i ( L ( t ) , L ( t ′ ));otherwise set R ( t ), R ( t ′ ) to be the trees obtained by erasing L ( t ) , L ( t ′ ) from t , t ′ (as well as theedge joining the root vertex to its leftmost child); we then have | R ( t ) | = | R ( t ′ ) | + = i for some i > f n ( t , t ′ ) = i ( i + n − i − n − n ( n + f i ( R ( t ) , R ( t ′ )) . The main reason why the collection of mappings f n can be used to construct mappings g n which satisfy the properties we require is the following: Lemma 5.1. If ( t , t ′ ) ∈ LT n × LT n − , ε = ± and f n ( t , t ′ ) > , then F ( φ ( t ′ , ε )) is obtained from F ( φ ( t , ε )) by collapsing a face, where F : S i ≥ Q • i → S i ≥ Q i is the mapping which forgets the distinguished vertex.Proof. The proof does of course rely on the specific definition of φ (and is the only part of thispaper that does). The quadrangulation φ ( t , ε ) can be drawn using the vertex set of t and an addedvertex δ as follows: consider a counterclockwise (cyclic) contour c , . . . , c n of the one face of t ;for each i , draw a quadrangulation edge joining c i to its “target” corner, which we will take to bethe next corner in the contour whose vertex has strictly smaller label than the vertex of c i , or thecorner around δ if the label of the vertex that c i belongs to is minimal.Suppose t ′ is obtained from t by erasing a leaf v and the edge ( v , p ( v )). If l ( v ) = l ( p ( v )) or l ( v ) = l ( p ( v )) +
1, then the the two quadrangulation edges issued from the corner of t right beforethe one around v and the one right after v have the same “target” corner and enclose a degenerateface of the quadrangulation φ ( t , ε ) (see Figure 10). Erasing v collapses those two edges into asingle edge; targets for corners other than the one around v (which is eliminated) are una ff ected.Furthermore, there is no issue with the rooting: φ ( t , ε ) is rooted in the edge issued by the rootcorner of t , with an orientation given by ε : such an edge does correspond to the edge issued fromthe root corner of t ′ .If l ( v ) < l ( p ( v )), then the matter slightly more complicated. The contour of t ′ has two fewercorners than the contour of t : the quadrangulation edges e and e issued from the corner before v and the corner around v are eliminated. Let c be the target corner of the corner immediately after , which must be around a vertex labelled l ( v ). Suppose c is not the corner of v ; then all cornershaving the corner of v as a target in t have c as a target in t ′ : equivalently, all edges adjacent to v in φ ( t , ε ) become adjacent to the vertex of c in φ ( t ′ , ε ). Eliminating edges e , e and reroutingall edges adjacent to v to the vertex of c exactly amounts to collapsing the quadrangulation facewhich encloses the tree edge ( v , p ( v )). If ˜ c is the quadrangulation corner in φ ( t , ε ) correspondingto the edge e , oriented towards v , the quadrangulation F ( φ ( t ′ , ε )) is coll( F ( φ ( t , ε )) , ˜ c ) (again, therooting is correctly preserved).If c is the corner around v , then l ( v ) is minimal and v is the unique vertex carrying label l ( v ); inthat case, the face enclosing the edge ( v , p ( v )) is again degenerate and contains the vertex δ , whichis the furthest one from the origin in the quadrangulation φ ( t , ε ). In this case, the quadrangulation φ ( t ′ , ε ) can be obtained from t and φ ( t , ε ) by simply eliminating the original pointed vertex δ from φ ( t , ε ), erasing the tree edge ( v , p ( v )) and renaming vertex v to δ : that way, all one needs todo is erase the two quadrangulation edges that were drawn from the corner of v and from thecorner after v , which amounts to collapsing the degenerate face that corresponded to the tree edge( v , p ( v )). The quadrangulation φ ( t ′ , ε ) also has its pointing “moved” (which is natural, since φ ( t , ε )was pointed in a vertex within the face to be collapsed), but this has no bearing on F ( φ ( t , ε )). (cid:3) Lemma 5.2.
The collection of mappings g n : Q n × Q n − → R defined asg n ( q , q ′ ) = n + X v ∈ V ( q ) X v ′ ∈ V ( q ′ ) ε q , v = ε q ′ , v ′ f n ( t q , v , t q ′ , v ′ ) , where f n : LT n × LT n − → R ≥ is defined recursively as described before and where φ ( t q , v , ε q , v ) = ( q , v ) and φ ( t q ′ , v ′ , ε q ′ , v ′ ) = ( q ′ , v ′ ) , satisfies properties (i), (ii) and (iii).Proof. This is straightforward from the properties of f n .Indeed, property (i) for g n is a consequence of Lemma 5.1: if there are v , v ′ in V ( q ) and V ( q ′ )respectively such that φ − ( q , v ) = ( t , ε ) and φ − ( q ′ , v ′ ) = ( t ′ , ε ), where t ∈ LT n and t ′ ∈ LT n − are suchthat f n ( t , t ′ ) ,
0, then, since t and t ′ di ff er by erasing a leaf, the quadrangulations q = F ( φ ( t , ε )) and q ′ = F ( φ ( t ′ , ε )) di ff er by collapsing a face.As for property (ii), we have X q ′ ∈ Q n − g n ( q , q ′ ) = n + X v ∈ V ( q ) X ( q ′ , v ′ ) ∈ Q • n − ε q , v = ε q ′ , v ′ f n ( t q , v , t q ′ , v ′ ) == n + X v ∈ V ( q ) X ( t ′ ,ε ) ∈ LT n − ×{ , − } ε q , v = ε f n ( t q , v , t ′ ) = n + X v ∈ V ( q ) X t ′ ∈ LT n − f n ( t q , v , t ′ ) == n + X v ∈ V ( q ) = . Similarly, for (iii) one has X q ∈ Q n g n ( q , q ′ ) = n + X v ′ ∈ V ( q ′ ) X t ∈ LT n f n ( t , t q ′ , v ′ ) = | LT n | ( n + | LT n − | ( n + = | Q n || Q n − | . (cid:3) Canonical paths
Given two quadrangulations q , q ′ ∈ Q n , we intend to build a random canonical path from q to q ′ ,that is a probability measure P q → q ′ on the set Γ q → q ′ of all sequences ( q i , e i , s i ) Ni = such that • for all i = , . . . , N , we have q i ∈ Q n and e i ∈ E ( q i ) \ { ρ } , where ρ is the root edge of q i , while s i = ± ; • q i + = q e i , s i i for i = , . . . , N − • q = q and q e N , s N N = q ′ .Note that our aim is to construct these paths in such a way that, given an edge flip ( q , e , s ), thequantity P q , q ′ ∈ Q n P q → q ′ { γ ∈ Γ q → q ′ | ( q , e , s ) appears in γ } is as small as possible.The main idea of the construction is to have a canonical way of splitting intermediate quad-rangulations in the path into two parts: ideally, we want what we shall call the right part , whichshrinks with time, to retain as much memory of the initial quadrangulation q as possible, whilethe left part is a growing, increasingly accurate version of q ′ (see Figure 11 for the decomposition).Because, however, our canonical split requires an external face to act as a “separator” betweenthe left and right parts, it is not possible – or at least it is not convenient – to grow the completequadrangulation q ′ on the left, since we we have space for a quadrangulation of size at most n −
1. That is why we select a mapping F : Q n → Q n − (with certain properties) and construct therandom path from q to q ′ as • a random path from q to a quadrangulation whose left part is F ( q , q ′ ) and whose right part isempty, distributed according to a probability which will later be called P F ( q , q ′ ) q ; • a random path from the final quadrangulation of the path above to q ′ , whose reverse path isdistributed according to the probability P F ( q , q ′ ) q ′ .Our objective will be to describe a random flip path distributed according to the probabilitymeasure P F ( q , q ′ ) q ; this will consist of n concatenated flip subpaths, of which • the first is special: it collapses one appropriately chosen random face of q and establishesa “separating face” to the right of the root edge; at the end of this flip sequence, the facedirectly to the right of the root edge separates an empty left quadrangulation L from a rightquadrangulation R of size n − • the ( i + i = , . . . , n −
1) turns a quadrangulation with left part L i − and right part R i − into a quadrangulation with right part R i = coll( R i − , c ) for some c , andleft part L i , where L i has an additional face with respect to L i − (in the strong sense that L i − = coll( L i , c ′ ) for some corner c ′ of L i ). Given ( L i − , R i − ), the quadrangulations L i , R i arerandom, distributed in a way that is based on the growth algorithm from Section 5. Thesequence of flips constituting this subpath will be later denoted by P (( L i − , R i − ) , ( L i , R i )), anditself consists of three distinct phases: – right phase : the face of R i − containing c is replaced, via a local sequence of flips, bya degenerate face, which is then moved within R i − until it becomes adjacent to the“separating face”; – central phase : this is a very short sequence of just 4 edge flips which move the extradegenerate face from one side of the “separating face” to the other, making it now partof the left portion of the quadrangulation; R L · R Figure 11: From two quadrangulations L ∈ Q and R ∈ Q to a quadrangulation L · R ∈ Q (whose root is the red one on the right, while the marked blue oriented edge is forgotten).Notice that it is possible to recover L and R from L · R , by splitting the two cycles whichform the boundary of the face containing the root corner, erasing the dashed edges androoting appropriately. – left phase : the extra degenerate face is moved to the appropriate location in L i − andthen possibly replaced by a non-degenerate face via local flips in order to create the leftquadrangulation L i .In conclusion, the full canonical path from q to q ′ will consist of • a flip sequence modifying q to have a separating face, with a quadrangulation R of size n − L on the left; • for each i = , . . . , n −
1, a right phase, central phase and left phase, after which a face hasmoved from R i − into L i − , thus yielding left and right parts L i , R i , where | R i | = n − i − = n − − | L i | , on either side of a separating face. At the end of this whole process R n − is emptyand L n − is F ( q , q ′ ); • n − n − • a final sequence which “dismantles” the separating face and moves it to the appropriatelocation to yield q ′ .The next subsection will formalise the idea of a “separating face” and give the description of ourcanonical left-right decomposition, as well as the law of the sequence ( L i , R i ) n − i = ∈ Q n − i = Q i × Q n − − i as a function of the pair ( q , ˜ q ) ∈ Q n × Q n − .Section 6.2 describes the flip paths used to “collapse” a face by turning it into a degenerate faceand those that move a degenerate face from one location to another within a quadrangulation.Section 6.3 finally explains how to build subpaths of the form P (( L i − , R i − ) , ( L i , R i )) (which willturn out to be deterministic given L i − , L i , R i − , R i ) by assembling flip sequences from Section 6.2into a right phase, central phase and left phase, and establishes our desired estimates. .1 Basic structure of canonical paths In order to describe the general structure of our canonical paths, it will be useful to introducecertain “surgical operations” that will enable us to assemble multiple quadrangulations into asingle larger one. Given two quadrangulations L ∈ Q l and R ∈ Q r , where l , r ≥
1, we shall write L · R for the quadrangulation in Q l + r + obtained as follows (Figure 11): first “double” the rootedges of L and R by attaching a degree two face directly to their right; for convenience, draw thisdegree two face as the infinite face in the plane, so that L and R are each “enclosed” within a cycleof length 2 containing the root edge; now draw both quadrangulation in the plane, identifyingtheir origins, in such a way that both root edges are oriented clockwise (with respect to the infiniteface); finally, forget the rooting of L to obtain L · R . It will be convenient to also consider the casewhere l = r = Q = {→} ); we will set → · q , for any q with | q | ≥
1, to be the quadrangulation obtained by adding a degenerate face directly to the rightof the root edge of q (equivalently, the operation described above is performed without actuallydoubling the root edge of → ). The quadrangulation q · → is → · q , rerooted in the edge within theadded degenerate face, so as not to change the origin.We shall write Q l · Q r for the subset { L · R | ( L , R ) ∈ Q l × Q r } of Q l + r + . Notice that, given q ∈ Q l · Q r such that q = L · R , one can quite simply reconstruct L and R , since the rooting of L , which is theonly information not trivially encoded, can still be recovered by following the contour of the facecontaining the root corner of q .As previously described, the idea behind our canonical paths will be to “destroy” the startingquadrangulation q on the right while “growing” a new quadrangulation on the left.Before dealing with the general case, we shall focus on the case where the “final” quadrangu-lation q ′ is of the form ˜ q · → for some ˜ q ∈ Q n − . Furthermore, we shall not yet build the full randomcanonical path from q to ˜ q · → , but a random sequence of quadrangulations of the form ( L i · R i ) n − i = ,taking values in Q n − i = ( Q i × Q n − i − ), that our random canonical path will “go through”. Given thissequence, the path will actually be deterministic, as detailed within Sections 6.2 and 6.3.Given q ∈ Q n and q ′ · →∈ Q n , consider the probability distribution P q ′ q on the set Q n − i = Q i · Q n − i − defined as follows. Given ( L i · R i ) n − i = ∈ Q n − i = Q i · Q n − i − , set P q ′ q (( L i · R i ) n − i = ) = g n ( q , R )1 L n − = q ′ n − Y i = g n − i − ( R i , R i + ) g i + ( L i + , L i ) . It should be clear that P q ′ q is a probability distribution: a random sequence ( λ i · ρ i ) n − i = dis-tributed according to P q ′ q is simply built in such a way that λ n − , λ n − , . . . , λ and q , ρ , . . . , ρ n − areindependent sequences of random quadrangulations, started at q ′ and q respectively, built so asto collapse one random face according to the probability distribution given by g i ( − , · ) at each step.The key feature of the probability distribution P q ′ q which we will use to complete the necessaryestimates on the congestion given by our random canonical paths is expressed in the followinglemma: Lemma 6.1.
Given positive integers n , a < n − , b < n and quadrangulations l ∈ Q a , r ∈ Q b , we have X q ∈ Q n , q ′ ∈ Q n − P q ′ q (cid:16) { ( L i , R i ) n − i = | L a = l , R n − b − = r } (cid:17) ≤ n − b − a − . roof. The expression in the statement can be rewritten as X q ∈ Q n X ( R i ) n − b − i = ∈ Q n − b − i = Q n − i − R n − b − = r ( R i ) n − i = n − b ∈ Q n − i = n − b Q n − i − g n ( q , R ) n − Y i = g n − i − ( R i , R i + ) X q ′ ∈ Q n − X ( L i ) a − i = ∈ Q a − i = Q i L a = l ( L i ) n − i = a + ∈ Q n − i = a + Q i L n − = q ′ n − Y i = g i + ( L i + , L i ) . Let us give an upper bound for the second factor above: the computations involved in boundingthe first factor will be entirely similar.By appropriately exchanging sums and products, we can rewrite it as X ( L i ) n − i = a + ∈ Q n − i = a + Q i g a + ( L a + , l ) n − Y i = a + g i + ( L i + , L i ) X ( L i ) a − i = ∈ Q a − i = Q i g a ( l , L a − ) a − Y i = g i + ( L i + , L i );the entire internal sum is equal to 1 by property (ii) of the mappings g , . . . , g a ; the external sumcan thus be evaluated by using property (iii) of the mappings g a + , . . . , g n − (and by summing over L n − , L n − , . . . , L a + separately, in turn). We obtain that the above is n − Y i = a | Q i + || Q i | ≤ n − a − , where we have used the simple fact that, for all i ≥ | Q i + | = i + Cat( i + ≥ · · i Cat( i ) = | Q i | .As for the first factor above, a similar argument yields that it is equal to Q n − i = b | Q i + || Q i | , and thereforebounded above by 12 n − b , which concludes the proof of the lemma. (cid:3) Consider now the general case of a pair of quadrangulations q , q ∈ Q n : we are almostready to construct our probability measure P q → q on the set of all possible paths Γ q → q . Thiswill require three fundamental ingredients: one is the family of probability spaces ( Γ q ′ q , P q ′ q ) (for q ∈ Q n , q ′ ∈ Q n − ) we just built and discussed; one is a mapping F : Q n → Q n − , which we will useto assign to the pair q , q the probability space (cid:16) Γ F ( q , q ) q × Γ F ( q , q ) q , P F ( q , q ) q ⊗ P F ( q , q ) q (cid:17) ; the last one isa mapping Ψ q , q : Γ F ( q , q ) q × Γ F ( q , q ) q → Γ q → q , which will enable us to simply define P q → q as thepush-forward via Ψ q , q of the probability measure P F ( q , q ) q ⊗ P F ( q , q ) q .For the mapping F , we may choose any which satisfies the condition that, given q ∈ Q n and q ′ ∈ Q n − , we have (cid:12)(cid:12)(cid:12)(cid:8) ˜ q ∈ Q n | F ( q , ˜ q ) = q ′ (cid:9)(cid:12)(cid:12)(cid:12) ≤
12, and similarly (cid:12)(cid:12)(cid:12)(cid:8) ˜ q ∈ Q n | F ( ˜ q , q ) = q ′ (cid:9)(cid:12)(cid:12)(cid:12) ≤
12. The factthat such a mapping exists is an immediate consequence of the fact that | Q n | ≤ | Q n − | : we shallfrom here on use F : Q n → Q n − under the assumption that we have chosen one such mapping.The next section will be devoted to the construction of a mapping Ψ q , q : Γ F ( q , q ) q × Γ F ( q , q ) q → Γ q → q , which will consist in essentially “interpolating” sequences ( L i · R i ) n − i = ∈ Γ F ( q , q ) q and ( L i · R i ) n − i = ∈ Γ F ( q , q ) q by filling in the “gap” between successive quadrangulations via sequences of edgeflips and making sure to run the complete flip sequence constructed from ( L i · R i ) n − i = forward,then the one constructed from ( L i · R i ) n − i = ∈ Γ F ( q , q ) q backwards. This needs to be done with somecare: in particular, our aim is to be able to give an upper bound for the quantity X q , q ∈ Q n P q → q ( { γ ∈ Γ q → q containing ( q , e , s ) } ) q ∈ Q n , drawn in the plane so that the infiniteface lies directly to the right of the root edge, with a marked corner c within a face f . To theright, the quadrangulation → · coll( q , c ): the face f is “collapsed” and a degenerate face isadded directly to the right of the root edge. If ρ is the root corner of the quadrangulation q ′ drawn on the right, coll( q ′ , ρ ) is coll( q , c ). independent of the flip ( q , e , s ) by invoking Lemma 6.1. Indeed, we wish to build Ψ q , q in sucha way that knowing a flip ( q , e , s ) appears in a path Ψ q , q (( L i · R i ) n − i = , ( L i · R i ) n − i = ) gives as muchinformation as possible about the actual quadrangulations L ji , R ji . q to → · coll( q , c ) We now begin the task of constructing our mappings Ψ q , q , for q , q ∈ Q n . In order to do this,given ( L i · R i ) n − i = ∈ Γ q ′ q we wish to construct flip sequences leading from the quadrangulation L i · R i to the quadrangulation L i + · R i + , plus a flip path from q to L · R . Notice that with probability 1(according to P q ′ q ) the quadrangulation R i + di ff ers from R i by collapsing a face; the same is truefor L i and L i + and for R and q . We may therefore assume this is the case when constructing Ψ q , q .First of all, we shall construct the very first part of the flip path, which will transform aquadrangulation q into L · R , where | L | = L = → ) and R is of the form coll( q , c ) forsome corner c of q . Once this construction is made, all others will be rather straightforwardgeneralisations of it.Hence our objective is this: given a quadrangulation q ∈ Q n and a corner c of q , we shall build aunique canonical path that, through a sequence of edge flips, transforms q into the quadrangulation → · coll( q , c ) (see Figure 12 for a representation of a quadrangulation of the form → · coll( q , c )).We shall say that such a path has two phases: the first phase has the aim of replacing the face f c containing c with a degenerate face in such a way that the appropriate vertices of q are identified;the second phase consists in “moving” the degenerate face so that it ends up lying directly to theright of the root edge. We shall first concern ourselves with the second phase, that is, build acanonical path P ( q , c ) from q to → · coll( q , c ) in the case where c is a corner within a degenerateface; note that the specific case where the internal edge of this face is the root edge of q is a littledi ff erent and will be dealt with separately. Lemma 6.2.
Let c be a corner within a degenerate face f of a quadrangulation q ∈ Q n and suppose the rootedge of q is not the internal edge of f . Define the path P ( q , c ) = ( q i , e i , + ) Ni = recursively as follows (it maybe useful to refer to Figure 13): • set q = q. i w i η i ∅ w i + v i + η i + η i + η i + η i + η i + η i + η i + η i + η i + η i + η i + η i + η i + η i + η i + η i + η i + ∅ v i w i η i ˜ η i ˜ η i ˜ η i ˜ η i ˜ η i ˜ η i ˜ η i ˜ η i ˜ η i ˜ η i ˜ η i ˜ η i ˜ η i ˜ η i ˜ η i ˜ η i ∅ v i + w i + ∅ Figure 13 • Let f = f and η be the internal edge of f ; let f i , for i ≥ , be the face of q i that contains the (possiblyflipped) image of edge η i − in q i , and η i the internal edge of f i (which will automatically be a degenerateface). Let v i be the vertex on the boundary of f i that is an endpoint of η i , let w i be the other vertexon the external boundary of f i and let ˜ η i be the edge immediately after η i in counterclockwise orderaround vertex v i . • If d gr ( v i , ∅ ) > d gr ( w i , ∅ ) (where d gr is the graph distance on the vertex set of q i ), set e i = η i . If, on theother hand, d gr ( v i , ∅ ) < d gr ( w i , ∅ ) , set e i = ˜ η i . • Set q i + = q e i , + i . • Set N to be the first non-negative integer for which q e N , s N N is the quadrangulation → · coll( q , c ) .The path above is well defined, in the sense that f i is always degenerate (so that the construction can beperformed), e i is never the root edge of f and N is a positive integer.Furthermore, we have | P ( q , c ) | = N ≤ n and, for i = , . . . , N, we have coll( q i , c i ) = coll( q , c ) , where c i is the corner corresponding to the edge e i , oriented towards v i .Proof. The fact that f i is degenerate is easily shown by induction. Indeed, flipping η i does notchange the fact that it is an internal edge in a degenerate face. On the other hand, suppose d gr ( v i , ∅ ) < d gr ( w i , ∅ ) and let u i be the endpoint of η i that is di ff erent from v i . Then flipping ˜ η i clockwise does not increase the degree of u i , so that η i remains within a degenerate face in q i + .Now, since η is not the root edge of q , the edge η i (which is the image of η after multiple flipsin the path) cannot at any point be the root edge. On the other hand, if the root edge were ˜ η i andwe had d gr ( v i , ∅ ) < d gr ( w i , ∅ ), hence v i = ∅ , we would actually have q i = → · coll( q , c ).The fact that N is finite can be seen as a consequence of the fact that d gr ( v i , ∅ ) is weaklydecreasing (since it is not increased by the flip of ˜ η i and is decreased when flipping η i ). After wehave v i = ∅ , flipping ˜ η i repeatedly will eventually make f i the face immediately to the right of theroot edge, yielding exactly the quadrangulation → · coll( q , c ).Let us now check the bound on N . Consider a step ( q i , e i , + ) in the path, where e i , η i and e i − , η i − ; the edge e i , which is then ˜ η i , has never been flipped before (i.e. it is not the image in q i of any e j for j < i ). On the other hand, { i ≤ N | e i = η i } ≤ d q gr ( v , ∅ ) ≤ n , hence the bound.Finally, c i is a corner of the degenerate face f i (since f i contains η i and lies directly to the right tothe right of ˜ η i ), and we can show that coll( q i , c i ) = coll( q i + , c i + ). This is obvious if e i = η i ; if e i = ˜ η i , u we e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e v u we e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e v u we e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e v u we e e e e e e e e e e e e e e e e v u we e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e v uuuuuuuuuuuuuuuuu we e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Figure 14 the quadrangulation q i + di ff ers by q i only by the fact that the degenerate face f i is “rotated” ontothe edge after ˜ η i in counterclockwise order around w i , then labelled f i + : collapsing it after theprocedure will still yield coll( q i , c i ). (cid:3) We shall then perform an ad hoc construction in the case where the root edge is the internaledge within the degenerate face of q containing c : Lemma 6.3.
Let q ∈ Q n be a quadrangulation whose root edge ρ is the internal edge within a degenerateface f and let c be a corner within f . Let u be the degree one endpoint of ρ , let v be its other endpoint andlet w be the third vertex adjacent to f .If u is the origin of q, let e , . . . , e deg( w ) be the edges incident to w, in counterclockwise order, indexedin such a way that e and e deg( v ) are the boundary edges of q. Set P ( q , c ) = ( q i , e i , + ) Ni = , where q = q andq i + = q e i , + i for i = , . . . , deg( w ) − = N (Figure 14, above).If u is not the origin of q (hence v is), let e , . . . , e deg( w ) be the edges incident to w, in clockwise order,indexed in such a way that e and e deg( w ) are the boundary edges of q. Set P ( q , c ) = ( q i , e i , − ) Ni = , whereq = q and q i + = q e i , + i for i = , . . . , deg( w ) − = N (Figure 14, below).We then have q e N , s N N = → · coll( q , c ) , where s N = + in the first case and s N = − in the second. Noticethat in any case we have N < n.In the first case, let c i be the corner corresponding to edge e i in q i , oriented away from w; in the second, letc i be the corner corresponding to e i in q i , oriented towards w. In both cases, we have coll( q i , c i ) = coll( q , c ) .Proof. Notice that the root edge ρ does not have w as an endpoint, hence all flips we perform areallowed, and that N ≤ deg( w ) < n .Also remark that the quadrangulation → · coll( q , v ) can be obtained from q by “detaching” theedges e , . . . , e deg( w ) − from w and rerouting them to u , replacing e with an edge joining u to v insuch a way as to create a face containing w (which has now degree 1) directly on the right of theroot edge, and finally replacing e deg w with an edge between w and u in the case where u is theorigin of q .But indeed, this is exactly the e ff ect achieved by the sequence of flips given: when flipping e i we are erasing it in favour of an edge that is a version of e i + rerouted towards u rather than w .The flip of e deg( w ) − creates an edge between u and v enclosing w within a degenerate face, andflipping e deg( w ) in the case where u is the origin ensures that the degree 1 vertex w is a neighbourof the origin (see Figure 14). ∅ e e e e w ∅ e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e w ∅ e e e e e e e e e e e e e e e e e e e w ∅ e e e e e e e e e e e e e e e e e e Figure 15: The quadrangulations q , q , . . . , q deg w in the path P ( q , c ), where c is a corner notbelonging to a degenerate face. Indeed, one can identify u and w in q i by collapsing the face lying directly to the right of theroot and obtain the quadrangulation coll( q , c ); the corner c i is defined in such a way that this isexactly the e ff ect of taking coll( q , c i ). (cid:3) We will now construct a path of flips from q to → · coll( q , c ) in the case where c is a cornerwithin a non-degenerate face. Lemma 6.4.
Let c be a corner within a non-degenerate face f of a quadrangulation q ∈ Q n ; if f hasfour distinct vertices then let c = c , c , c , c be a clockwise contour of f , and let v , v , v , v be thecorresponding vertices. If f has three distinct vertices, then let c , c , c , c be a clockwise contour of f suchthat c and c are adjacent to the same vertex v , and let v , v be the vertices of c , c . If the root edge of qhas v as an endpoint, let w = v ; otherwise, let w = v .Let e , e , . . . , e deg( w ) be the edges adjacent to w, in clockwise order, indexed in such a way that e and e deg( w ) are on the boundary of f . Set P ( q , c ) = ( q i , e i , − ) deg( w ) − i = , where q = q and q i + = q e i , − i fori = , . . . , deg( w ) . Now set P ( q , c ) = P ( q deg( w ) , c ′ ) , where c ′ is any corner of the face containing the edgee deg( w ) in q deg( w ) (which is a degenerate face). Set P ( q , c ) to be the concatenation of P ( q , c ) and P ( q , c ) .Then P ( q , c ) = ( q i , e i , s i ) Ni = is well defined and we have q e N , s N N = → · coll( q , c ) .Moreover, we have N < n and, setting c ′ i to be the corner corresponding to the edge e i in q i , orientedtowards w for i = , . . . , deg w, and oriented towards the vertex v i from the construction of Lemma 6.2 forN ≥ i > deg w, we have coll( q i , c ′ i ) = coll( q , c ) .Proof. First of all, notice that the root edge does not appear in { e , e , . . . , e deg w } so that all of thefirst deg w − w then the root edge would haveboth v and v as endpoints; but this would create a cycle of length 3 in the quadrangulation q ,which is bipartite.We can show inductively that coll( q i , c ′ i ) = coll( q , c ) for i = , . . . , deg w − e , e deg w , η, η ′ be the edges forming the boundary of f in q , named in clockwise order. Thequadrangulation coll( q , c ) is obtained by identifying e with η ′ and e deg w with η , thus collapsing f ; equivalently, it is obtained by first erasing either e or η ′ (i.e. an edge adjacent to c in f ), andidentifying e deg w with η (the edges opposite c ).Consider now the quadrangulation q = q e , − ; the clockwise boundary of the face lying to theleft of the flipped oriented edge e is formed by e , e deg w , η, − e , with e and e being adjacent to c ′ . We thus have that coll( q , c ′ ) can be obtained by first erasing e , then identifying e deg w with η . But, since the map obtained from q by erasing e and the map obtained from q by erasing theflipped e are exactly the same (with all labels assigned to objects in the same way), it follows thatcoll( q , c ′ ) = coll( q , c ). he argument can be repeated to show that, for i ≤ deg w , coll( q i , c ′ i ) = coll( q i − , c ′ i − ) (becausethe two are obtained in the same way from the coinciding maps created by erasing e i − from q i − and q i ).Consider now the quadrangulation q deg w ; in it, the degree of w is 1, and therefore e w is theinternal edge of a degenerate face f ′ (and is not the root edge). Moreover, collapsing f ′ yieldscoll( q , c ). We can thus invoke Lemma 6.2, which tells us that P ( q deg w , c ′ ), where c ′ is a cornerof f ′ , is a flip path of length at most 6 n ending with → · coll( q deg w , c ′ ) = → · coll( q , c ), and thatcoll( q i , c ′ i ) = coll( q deg w , c ′ ) = coll( q , c ) for i > deg w .The estimate for | P ( q , c ) | follows from the fact that deg w < n = | E ( q ) | . (cid:3) Via the three lemmas above, for all pairs ( q , c ), where q ∈ Q n and c is a corner of q , wehave constructed a canonical path P ( q , c ) = ( q i , e i , s i ) Ni = such that q e N , s N N = → · coll( q , c ). The crucialproperty of these canonical paths is highlighted by the corollary below: Corollary 6.5.
Consider any triple ( q , e , s ) , where q ∈ Q n , e is an edge of q other than the root edge and s = ± .Suppose ( q , e , s ) appears in the sequence P ( q ′ , c ′ ) for some q ′ ∈ Q n and some corner c ′ of q ′ . Let c , c be thecorners of q that correspond to the two possible orientations of e; we have coll( q ′ , c ′ ) ∈ { coll( q , c ) , coll( q , c ) } . Ψ q , q Given ( L i · R i ) n − i = ∈ Γ q ′ q , we now wish to build a flip path turning the quadrangulation L i · R i into L i + · R i + . That is, given L ∈ Q a and R ∈ Q n − a − and two corners c and c ′ of L and R respectively,we wish to build a flip path from coll( L , c ) · R to L · coll( R , c ′ ).This we shall do by simply combining multiple constructions from the previous section. In-deed, consider P ( L , c ) = ( q Li , e Li , s Li ) N L i = and P ( R , c ′ ) = ( q Ri , e Ri , s Ri ) N R i = as constructed previously. Thoughthe edge e R is an edge of R , it can be uniquely identified with an edge of coll( L , c ) · R ; inductively,though e Ri is an edge of q Ri , we can see it as an edge of coll( L , c ) · q Ri = coll( L , c ) · ( q Ri − ) e Ri − , s Ri − .We may therefore consider the sequence of flips (coll( L , c ) · q Ri , e Ri , s Ri ) N R i = , which is such that(coll( L , c ) · q RN R ) e RNR , s RNR is equal to coll( L , c ) · ( → · coll( R , c ′ )).Now consider the face f lying directly to the right of the root in coll( L , c ) · ( → · coll( R , c ′ )), let η be the edge immediately after the root edge in the clockwise contour of f and let η ′ be the internaledge of the degenerate face adjacent to η within the “right” quadrangulation → · coll( R , c ′ ). Byalternatively flipping η and η ′ , one can have the degenerate face containing η ′ “slide” along theboundary of f . Consider in particular the sequence of four flips(coll( L , c ) · ( → · coll( R , c ′ )) , η, + )((coll( L , c ) · ( → · coll( R , c ′ ))) η, + , η ′ , + )(((coll( L , c ) · ( → · coll( R , c ′ ))) η, + ) η ′ , + , η, + )((((coll( L , c ) · ( → · coll( R , c ′ ))) η, + ) η ′ , + ) η, + , η ′ , + )as depicted in Figure 16. After the first flip, the degenerate face containing η ′ lies immediatelyto the left of the root edge in the “left quadrangulation” obtained as described in Section 6.1 andshown in Figure 11; the next three flips make it so that the degenerate face lies immediately to the right of the root edge of the “left quadrangulation”, with η ′ adjacent to the origin. The result ofthe four flips is therefore ( → · coll( L , c )) · coll( R , c ′ ).We can thus define the whole path from coll( L , c ) · R to L · coll( R , c ′ ), which we shall denote by P (coll( L , c ) · R , L · coll( R , c ′ )), by a concatenation of the following sequences of flips, which we willrefer to as the “right phase”, the “central phase” (consisting of 4 flips), the “left phase”: ηηηηηηηηηηηηηηηη ηηηηηηηηηηηηηηηηηη ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ ηηηηηηηηηηηηηηηηη η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ ηηηηηηηηηηηηηηηηηη ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ (coll( L , c ) · q Ri , e Ri , s Ri ) N R i = flip η clockwise flip η ′ clockwise ηηηηηηηηηηηηηηηηηη ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ ηηηηηηηηηηηηηηηηη η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ η ′ flip η clockwise flip η ′ clockwise (( q Li · coll( R , c ′ ) , e Li , s Li ) N L i = ) rev Figure 16: The path P (coll( L , c ) · R , L · coll( R , c ′ )); notice how the four flips in the “centralphase” of the path turn the result of the “right phase”, which is coll( L , c ) · ( → · coll( R , c ′ )),into the quadrangulation ( → · coll( L , c )) · coll( R , c ′ ), so that the “left phase” can begin andturn the quadrangulation into the desired L · coll( R , c ′ ). Note that the root edge of thequadrangulation is always the one marked in red appearing in the lower right part of thepicture; the arrow marked in blue represents the root edge of the “left quadrangulation”and is marked to help confirm the fact above. • right phase: (coll( L , c ) · q Ri , e Ri , s Ri ) N R i = • central phase: (coll( L , c ) · ( → · coll( R , c ′ )) , η, + )((coll( L , c ) · ( → · coll( R , c ′ ))) η, + , η ′ , + )(((coll( L , c ) · ( → · coll( R , c ′ ))) η, + ) η ′ , + , η, + )((((coll( L , c ) · ( → · coll( R , c ′ ))) η, + ) η ′ , + ) η, + , η ′ , + ) • left phase: (( q Li · coll( R , c ′ ) , e Li , s Li ) N L i = ) rev , where, given a flip path P = ( q i , e i , s i ) Ni = ∈ Γ q → q eN , sNN , we set P rev to be the flip path( q e N + − i , s N + − i N − i , e N + − i , − s N + − i ) Ni = in Γ q eN , sNN → q .We are now ready to fully describe the mapping Ψ q , q : given q , q ∈ Q n , consider any pair ofsequences (( L i · R i ) n − i = , ( L i · R i ) n − i = ) ∈ Γ F ( q , q ) q × Γ F ( q , q ) q that has nonzero probability according to P F ( q , q ) q ⊗ P F ( q , q ) q . Set Ψ q , q (( L i · R i ) n − i = , ( L i · R i ) n − i = ) to be the successive concatenation of P ( q , c ), where R = coll( q , c ); • P (( L i , R i ) , ( L i + , R i + )) for i = , . . . , n − • P (( L i − , R i − ) , ( L i , R i )) rev for i = n − , n − , . . . , • P ( q , c ′ ) rev , where R = coll( q , c ′ ).We also have all the setup necessary to show the following important estimate: Proposition 6.6.
Consider a quadrangulation q ∈ Q n , an edge e of q other than the root edge and anelement s ∈ { + , −} . We have X q , q ∈ Q n P q → q ( { γ ∈ Γ q → q containing ( q , e , s ) } ) ≤ · n + . Proof.
By our definition of P q → q , the expression we wish to estimate is X q , q ∈ Q n P F ( q , q ) q ⊗ P F ( q , q ) q ( { X ∈ Γ F ( q , q ) q × Γ F ( q , q ) q | Ψ q , q ( X ) contains ( q , e , s ) } ) . We will use as an upper bound the one we obtain by summing the terms corresponding to thefollowing three possibilities: • the flip ( q , e , s ) appears in P ( q , c ) or P ( q , c ′ ) rev , in which case R = coll( q , c ) ∈ { coll( q , x ) , coll( q , x ) } or R = coll( q , c ′ ) ∈ { coll( q e , y ) , coll( q e , y ) } , where x , x are the corners of q that correspondto the two possible orientations of e and y , y are the corners of q e that correspond to thetwo possible orientations of the flipped version of e , by Corollary 6.5. Now, by Lemma 6.1,we have X i = , X q , q P F ( q , q ) q ( R = coll( q , x i )) + P F ( q , q ) q ( R = coll( q e , y i )) == X i = , [ X q , q ′ |{ q | F ( q , q ) = q ′ }| · P q ′ q ( R = coll( q , x i ) , L = → ) + X q , q ′ |{ q | F ( q , q ) = q ′ }| · P q ′ q ( R = coll( q e , y i ) , L = → )] ≤≤ · · n − ( n − − = · n . • The flip ( q , e , s ) appears in P ( L i · R i , L i + · R i + ) for some i ; we shall consider some separatesubcases: – we have q = q L · q R ∈ Q l · Q r and e is the image of an edge other than the root edgein E ( q L ) (so that q e , s also lies in Q l · Q r , and in fact in Q l · q R ). Let c , c be the cornerscorresponding to the two possible orientations of e in q e . If ( q , e , s ) is a flip in the “centralphase” of the path, with e = η or e = η ′ (see Figure 16), then at least one of the corner c , c lies in the degenerate face that is in the process of being moved along the boundaryof the “left” quadrangulation; as a consequence, we have L i ∈ { coll( q eL , c ) , coll( q eL , c ) } ,hence i = l − R i + = q R . If not, then the flip happens in the “left phase” of the pathand Corollary 6.5 implies that L i ∈ { coll( q eL , c ) , coll( q eL , c ) } , hence i = l − R i + = q R .Thus we have the term X i ∈{ , } X q , q P F ( q , q ) q ( L l − = coll( q eL , c i ) , R l = q R ) ≤ · · n − ( l − − ( n − l − − = · n + . we have q = q L · q R ∈ Q l · Q r and e is the image of an edge other than the root edgein E ( q R ). This case is analogous: this time Corollary 6.5 gives L l = q L and R l + ∈{ coll( q R , c ) , coll( q R , c ) } , where c , c are the corners of q corresponding to the two possibleorientations of e . This yields another term of the form X i ∈{ , } X q , q P F ( q , q ) q ( L l = q L , R l + = coll( q R , c i )) ≤ · · n − l − ( n − l − − = · n + . – we have q ∈ Q l · Q r and q e , s ∈ Q l + · Q r − . This is the only case we are missing, i.e. theone where e is the edge right after the root edge of q in the clockwise contour of the facelying directly to the right of the root edge (it can be seen that, by construction, all otherflips in P ( L i · R i , L i + · R i + ) happen within q L or within q R . In this case, if q e , s = q ′ L · q ′ R ,we have L i = q L and R i + = q ′ R , hence i = l ; we get the term X q , q P F ( q , q ) q ( L l = q L , R l + = q ′ R ) ≤ · n − l − ( n − l − − = n + . Globally, this yields a term that can be upper bounded by 2 · n + . • The flip ( q , e , s ) appears in P ( L i · R i , L i + · R i + ) rev for some i ; clearly, this case is entirelyanalogous to the previous one, and will yield another term upper bounded by 2 · n + .Summing the three upper bounds above proves the lemma. (cid:3) All this being done, we can apply the technique of canonical paths of Diaconis and Salo ff -Coste[9] to bound the relaxation time of F n . Proof of Theorem 1.
The fact that ν n = µ n is an obvious consequence of Proposition 4.1. The upperbound for ν n can be proven in exactly the same way as the one in [8]: because the only di ff erencebetween the chains F n and ˜ F n is the fact that the root edge can no longer be flipped and that eachflip is assigned a probability of n − rather than n , the proof of Proposition 4.1 in [8] also appliesto the spectral gap ν n of ˜ F n .As for the lower bound, we have1 ν n ≤ max ( q , e , s ) π ( q ) p ( q , q e , s ) X q , q ∈ Q n X γ ∈ Γ q → q :( q , e , s ) ∈ γ | γ | P q → q ( γ ) π ( q ) π ( q ) , where π is the uniform measure on Q n , ( q , e , s ) varies among all possible flips ( q ∈ Q n , e ∈ E ( q ), s = ± ) and p ( q , q e , s ) is the transition probability according to ˜ F n .Now, all instances of π ( · ) can be replaced by | Q n | . Also, we have p ( q , q e , s ) ≥ n − (hence p ( q , q e , s ) ≤ n ) for all q ∈ Q n , e ∈ E ( q ), s = ± . Moreover, the length of our canonical paths asconstructed is at most 32 n . This can be checked by going through the final construction fromSection 6.3: each path of non-zero weight in Γ q → q is built as two sequences (one “straight” andone “reversed”) of • one path of the form P ( q , c ); • n − P (( L , R ) , ( L ′ , R ′ )). n turn, every path of the form P (( L , R ) , ( L ′ , R ′ )) is built as a concatenation of • one path of the form P ( q , c ); • • one path of the form P ( q , c ), reversed.By the three lemmas in Section 6.2, we know that the length of a path of the form P ( q , c ) is at most8 n , which yields the global upper bound of 32 n .Applying the bound given by Proposition 6.6 we then obtain1 ν n ≤ n · n · · n + | Q n | . Since | Q n | n ≥ Cn / , we have 1 ν n ≤ n · n · · · Cn / ≤ C n / for some appropriate constant C , as desired. (cid:3) References [1] D. A ldous , Triangulating the circle, at random. , Amer. Math. Monthly, 101 (1994).[2] D. A ldous , Mixing time for a markov chain on cladograms , Combinatorics, Probability andComputing, 9 (2000), p. 191–204.[3] T. B udzinski , On the mixing time of the flip walk on triangulations of the sphere , Comptes RendusMathematique, 355 (2017), pp. 464 – 471.[4] S. C annon , D. A. L evin , and A. S tauffer , Polynomial mixing of the edge-flip Markov chain forunbiased dyadic tilings , Combinatorics, Probability and Computing, 28 (2019), pp. 365–387.[5] S. C annon , S. M iracle , and D. R andall , Phase transitions in random dyadic tilings and rectan-gular dissections , SIAM Journal on Discrete Mathematics, 32 (2018), pp. 1966–1992.[6] P. C aputo , F. M artinelli , A. S inclair , and A. S tauffer , Random lattice triangulations: Structureand algorithms , Ann. Appl. Probab., 25 (2015), pp. 1650–1685.[7] ,
Dynamics of lattice triangulations on thin rectangles , Electron. J. Probab., 21 (2016), p. 22.[8] A. C araceni and
A. S tauffer , Polynomial mixing time of edge flips on quadrangulations , Proba-bility Theory and Related Fields, (2019).[9] P. D iaconis and
L. S aloff -C oste , Logarithmic Sobolev inequalities for finite Markov chains , Ann.Appl. Probab., 6 (1996), pp. 695–750.[10] L. M c S hine and P. T etali , On the mixing time of the triangulation walk and other Catalan structures ,in Randomization Methods in Algorithm Design, Proceedings of a DIMACS Workshop,Princeton, New Jersey, USA, December 12-14, 1997, 1997, pp. 147–160.[11] M. M olloy , B. R eed , and W. S teiger , On the mixing rate of the triangulation walk , in Random-ization methods in algorithm design (Princeton, NJ, 1997), vol. 43 of DIMACS Ser. DiscreteMath. Theoret. Comput. Sci., Amer. Math. Soc., Providence, RI, 1999, pp. 179–190.[12] G. S chaeffer , Conjugaison d’arbres et cartes combinatoires aléatoires. PhD thesis , (1998).
13] A. S tauffer , A Lyapunov function for Glauber dynamics on lattice triangulations , ProbabilityTheory and Related Fields, 169 (2017), p. 469–521.[14] W. T. T utte , A census of planar maps , Canad. J. Math., 15 (1963), pp. 249–271., Canad. J. Math., 15 (1963), pp. 249–271.