Chen Linxiao
email: linxiao.chen@math.ethz.ch

March

Enumeration of fully parked trees

come

generic :

non-generic

empty generic + dilute - B (ρ) = ∞ B (ρ) < ∞ B (ρ) < ∞ B (ρ) = ∞ Yc < ρ x (ρ) < 0 x(Y ) Y Yc = ρ x (ρ) = 0 x(Y ) Y Yc = ρ x (ρ) > 0 x(Y ) Y B (ρ) = ∞ or ρ = ∞ or dilute :
non-generic dense :

2 < α < 3 α > 3 α ρ = ∞ or Figure :
Left: the characterizations of the generic, dilute and dense phases. While the graphs seem to suggest that x is concave, this is not necessarily true. We only prove a weaker property: see Lemma .

Right: a phase diagram indicating the relation between generic + , dilute -and the other phases of the model. The conditions at the bottom give the general de ntions of the three columns of the diagram, while the axis at the top gives the corresponding ranges of ã under the assumption ( * ).

Theorem (Coe cient asymptotics). Under Assumption ( * ) and in the generic and the dilute phases, we have

𝐹 𝑝 (𝑥 𝑐 ) ∼ 𝑝→∞ C 𝐹 Γ(-𝛾 0 ) • 𝑌 -𝑝 𝑐 • 𝑝 -𝛾 0 -1 𝜕 𝑥 𝐹 𝑝 (𝑥 𝑐 ) ∼ 𝑝→∞ 𝛼 2𝜇𝑥 𝑐 C 𝐹 Γ(-𝛾 1 ) • 𝑌 -𝑝 𝑐 • 𝑝 -𝛾 1 -1 ( ) 𝐹 𝑛,𝑝 ∼ 𝑛→∞ 𝐺 𝑝 Γ(-𝛽 0 ) • 𝑥 -𝑛 𝑐 • 𝑛 -𝛽 0 -1 𝐺 𝑝 ∼ 𝑝→∞ 𝛼 -1 2 𝜇 𝛽 0 C 𝐹 Γ(-𝛽 1 ) • 𝑌 -𝑝 𝑐 • 𝑝 -𝛽 1 -1 ( )
and when 𝑛 ∼ 𝑣 • 𝑝 1/𝜃 for some 𝑣 ∈ (0, ∞) :

𝐹 𝑛,𝑝 ∼ 𝑛,𝑝→∞ 𝜇 𝐶 𝐹 • 𝐼 𝛼 (𝜇𝑣) • 𝑥 -𝑛 𝑐 𝑌 -𝑝 𝑐 • 𝑝 -(𝛾 0 +1+1/𝜃 ) . ( )
where the exponents 𝛾 0 , 𝛾 1 , 𝛽 0 , 𝛽 1 and 𝜃 are universal (i.e. they only depend on 𝛼), and are given by

𝛾 0 = 𝛼 2 , 𝛾 1 = 1 - 𝛼 2 , 𝛽 0 = 𝛼 𝛼 -1 , 𝛽 1 = - 𝛼 2 and 𝜃 = 1 𝛼 -1 .
( )

The scaling function 𝐼 𝛼 : R >0 → R >0 is also universal. Its expression is given (without proof) in the remark below. .

( )

Remark. In an upcoming paper, we will explain how to carry out singularity analysis and to compute 𝐼 𝛼 for a fairly large class of bivariate generating functions. By applying this method to 𝐹 (𝑥, 𝑦), we obtain that

𝐼 𝛼 (𝜆) = ∞ ∑ 𝑛=0 𝑐 𝑛 Γ(𝜎 𝑛 -𝛾 0 )Γ(-𝜃𝜎 𝑛 ) 𝜆 -𝜃𝜎 𝑛 -1 ( )
where the constants 𝜎 𝑛 , 𝑐 𝑛 ∈ R are determined by

∞ 𝑛=0
𝑐 𝑛 𝑥 𝜎 𝑛 = √ 1 -𝛼𝑥 𝛼-1 + (𝛼 -1)𝑥 𝛼 as 𝑥 → 0. Or, explicitly

𝐼 𝛼 (𝜆) = ∞ ∑ 𝑝=0 ∞ ∑ 𝑞=1 (-1) 𝑞+1 2 √ 𝜋 Γ 𝑝 + 𝑞 -1 2 Γ (𝛼 -1)𝑝 + 𝛼 (𝑞 -1 2 ) Γ -𝑝 -𝛼 𝛼-1 𝑞 𝛼 𝑝 (𝛼 -1) 𝑞 𝑝! 𝑞! 𝜆 -(𝑝+ 𝛼 𝛼 -1 𝑞)-1 . ( )
Assuming the increasing order 𝜎 0 < 𝜎 1 < • • • , it is an elementary exercise to show that lim sup 𝑛→∞ |𝑐 𝑛 | 1 𝜎𝑛 < ∞. On the other hand, by Euler's re exion formula, we have 1 Γ (-𝜃𝜎 𝑛 ) = sin(𝜋𝜃𝜎 𝑛 ) 𝜋 Γ(𝜃𝜎 𝑛 + 1). It follows that lim sup

𝑛→∞ 𝑐 𝑛 Γ(𝜎 -𝛾 0 )Γ(-𝜃𝜎 𝑛 ) 1 𝜎𝑛 ≤ lim sup 𝑛→∞ |𝑐 𝑛 | 1 𝜎𝑛 • lim sup 𝑛→∞ Γ(𝜃𝜎 𝑛 + 1) Γ(𝜎 𝑛 -𝛾 0 ) 1 𝜎𝑛 ( ) Since 𝜃 = 1
𝛼-1 < 1, Stirling's formula implies that the second limsup on the right hand side is equal to zero. By a harmless generalization of the root test, we see that the series of power functions which de nes 𝐼 𝛼 is absolutely convergent for all 𝜆 ∈ C \ {0}.

The various asymptotic formulas in Theorem are related to each other by a number of heuristic scaling relations. For instance, by plugging the second asymptotics of ( ) into the rst one, we see that

𝐹 𝑛,𝑝 ∼ 𝑛,𝑝→∞ 𝛼 -1 2 𝜇 C 𝐹 Γ(-𝛽 0 )Γ(-𝛽 1 ) • 𝑥 -𝑛 𝑐 𝑌 -𝑝 𝑐 • (𝜇𝑛) -𝛽 0 -1 𝑝 -𝛽 1 -1 ( )
when 𝑛 tends to ∞ su ciently fast compared to 𝑝. On the other hand, the asymptotics ( ) can be rewritten as

𝐹 𝑛,𝑝 ∼ 𝑛,𝑝→∞
𝜇 𝐶 𝐹 • (𝜇𝑣) (𝛾 0 -𝛽 1 )𝜃 +1 𝐼 𝛼 (𝜇𝑣) • 𝑥 -𝑛 𝑐 𝑌

-𝑝 𝑐 • (𝜇𝑛) -(𝛾 0 -𝛽 1 )𝜃 -1 𝑝 -𝛽 1 -1 ( )

when 𝑛 ∼ 𝑣 •𝑝 1/𝜃 for 𝑣 ∈ (0, ∞). Although the two asymptotics are valid for di erent regimes of the limit 𝑛, 𝑝 → ∞, they sugguest heuristically the scaling relations 𝛽 0 = (𝛾 0 -𝛽 1 )𝜃 and lim 𝜆→∞ 𝜆 𝛽 0 +1 𝐼 𝛼 (𝜆) =

𝛼-1 2Γ (-𝛽 0 )Γ (-𝛽 1 ) . Both relations can be veri ed using the explicit expression of the exponents and of 𝐼 𝛼 . Another heuristic scaling relation is 𝛾 1 = 𝛾 0 -1 𝜃 . It is a bit harder to explain, and will be discussed in the upcoming paper containing the derivation of the expression of 𝐼 𝛼 .

One last result that we would like to mention here is a variational method for nding equations which constrain the dominant singularities of the inverse of an analytic function. It is used in the proof of Theorem , but applies in a general setting. We explain this method in detail in Appendix A.

. Discussions and corollaries

About on the technical assumption ( * ). Since we assumed 𝑏 0 > 0, the series 𝐵(𝑦) is periodic if and only if supp 𝐵 ⊆ 𝑚Z for some 𝑚 ≥ 2. A simple rewriting of the de nition of the surplus gives that ∑ 𝑣 ∈𝑉 (𝔱) ℓ (𝑣) = |𝑉 (𝔱)| + s(𝔱, ℓ) .

( ) So if supp 𝐵 ⊆ 𝑚Z, then all fully packed tree (𝔱, ℓ) such that |𝑉 (𝔱)| + s(𝔱, ℓ) ∉ 𝑚Z would have zero weight. This would cause complications in the asymptotic analysis of the coe cients of 𝐹 (𝑥, 𝑦), which we prefer to avoid. Both the aperiodicity of 𝐵(𝑦) and the assumption |supp 𝐵| = ∞ are only used in Section to prove the uniqueness of dominant singularity of the series Ŷ = x-1 . The above discussion shows that the aperiodicity is necessary for that conclusion to be true. On the other hand, we do not believe that the condition |supp 𝐵| = ∞ is necessary. But currently we do not have a proof that bypasses it. The assumptions in the non-generic phase contain two parts: First, we assume that 𝐵 is Δ-analytic and that the expansions ( ) hold in 𝜌 . This is necessary for having the corresponding Δ-analyticity and asymptotic expansion in 𝜌 of the function 𝑦 ↦ → 𝐹 (𝑥, 𝑦). Our asymptotic analysis of 𝐹 𝑛,𝑝 relies heavily on these ingredients. Second, we assume that the dominant singular term 𝐵 s (𝑦) in the asymptotic expansion of 𝐵 is a power function. While our method is applicable to more general 𝐵 s (𝑦) (e.g. power function with logarithmic corrections), allowing such terms would greatly complicate the singularity analysis of 𝐹 (𝑥, 𝑦) with little bene t. So we choose not to do so. Finally, remark that the assumption α > 2 is not restrictive, since according to Proposition , we must have 𝐵 (𝜌) < ∞ in the non-generic phase. That is, the weight sequences b and b de ne the same family of probability measures up to a change of indices. We say that they are equivalent. Alternatively, two weight sequences are equivalent if and only if B(𝑦) = 𝜆𝐵(𝑟𝑦) for some 𝜆, 𝑟 > 0. It is not hard to see that equivalent weight sequences always belong to the same phase.

Probabilistic characterization of the phases. Under the assumptions 𝜌 > 0 and 𝑏 0 > 0, every weight sequence has an equivalent in exactly one of the three categories: (𝜌 = ∞ and 𝐵(1) = 1), (𝜌 = 1 and 𝐵(1) = ∞), or 𝜌 = 1 = 𝐵(1). Proposition ensures that the weight sequences in the rst two categories are always in the generic phase. On the other hand, each weight sequence satisfying 𝜌 = 1 = 𝐵(1) de nes a probability measure on {0, 1, 2, • • • } with sub-exponential tail (in the sense that 𝑏 𝑛 = 𝑜 (𝑟 𝑛 ) and 𝑏 𝑛 ≠ 𝑂 (𝑟 -𝑛 ) as 𝑛 → ∞ for all 𝑟 > 1). Moreover, this probability distribution has a nite second moment if and only if 𝐵 (1) < ∞, and when this is the case, the condition x (1) < 0 simplies to 2(𝐵 (1) + 𝐵 (1) -𝐵 (1) 2 ) + 𝐵 (1) 2 > 1. This gives the following probabilistic reformulation of Proposition .

Corollary . Assume that b ≡ (𝑏 𝑙 ) 𝑙 ≥0 is a probability distribution on {0, 1, 2, • • • } with a sub-exponential tail. If b has in nite second moment, the it is in the generic phase. Otherwise, it is in the generic (resp. dilute, dense) phase if and only if 2𝜎 2 + 𝑚 2 > 1 (resp. = 1, < 1), where 𝑚 and 𝜎 2 are the mean and the variance of b.

Using Corollary , one can easily construct examples of weight sequences in each of the three phases, or a continuous family of weight sequences that passes through the generic, dilute and dense phases consecutively. We leave the reader to verify the following particular construction.

Corollary . Let b (0) and b (1) be two probability distributions on {0, 1, 2, • • • } with sub-exponential tails such that b (0) is generic and b (1) is dense. Then there exists 𝑝 𝑐 ∈ (0, 1), such that the weight sequence b (𝑝) := (1-𝑝)b (0) +𝑝b (1) is generic, dilute and dense when 𝑝 ∈ [0, 𝑝 𝑐 ), 𝑝 = 𝑝 𝑐 and 𝑝 ∈ (𝑝 𝑐 , 1], respectively.

If we drop the condition of sub-exponential tail (that is, 𝜌 = 1) in Corollary , then the sign of 2𝜎 2 + 𝑚 2 -1 and the phase of b no longer determine each other. However, not all combinations of these two properties are possible: For any b representing a probability distribution, we have 𝜌 ≥ 1. By de nition, x (𝑌 ) > 0 for all 𝑌 ∈ [0, 𝜌) in the non-generic phase, which implies that either x (1) > 0, or 𝜌 = 1 and x (0) = 0 and b is dilute. After simple rearrangements, the previous statement is equivalent to: for any probability distribution b on N :

• If 2𝜎 2 + 𝑚 2 < 1, that is, x (1) > 0, then b can be in the generic, dilute or dense phase.

• If 2𝜎 2 + 𝑚 2 = 1, that is, x (1) = 0, then b can be in the generic or the dilute phase.

• If 2𝜎 2 + 𝑚 2 > 1 (or if 𝐵 (1) = ∞), then b can only be in the generic phase.

It is also worth noting that while all the other ve cases can be realized by a probability distribution with exponential tail (that is, 𝜌 > 1), we must have 𝜌 = 1 to realize the case where 2𝜎 2 + 𝑚 2 = 1 and b is dilute. We will explain in the next subsection the signi cance of the above observations in the context of the phase transition of parking processes on trees.

.

Motivation and background

Fully packed trees as fully parked trees. This work is motivated by the following interpretation of labeled trees as the initial con guration of a parking process on trees: Given a labeled tree (𝔱, ℓ), we view each vertex 𝑣 ∈ 𝑉 (𝔱) as a parking spot that can accommodate at most one car. The label ℓ (𝑣) represents the number of cars that arrive at the vertex 𝑣. The parking process assumes that each car attempts to park at its vertex of arrival, and if that vertex is occupied, travels towards the root until it nds an unoccupied vertex. If all vertices on its way are occupied, then the car exits the tree through the root. The nal con guration of the parking process is encoded by the function 𝜒 : 𝑉 (𝑡) → N, where 𝜒 (𝑣) is the total number of cars that visited the vertex 𝑣 (either parking there, or passing by) after all the cars have either parked or exited the tree. An important observation is that 𝜒 does not depend on the order in which one chooses to park the cars. Indeed, one can check that 𝜒 satis es the recursion relation

𝜒 (𝑣) = ℓ (𝑣) + ∑ 𝑢 ∈ℭ 𝑣 (𝜒 (𝑢) -1) + ( )
where ℭ 𝑣 denotes the set of children of the vertex 𝑣, and (𝑥) + ≡ max(𝑥, 0) is the positive part of a real number 𝑥.

Since the tree is nite, the above recursion relation completely determines 𝜒. Another presentation of the nal con guration consists of recording whether each vertex 𝑣 is occupied at the end of the parking process, and the ux 𝜑 (𝑣) of cars that went from 𝑣 to its parent vertex during the process. The relation between the two presentations is simple: a vertex 𝑣 is occupied at the end if and only if 𝜒 (𝑣) ≥ 1, and we have 𝜑 (𝑣) = (𝜒 (𝑣) -1) + .

The ux of cars 𝜑 (∅) going out from the root vertex ∅ is called the over ow of the parking process.

With a bit of thought, it is not hard to see that a labeled tree is fully packed if and only if the corresponding parking con guration 𝜒 is fully parked, that is, every vertex is occupied, or equivalently, 𝜒 (𝑣) ≥ 1 for all 𝑣. In this case, the parking con guration 𝜒 and the ux 𝜑 are related to the surplus by 𝜒 (𝑣) -1 = 𝜑 (𝑣) = s(𝔱 𝑣 , ℓ), where 𝔱 𝑣 is the subtree rooted at 𝑣.

Previous works on the parking process on trees. The parking problem was rst introduced by Konheim and Weiss [ ] to model the linear probing scheme of hash collision resolution in computer science. In their model, the parking process takes place on a directed linear graph (i.e. a rooted tree with a single branch). Parking processes on non-degenerate trees was introduced more recently by Lackner and Panholzer [ ], who enumerated the parking functions on Cayley trees of size 𝑛. In our terminology, a parking function is an initial con guration of the parking process which produces no over ow at the root, and in which the cars are labeled from 1 to 𝑚. It is represented by a function from {1, . . . , 𝑚} to the vertex set of the tree, thus the name. Using analytic combiantorics methods, it was shown in [ ] that when 𝑚 = 𝛼𝑛 labeled cars arrive independently at uniformly chosen vertices of a random Cayley tree with 𝑛 vertices, the probability that there is no macroscopic over ow at the root undergoes a continuous phase transition. More precisely, as 𝑛 → ∞, this probability converges to a continuous limit 𝑝 (𝛼) which is positive if 𝛼 < 𝛼 𝑐 , and zero if 𝛼 ≥ 𝛼 𝑐 , for some 𝛼 𝑐 ∈ (0, 1). This result was later generalized to many other classes of trees and to variants of the parking functions [ ].

A probabilistic explanation of the phase transition in [ ] was given by Goldschmidt and Przykucki [ ] using the objective method [ ]. Their key observation is that the parking process of [ ] has a nice local limit in distribution, and that the probability of having no macroscopic over ow at the root is continuous with respect to this limit. More precisely, the limit parking process lives on a Kesten's tree (i.e. critical Galton-Watson tree conditioned to survive forever, see [ ]), and an i.i.d. number of cars arrives at each vertex of this tree. Chen and Goldschmidt [ ] later used the same idea to study the parking process on uniform random rooted plane trees, which also gives rise to a limit process on a Kesten's tree, but with a di erent o spring distribution. This motivates the study of parking processes with i.i.d. car arrivals on general critical Galton-Watson trees. Interestingly, the same type of parking processes was also proposed and studied independently as a good model of rainfall runo from hillsides, where the aforementioned phase transition is of practical importance. See Jones [ ] and the references therein.

In both [ ] and [ ], the derivation of the phase transition relies on computing explicitly the probability of macroscopic over ow in the limit model. The o spring distributions of critical Galton-Watson trees involved are Poissonian and geometric respectively, while the car arrival distribution is Poissonian in both cases. Using a more exible argument involving the spinal decomposition of Galton-Watson trees [ , Chapter . ], Curien and Hénard [ ] generalized these phase transition results to parking processes on critical Galton-Watson trees of any o spring distribution 𝜈 and with any car arrival distribution 𝜇. They also found a simple algebraic characterization the phase transition involving only the rst and second moments of 𝜈 and 𝜇. The results in [ ] are later further generalized by Contat [ ] to the case where the car arrival distribution may depend on the degree of the vertex, and re ned with some large deviation estimates for the sharpness of the phase transition. There has also been a recent work [ ] focusing on parking processes with i.i.d. car arrivals on a supercritical Galton-Watson tree, which makes an interesting link with the Derrida-Retaux model [ ].

Phases of the unconditioned parking process. The fully parked trees in this paper are derived from a special case of the parking model of [ ] described in the previous paragraph. More precisely, if we choose the geometric o spring distribution 𝜈 𝑘 = 2 -𝑘-1 and let 𝜇 𝑘 = 𝑏 𝑘 for the law of car arrivals, then the parking process of [ ], conditioned on the event that its nal con guration is fully parked, follows the law P 𝑥,𝑦 b de ned in ( ) with (𝑥, 𝑦) = (1/4, 1). (Notice that when 𝑥 < 1/4, the model of fully parked trees in this paper is derived from a parking process with i.i.d. car arrivals on a subcritical Galton-Watson tree with geometric o spring distribution. But we shall not pursue this link further here.)

In [ ], the parking process is called supercritical if the over ow of cars at the root of a Galton-Watson tree conditioned to have 𝑛 vertices (without the conditioning of being fully parked) scales linearly with 𝑛 as 𝑛 → ∞. The process is called critical if the over ow scales sublinearly but is unbounded, and subcritical if it is bounded. (Other descriptions of the phase transition are also available in [ ].) In our special case of geometric o spring distribution, the characterization of phase transition in [ ] simpli es to:

Theorem A ([ ]). The model is subcritical (resp. critical, supercritical) if and only if 2𝜎 2 + 𝑚 2 < 1 (resp. = 1, > 1),
where 𝑚 and 𝜎 2 denote the mean and the variance of the car arrival distribution.

Comparing this result to Corollary , we see that when the law b of car arrivals has a subexponential tail, the subcritical, critical, and supercritical phases described in [ ] correspond precisely to the generic, dilute, and dense phases de ned in this paper. When b does not have a subexponential tail, the possible combinations of the two notions of phase are given in the discussion after Corollary .

Relation to upcoming works, and the reason to skip the dense phase. Of course, the behaviors of the parking process decribed in [ ] do not directly apply to the model conditioned to be fully parked. Instead, fully parked trees appear as geometric building blocks of the nal con guration of an unconditioned parking process on Galton-Watson trees. More precisely, the clusters of occupied vertices in a such con guration are distributed according to the law of a fully parked tree with no over ow (i.e. P 𝑥,0 b for some 𝑥 > 0, the cluster of the root requires some special treatment since it may have a nonzero over ow). The full con guration can then be generated as a multitype Galton-Watson tree whose vertices represent either an unoccupied vertex or a fully parked cluster of the original parking process. This decomposition will be used in an upcoming work to study the scaling limit of the parking process on Galton-Watson trees. This paper provides the necessary asymptotic enumeration results in order to understand the law of the fully parked clusters in its nal con guration. As explained in the concluding remarks of [ ], this scaling limit of the parking process is most interesting when the car arrival distribution 𝜈 = b is critical. According to the discussion below Corollary , the fully parked trees can only be in the generic or the dilute phases in this case. This explains why we decide to skip the dense phase at rst approach: while interesting from a combinatorial point of view, the asymptotic enumeration of fully parked trees in the dense phase is not relevant for the study of critical parking processes.

Outline of sections. Section derives the parametrization of 𝐹 (𝑥, 𝑦) in Proposition from its combinatorial de nition. Section proves the characterization of the generic, dilute and dense phase given in Proposition . Section gathers some useful algebraic properties (Lemma ) of the parametrization of 𝐹 (𝑥, 𝑦). Based on these properties, Section derives the asymptotic expansions (Proposition ) of 𝐹 (𝑥, 𝑦), which is then used in Section to prove the coe cient asymptotics in Theorem . The proof assumes that 𝐹 (𝑥, 𝑦) has a double Δ-analyticity property (Proposition ). This assumption is veri ed in Section , with the proof of a technical lemma (Lemma ) being deferred to Section . Finally, Appendices A and B contains some analysis results that are used in the proofs of Lemmas and . As mentioned before, Appendix A provides a variational method for nding equations which constrain the dominant singularities of the inverse of an analytic function, which is considered another main result of this paper. Appendix B provides modi ed versions of the (analytic) inverse function theorem and implicit function theorem, in situations where the conditions of the classical versions break down. problem on random trees to me and for sharing the progress of their recent and ongoing works. Their insight into the probabilistic properties of the model greatly helped the formulation of this work. I am also grateful to Mireille Bousquet-Mélou for explaining to me how the generalized kernel method can be applied to this problem, and for many other discussions. The author was a liated to the Univerity of Helsinki during the initial stage of this work, and would like to thank the hospitality of his colleagues there. This work has been supported by the ERC Advanced Grant (QFPROBA) and Swiss National Science Foundation (SNF) Grant .

Parametrization of the generating function 𝐹 (𝑥, 𝑦)

In this section, we rst derive the following functional equation on 𝐹 (𝑥, 𝑦) from the recursive decomposition of labeled trees:

𝐹 (𝑥, 𝑦) = 𝑥 𝑦 𝐵(𝑦) 1 -𝐹 (𝑥, 𝑦) - 𝑏 0 1 -𝐹 0 (𝑥) ( )
where 𝐹 0 (𝑥) = 𝐹 (𝑥, 0). Then, we solve the above equation using the generalized kernel method explained in [ ] in order to deduce the parametrization of 𝐹 (𝑥, 𝑦) in Proposition .

Derivation of ( ). Recall that FT is the set of all fully packed trees, i.e., labeled (rooted plane) trees (𝔱, ℓ) such that s(𝔱 , ℓ) ≥ 0 for all 𝔱 ⊆ 𝔱, where s(𝔱 , ℓ) := 𝑣 ∈𝑉 (𝔱 ) (ℓ (𝑣) -1) is the surplus of a subtree 𝔱 . To expand the generating function of FT using the recursive decomposition of trees, let us consider the slightly larger class

FT † := {(𝔱, ℓ) | s(𝔱 , ℓ) ≥ 0 for all proper subtree 𝔱 𝔱 } . ( )
It is clear that a labeled tree belongs to FT † if and only if the subtrees rooted at the children of the root vertex are all fully packed. Therefore

FT † ({∅} × N) × SEQ(FT) ( )
where {∅} × N represents the root vertex ∅ with its integer label ℓ (∅), and SEQ(FT) is the class of ( nite) sequences of fully packed trees. The sign denotes an equivalence of combinatorial classes, that is, there is a bijection between the two sides that preserves the vertex count (𝔱, ℓ) ↦ → |𝑉 (𝔱)|, the surplus (𝔱, ℓ) ↦ → s(𝔱, ℓ), and the weight function (𝔱, ℓ) ↦ → 𝑤 b (𝔱, ℓ). In terms of generating functions, ( ) translates to

∑ (𝔱,ℓ) ∈FT † 𝑤 b (𝔱, ℓ) • 𝑥 |𝑉 (𝔱) | 𝑦 s(𝔱,ℓ) =: 𝐹 † (𝑥, 𝑦) = 𝑥𝐵(𝑦) 𝑦 • 1 1 -𝐹 (𝑥, 𝑦) ( )
where

𝑥𝐵 (𝑦) 𝑦
is the generating function of the class {∅} × N with the surplus being de ned by s(∅, 𝑙) = 𝑙 -1. (We refer readers unfamiliar with the formalism to [ , Chapter I. ].)

On the other hand, FT is simply the subset of FT † de ned by the condition s(𝔱, ℓ) ≥ 0. Moreover, the elements in its complement all satisfy s(𝔱, ℓ) = -1. Therefore FT = FT † \ {(𝔱, ℓ) ∈ FT † | s(𝔱, ℓ) = -1}, or in terms of the generating function, 𝐹 (𝑥, 𝑦) = 𝐹 † (𝑥, 𝑦) -[𝑦 -1 ]𝐹 † (𝑥, 𝑦). With ( ), this gives ( ) in the sense of formal power series.

Solution of ( ). First, notice that the coe cient of [𝑥 𝑛 ] of the right hand side of ( ) only depends on the coe cients of 𝐹 (𝑥, 𝑦) up to order [𝑥 𝑛-1 ] in 𝑥. Therefore Equation ( ) uniquely determines the series 𝐹 (𝑥, 𝑦) order by order in 𝑥 (with the initial condition 𝐹 (0, 𝑦) = 0). Equation ( ) involves not only the unknown function 𝐹 (𝑥, 𝑦), but also its specialization at 𝑦 = 0. Equations of this form are called equations with one catalytic variable 𝑦, and a general method for solving them -which is a generalization of the kernel method and the quadratic method -is given in [ ]. In the following, we apply this method to solve Equation ( ), while keeping the presentation self-contained.

Let Φ(𝑓 , 𝑓 0 , 𝑥, 𝑦) = 𝐵 (𝑦) 1-𝑓 -𝑏 0 1-𝑓 0 -𝑥 -1 𝑦 𝑓 . Then Equation ( ) is equivalent to Φ(𝐹 (𝑥, 𝑦), 𝐹 0 (𝑥), 𝑥, 𝑦) = 0. Its partial derivative with respect to 𝑦 gives 𝜕 𝑦 𝐹 (𝑥, 𝑦) • 𝜕 𝑓 Φ 𝐹 (𝑥, 𝑦), 𝐹 0 (𝑥), 𝑥, 𝑦 + 𝜕 𝑦 Φ 𝐹 (𝑥, 𝑦), 𝐹 0 (𝑥), 𝑥, 𝑦 = 0 .

( )

Assume that there exists a formal power series Ŷ (𝑥) ∈ C[[𝑥]] such that

𝜕 𝑓 Φ 𝐹 (𝑥, Ŷ (𝑥)), 𝐹 0 (𝑥), 𝑥, Ŷ (𝑥) = 0. ( )
Then the formal power series 𝐹 ≡ 𝐹 (𝑥, Ŷ (𝑥)), 𝐹 0 ≡ 𝐹 0 (𝑥) and 𝑌 ≡ Ŷ (𝑥) must satisfy the system of equations 𝜕 𝑓 Φ(𝐹, 𝐹 0 , 𝑥, 𝑌 ) = 0 , 𝜕 𝑦 Φ(𝐹, 𝐹 0 , 𝑥, 𝑌 ) = 0 , and Φ(𝐹, 𝐹 0 , 𝑥, 𝑌 ) = 0 , ( ) or, explicitly

𝐵(𝑌 ) (1 -𝐹 ) 2 = 𝑥 -1 𝑌 , 𝐵 (𝑌 ) 1 -𝐹 = 𝑥 -1 𝐹 , and
𝐵(𝑌 ) 1 -𝐹 - 𝑏 0 1 -𝐹 0 = 𝑥 -1 𝑌 𝐹 .
( )

The rst equation, which is equivalent to ( ), determines the coe cients of Ŷ (𝑥) inductively starting from Ŷ (0) = 0 in the same way that ( ) determines 𝐹 (𝑥, 𝑦). This shows the existence of the series Ŷ (𝑥) assumed above. Moreover, since the expansion of

𝐵 (𝑌 ) (1-𝐹 ) 2 ≡ 𝐵 (𝑌 )
(1-𝐹 (𝑥,𝑌 )) 2 as a power series of 𝑥 and 𝑌 has nonnegative coe cients, the above inductive de nition shows that all the coe cients of Ŷ (𝑥) are nonnegative.

One can eliminate 𝐹 = 𝐹 (𝑥, 𝑌 ) from the system ( ), and express 𝑥 and 𝐹 0 (𝑥) explicitly in terms of 𝑌 = Ŷ (𝑥):

𝑥 = x (𝑌 ) := 𝑌 𝐵(𝑌 ) • (1 + 𝜓 (𝑌 )) 2 and 𝐹 0 (𝑥) = F0 (𝑌 ) := 1 - 𝑏 0 𝐵(𝑌 ) • (1 -𝜓 (𝑌 ) 2 ) ( )
where 𝜓 (𝑌 ) = 𝑌 𝐵 (𝑌 ) 𝐵 (𝑌 ) . Plugging these into the original combinatorial equation ( ) gives a quadratic equation for 𝐹 (𝑥, 𝑦), whose two solutions are

𝐹 (𝑥, 𝑦) = 1 2 + ± √ 𝑄 (𝑌 , 𝑦) -𝜙 (𝑌 ) 2𝑦 where 𝑄 (𝑌 , 𝑦) := (𝜙 (𝑌 ) + 𝑦) 2 -4𝑦𝐵(𝑦) • x (𝑌 ) ( )
and

𝜙 (𝑌 ) = 𝑌 1-𝜓 (𝑌 )
1+𝜓 (𝑌 ) . One can check directly that 𝑄 (𝑌 , 𝑌 ) = 𝜕 𝑦 𝑄 (𝑌 , 𝑌 ) = 0 for all 𝑌 (see also Lemma ( )). This means that 𝑄 (𝑌 , 𝑦) = (𝑌 -𝑦) 2 𝑞(𝑌 , 𝑦) for some series

𝑞(𝑌 , 𝑦) ∈ C[[𝑌 ]] [[𝑦]]. Moreover, we have 𝑞(0, 0) = 1 2 𝜕 2 𝑦 𝑄 (0, 0) = 1.
Therefore the square root of 𝑄 (𝑌 , 𝑦) has two analytic determinations in a neighborhood of (0, 0), given by ±(𝑌 -𝑦) √ 𝑞(𝑌 , 𝑦). Since 𝐹 (𝑥, 𝑦) is a power series of 𝑦, we must choose the plus sign, which gives the parametrization of 𝐹 (𝑥, 𝑦) in Proposition . This nishes the proof of Proposition .

Notice that F (𝑌 , 0) = 1 -𝑏 0

x (𝑌 ) 𝜙 (𝑌 ) , and this is in agreement with the second equation in ( ). From now on, we make the following distinction between the notations 𝑌 and Ŷ (𝑥): we treat 𝑌 as an independent formal or complex variable, and treat Ŷ (𝑥) as a formal power series or complex function of 𝑥. Notice that Ŷ (0) = 0, and Ŷ is the functional inverse of x in the sense that Ŷ ( x (𝑌 )) = 𝑌 and x ( Ŷ (𝑥)) = 𝑥 as formal power series.

The phase diagram

In this short section, we prove the characterization of the phases given in Proposition . As illustrated in Figure , Proposition would follow almost directly from the de nition of the phases if the function x was concave. While x is not always concave, the following lemma gives a weaker property (i.e. local concavity at the critical points) of x, which su ces for the proof of Proposition . It will also be used in the proof of Lemma to show that the asymptotic expansion of x in the generic phase is indeed generic.

Recall that x (𝑌 ) = 𝑌 𝐵 (𝑌 )

(𝐵 (𝑌 )+𝑌 𝐵 (𝑌 )) 2 and the model is said to be in the generic phase if x vanishes on (0, 𝜌).

Lemma . If x (𝑌 ) = 0 for some 𝑌 ∈ (0, 𝜌), then x (𝑌 ) < 0. When 𝜌 < ∞, the same implication holds for 𝑌 = 𝜌.

Proof. Notice that x is a rational function of 𝑌 , 𝐵(𝑌 ) and 𝐵 (𝑌 ). Therefore x (𝑌 ) depends linearly on 𝐵 (𝑌 ).

More precisely,

x (𝑌 ) = (𝐵(𝑌 ) -𝑌 𝐵 (𝑌 )) 2 (𝐵(𝑌 ) + 𝑌 𝐵 (𝑌 )) 3 - 2𝑌 2 𝐵(𝑌 ) (𝐵(𝑌 ) + 𝑌 𝐵 (𝑌 )) 3 • 𝐵 (𝑌 ) ( )
By solving 𝐵 (𝑌 ) from the equation x (𝑌 ) = 0 and plugging the result into the expression of x (𝑌 ), we obtain after simpli cation:

x (𝑌 ) = - 3(𝐵(𝑌 ) -𝑌 𝐵 (𝑌 )) 2 + 2𝑌 3 𝐵 (𝑌 ) 𝑌 𝐵(𝑌 ) • (𝐵(𝑌 ) + 𝑌 𝐵 (𝑌 )) 3 < 0 .
Proof of Proposition . Lemma implies that x vanishes at most once on (0, 𝜌), and when it does, it changes sign. Therefore the model is in the generic (resp. dilute, dense) phase if and only if x (𝜌 -) := lim 𝑥→𝜌 -x (𝑌 ) < 0 (resp. = 0, > 0). This proves Proposition when 𝜌 < 0 and 𝐵 (𝜌) < ∞, where x (𝜌) is well-de ned and nite.

When

𝐵 (𝜌) = ∞ but 𝐵 (𝜌) < ∞, the expression ( ) shows that x (𝜌 -) = -∞. When 𝐵 (𝜌) = ∞ or 𝜌 = ∞, we have 𝐵 (𝜌 -) = ∞. (Recall that the case 𝐵(𝑌 ) = 𝑏 0 + 𝑏 1 𝑌 is excluded by assumption.) This implies x (𝜌 -) = 0, because x (𝑌 ) = 𝜓 (1+𝜓 ) 2 1 𝐵 (𝑌 ) ≤ 1 4 1 𝐵 (𝑌 ) for all 𝑌 ∈ (0, 𝜌), where 𝜓 = 𝑌 𝐵 (𝑌 )
𝐵 (𝑌 ) . Combining the two cases, we see that x vanishes at least once on (0, 𝜌) when 𝐵 (𝜌) = ∞ or 𝜌 = ∞, so the model is in the genric phase.

Basic algebraic properties of the parametrization of 𝐹 (𝑥, 𝑦)

In this section, we gather some useful algebraic properties of the parametrization of 𝐹 (𝑥, 𝑦) in Proposition . All of these properties can be veri ed by direct computation. However, we will provide a proof that relies as little as we can on the explicit expressions of the parametrization, with the hope that it would shed some light on the combinatorial origin of these properties.

To help organize the statement and the proof of these properties, we introduce several di erential operators: For any function f (𝑌 , 𝑦), de ne

/ 𝜕 𝑥 f (𝑌 , 𝑦) := 𝜕 𝑌 f (𝑌 , 𝑦) x (𝑌 ) . ( )
This operator has a simple meaning: if a function 𝑓 (𝑥, 𝑦) is parametrized by 𝑥 = x (𝑌 ) and 𝑓 (𝑥, 𝑦) = f (𝑌 , 𝑦), then its 𝑥-derivative is parametrized by 𝑥 = x (𝑌 ) and 𝜕 𝑥 𝑓 (𝑥, 𝑦) = / 𝜕 𝑥 f (𝑌 , 𝑦). Now consider a function of the form

𝑓 (𝑌 , 𝑦) = 𝑅 𝑓 (𝑌 , 𝐵(𝑌 ), 𝐵 (𝑌 ), . . . , 𝑦, 𝐵(𝑦), 𝐵 (𝑦), . . .) ( )
where 𝑅 𝑓 (𝑌, 𝑈 0 , 𝑈 1 , . . . , 𝑦, 𝑢 0 , 𝑢 1 , . . .) is some algebraic function that depends only on nitely many of the variables 𝑈 𝑘 and 𝑢 𝑘 . (In practice we will only need 𝑘 ≤ 2.) We de ne

/ 𝜕 𝑈 𝑘 𝑓 (𝑌 , 𝑦) := 𝜕 𝑈 𝑘 𝑅 𝑓 (𝑌 , 𝐵(𝑌 ), . . . ; 𝑦, 𝐵(𝑦), . . .) ( )
For generic values of 𝑌 , 𝑦 and (𝑏 𝑙 ) 𝑙 ≥0 , the variables 𝑌 , 𝐵(𝑌 ), 𝐵 (𝑌 ), . . ., and 𝑦, 𝐵(𝑦), 𝐵 (𝑦), . . . are algebraically independent. Hence the representation ( ) of 𝑓 is unique, and the above de nition of / 𝜕 𝑈 𝑘 is non-ambiguous. We de ne / 𝜕 𝑌 𝑓 , / 𝜕 𝑢 𝑘 𝑓 and / 𝜕 𝑦 𝑓 similarly. We have the operator relations

𝜕 𝑌 = / 𝜕 𝑌 + 𝑈 1 / 𝜕 𝑈 0 + 𝑈 2 / 𝜕 𝑈 1 + • • • and 𝜕 𝑦 = / 𝜕 𝑦 + 𝑢 1 / 𝜕 𝑢 0 + 𝑢 2 / 𝜕 𝑢 1 + • • • ( )
Notice that while the operators / 𝜕 𝑈 0 , / 𝜕 𝑈 1 , . . . commute with each other and with / 𝜕 𝑌 , they do not commute with 𝜕 𝑌 . The same remark holds for the operators / 𝜕 𝑢 0 , / 𝜕 𝑢 1 , . . . and / 𝜕 𝑦 , 𝜕 𝑦 .

Lemma (Algebraic properties of the parametrization of 𝐹 (𝑥, 𝑦)).

( ) / 𝜕 𝑥 𝜙 (𝑌 ) = 𝐵(𝑌 ) + 𝑌 • 𝐵 (𝑌 ).
( ) For all 𝑌 , we have

𝑄 (𝑌 , 𝑌 ) = / 𝜕 𝑈 1 𝑄 (𝑌 , 𝑌 ) = / 𝜕 𝑥 𝑄 (𝑌 , 𝑌 ) = 𝜕 𝑌 𝑄 (𝑌 , 𝑌 ) = 𝜕 𝑦 𝑄 (𝑌 , 𝑌 ) = 0. On the other hand, 𝜕 𝑌 / 𝜕 𝑥 𝑄 (𝑌,𝑌 ) = -𝜕 𝑦 / 𝜕 𝑥 𝑄 (𝑌,𝑌 ) = 2/ 𝜕 𝑥 𝜙 (𝑌 ), so 𝜕 2 𝑌 𝑄 (𝑌 , 𝑌 ) = -𝜕 𝑦 𝜕 𝑌 𝑄 (𝑌 , 𝑌 ) = 2𝜙 (𝑌 ). ( ) 𝑞(𝑌, 𝑦) = - ˆ1 0 ˆ𝜆1 0 𝜕 𝑌 𝜕 𝑦 𝑄 𝑌 + 𝜆 2 (𝑦 -𝑌 ), 𝑌 + 𝜆 1 (𝑦 -𝑌 ) • d𝜆 2 d𝜆 1 . In particular, 𝑞(𝑌 , 𝑌 ) = 𝜙 (𝑌 ). ( ) 𝜕 𝑌 (𝑌 -𝑦) √ 𝑞(𝑌 , 𝑦) = 𝜕 𝑌 𝑄 (𝑌 ,𝑦) 2(𝑌 -𝑦) √ 𝑞 (𝑌 ,𝑦)
and

𝜕 𝑦 (𝑌 -𝑦) √ 𝑞(𝑌 , 𝑦) = - 𝜕 𝑦 𝑄 (𝑌 ,𝑦) 2(𝑌 -𝑦) √ 𝑞 (𝑌 ,𝑦)
.

Proof.

( ) By de nition, we have 𝜙 = 𝑌 1-𝜓 1+𝜓 and x = 𝑌 (1+𝜓 ) 2 𝐵 with 𝜓 = 𝑌 𝐵 𝐵 . By comparing the logarithmic derivatives 𝜙 𝜙

= 1 𝑌 - 2 1 -𝜓 2 • 𝜓 and x x = 1 -𝜓 𝑌 - 2 1 + 𝜓 • 𝜓 , ( ) we see that / 𝜕 𝑥 𝜙 = 𝜙 (1-𝜓 ) x = 𝐵 + 𝑌 𝐵 .
( ) Notice that 𝑄 (𝑌 , 𝑦) is the discriminant of the quadratic equation ( ) satis ed by F (𝑌 , 𝑦) = 𝐹 ( x (𝑌 ), 𝑦).

When applying the generalized kernel method in Section , we have seen that this quadratic equation and its derivative share the same solution

𝐹 ≡ F (𝑌 , 𝑌 ) if 𝑦 is set to 𝑌 . It follows that 𝑄 (𝑌 , 𝑌 ) ≡ 0.
By di erentiating the function Δ(𝑌 ) := 𝑄 (𝑌 , 𝑌 ) ≡ 0, we see that

Δ (𝑌 ) = / 𝜕 𝑌 𝑄 (𝑌 , 𝑌 ) + / 𝜕 𝑦 𝑄 (𝑌 , 𝑌 ) ≡ 0 and / 𝜕 𝑈 1 Δ(𝑌 ) = / 𝜕 𝑈 1 𝑄 (𝑌 , 𝑌 ) + / 𝜕 𝑢 1 𝑄 (𝑌 , 𝑌 ) ≡ 0 . ( ) Since 𝑄 (𝑌 , 𝑦) is independent of the variable 𝑢 1 = 𝐵 (𝑦), the second identity is reduced to / 𝜕 𝑈 1 𝑄 (𝑌 , 𝑌 ) = 0. By plugging the operator relation 𝜕 𝑌 = / 𝜕 𝑌 + 𝑈 1 / 𝜕 𝑈 0 + • • • into the identity 𝜕 𝑌 𝑄 = / 𝜕 𝑥 𝑄 • x , we get (/ 𝜕 𝑌 + 𝑈 1 / 𝜕 𝑈 0 + 𝑈 2 / 𝜕 𝑈 1 )𝑄 = / 𝜕 𝑥 𝑄 • (/ 𝜕 𝑌 + 𝑈 1 / 𝜕 𝑈 0 + 𝑈 2 / 𝜕 𝑈 1 ) x . ( ) According Point ( ), / 𝜕 𝑥 𝑄 (𝑌 , 𝑦) = 2(𝜙 (𝑌 ) +𝑦) • / 𝜕 𝑥 𝜙 (𝑌 ) + 4𝑦𝐵(𝑦) is independent of the variable 𝑈 2 = 𝐵 (𝑌 ).
Hence we can extract the coe cient of 𝑈 2 from both sides of the equation, which gives /

𝜕 𝑈 1 𝑄 = / 𝜕 𝑥 𝑄 • / 𝜕 𝑈 1 x. Since /
𝜕 𝑈 1 x is not identically zero, we must have / 𝜕 𝑥 𝑄 (𝑌 , 𝑌 ) ≡ 0. It follows that 𝜕 𝑌 𝑄 (𝑌 , 𝑌 ) = / 𝜕 𝑥 𝑄 (𝑌 , 𝑌 ) • x (𝑌 ) ≡ 0. Then, the rst identity of ( ) shows that 𝜕 𝑦 𝑄 (𝑌 , 𝑌 ) ≡ 0 as well.

The total derivative of the identity / 𝜕 𝑥 𝑄 (𝑌 , 𝑌 ) ≡ 0 gives that 𝜕 𝑌 / 𝜕 𝑥 𝑄 (𝑌 , 𝑌 ) = -𝜕 𝑦 / 𝜕 𝑥 𝑄 (𝑌 , 𝑌 ) for all 𝑌 . Thanks to the general formula

𝜕 𝑦 / 𝜕 𝑥 𝑄 (𝑌 , 𝑦) = 2/ 𝜕 𝑥 𝜙 (𝑌 ) -4/ 𝜕 𝑥 𝜙 (𝑦), we have 𝜕 𝑦 / 𝜕 𝑥 𝑄 (𝑌 , 𝑌 ) = -2/ 𝜕 𝑥 𝜙 (𝑌 ). Finally, the formula 𝜕 2 𝑌 𝑄 (𝑌 , 𝑌 ) = -𝜕 𝑦 𝜕 𝑌 𝑄 (𝑌 , 𝑌 ) = 2𝜙 (𝑌 )
follows from the previous one by the de nition of / 𝜕 𝑥 .

( ) Thanks to the identities 𝑄 (𝑌 , 𝑌 ) = 𝜕 𝑌 𝑄 (𝑌 , 𝑌 ) ≡ 0, we have

- ˆ1 0 ˆ𝜆1 0 𝜕 𝑌 𝜕 𝑦 𝑄 𝑌 + 𝜆 2 (𝑦 -𝑌 ), 𝑌 + 𝜆 1 (𝑦 -𝑌 ) • d𝜆 2 d𝜆 1 = ˆ1 0 𝜕 𝑌 𝑄 (𝑌 , 𝑌 +𝜆 1 (𝑦 -𝑌 )) -
), 𝑌 +𝜆 1 (𝑦 -𝑌 )) 𝑦 -𝑌 d𝜆 1 = 𝑄 (𝑌 , 𝑦) - 𝑄 (𝑌 , 𝑌 ) (𝑌 -𝑦) 2 = 𝑞(𝑌 , 𝑦) .
( ) The rst identity is simply the correct analytic branch of the formula

𝜕 𝑌 √ 𝑄 (𝑌 , 𝑦) = 𝜕 𝑌 𝑄 (𝑌 ,𝑦) 2 √ 𝑄 (𝑌 ,𝑦)
. It can be obtained by dividing 𝜕 𝑌 𝑄 (𝑌 , 𝑦) = 2(𝑌 -𝑦)𝑞(𝑌 , 𝑦) + (𝑌 -𝑦) 2 𝜕 𝑌 𝑞(𝑌 , 𝑦) by 2(𝑌 -𝑦) √ 𝑞(𝑌 , 𝑦). The second identity is proved similarly.

Remark. Most of the above proof does not rely on the explicit expression of parametrization of 𝐹 (𝑥, 𝑦). But the precise formulas of x and 𝜙 are crucial for / 𝜕 𝑥 𝜙 to not depend on 𝑈 2 = 𝐵 (𝑌 ) in Point ( ). Indeed, for a generic function φ (𝑌 ) that depends on (𝑌 , 𝑈 0 , 𝑈 1 ), the derivative / 𝜕 𝑥 φ will depend on 𝑈 2 :

/ 𝜕 𝑥 φ ≡ φ x = (/ 𝜕 𝑌 + 𝑈 1 / 𝜕 𝑈 0 ) φ + 𝑈 2 • / 𝜕 𝑈 1 φ (/ 𝜕 𝑌 + 𝑈 1 / 𝜕 𝑈 0 ) x + 𝑈 2 • / 𝜕 𝑈 1 x . ( )
So the fact that / 𝜕 𝑥 𝜙, and therefore / 𝜕 𝑥 F (𝑌 , 𝑦), does not depend on 𝑈 2 re ects some property that is proper to our model. Combinatorially, this means that the generating function 𝜕 𝑥 𝐹 (𝑥, 𝑦) of fully packed trees with a distinguished vertex also has a relatively simple expression under the parametrization 𝑥 = x (𝑌 ).

Asymptotic expansions of 𝐹 (𝑥, 𝑦)

In this section we compute the asymptotic expansions of 𝐹 (𝑥, 𝑦) necessary for establishing the coe cient asymptotics in Theorem . We start with the corresponding asymptotic expansions of x (𝑌 ) and F (𝑌 , 𝑦), then combine them to get the desired expansions of 𝐹 (𝑥, 𝑦).

To this end, we rst need to locate the singularities of 𝐹 (𝑥, 𝑦) that are relevant for its coe cient asymptotics. The de nition of the generic and non-generic phases already hints that the values 𝑥 𝑐 and 𝑌 𝑐 play a role. We will prove in Section that the bivariate function 𝐹 (𝑥, 𝑦) actually has a unique dominant singularity at (𝑥 𝑐 , 𝑌 𝑐 ). In this section, we take this information as granted, and focus on the asymptotics of 𝐹 (𝑥, 𝑦) when (𝑥, 𝑦) → (𝑥 𝑐 , 𝑌 𝑐 ) in 𝑥 𝑐 × 𝑌 𝑐 . By de nition, 𝑥 = 𝑥 𝑐 is parametrized by 𝑌 = 𝑌 𝑐 . We will use the following change of variables

𝑥 = 𝑥 𝑐 • (1 -𝑠) , 𝑦 = 𝑌 𝑐 • (1 -𝑡) , 𝑌 = 𝑌 𝑐 • (1 -𝑆) , ( )
so that the limit to be taken becomes (𝑠, 𝑡) → (0, 0), or (𝑆, 𝑡) → (0, 0) under the parametrization 𝑥 = x (𝑌 ).

Recall that the Δ-domain 𝑟 ≡ 𝜖,𝛿 𝑟 depends on two positive parameters 𝜖 and 𝛿. But we choose to often omit them from the notation, and their values may change from one place to another. De ne the cone

𝐾 𝛿 = {𝑧 ∈ C | 𝑧 ≠ 0 and arg(𝑧) ∈ (-𝜋/2 -𝛿, 𝜋/2 + 𝛿)} .
( ) Recall that we restrict our attention to the generic + and the dilute -phases, in both of which x (𝑌 𝑐 ) = 0. Recall also that we de ne 𝛼 = 3 in the generic + phase and 𝛼 = α ∈ (2, 3) in the dilute -phase, where α is the singular exponent in Assumption ( * ).

For
Lemma (Asymptotics of x (𝑌 )). When 𝑆 → 0 in the closed cone 𝐾 𝛿 , we have

1 - x (𝑌 ) 𝑥 𝑐 ∼ 𝜇 • 𝑆 𝛼-1 and x (𝑌 ) 𝑥 𝑐 ∼ 𝛼 -1 𝑌 𝑐 𝜇 • 𝑆 𝛼-2 ( )
where 𝜇 = -

𝑌 2 𝑐 2 x (𝑌 𝑐 ) 𝑥 𝑐 in the generic + phase and 𝜇 = αC 𝐵 𝑌 𝑐 • / 𝜕 𝑈 1 x (𝑌 𝑐 ) 𝑥 𝑐
in the dilute -phase. In both phases, 𝜇 > 0.

Proof. The de nition of x can be written as x (𝑌 ) = 𝑅 x (𝑌 , 𝐵(𝑌 ), 𝐵 (𝑌 )) with 𝑅 x (𝑌 , 𝑈 0 , 𝑈 1 ) = 𝑌𝑈 0 (𝑈 0 +𝑌𝑈 1 ) 2 . In the generic + phase, 𝐵(𝑌 ) is 𝐶 3 -continuous as 𝑌 → 𝑌 𝑐 in 𝑌 𝑐 . (Indeed, 𝐵(𝑌 ) is analytic at 𝑌 𝑐 in the generic case, while in the dilute but generic + case, 𝐶 3 -continuity follows from Assumption ( * ) and 𝐵 (𝜌) < ∞.) It follows that x is 𝐶 2 -continuous at 𝑌 𝑐 . Since x (𝑌 𝑐 ) =𝑥 𝑐 and x (𝑌 𝑐 ) = 0, the Taylor expansion of x gives

x (𝑌 ) = 𝑥 𝑐 + 1 2 x (𝑌 𝑐 ) • (𝑌 𝑐 𝑆) 2 + 𝑜 (𝑆 2 ) . ( ) That is, 1 -x (𝑌 ) 𝑥 𝑐 ∼ 𝜇 • 𝑆 2 with 𝜇 = - 𝑌 2 𝑐 2 x (𝑌 𝑐 )
𝑥 𝑐 . Similarly, the Taylor expansion of x (𝑌 ) gives x (𝑌 )

𝑥 𝑐 ∼ 2 𝑌 𝑐 𝜇 • 𝑆. In the dilute -phase, recall the decomposition 𝐵(𝑌 ) = 𝐵 r (𝑌 ) + 𝐵 s (𝑌 ) + 𝑜 (𝐵 s (𝑌 )) in Assumption ( * ). Let xr (𝑌 ) = 𝑅 x (𝑌, 𝐵 r (𝑌 ), 𝐵 r (𝑌 ))
. By expanding the rational function 𝑅 x around (𝑌 , 𝐵 r (𝑌 ), 𝐵 r (𝑌 )), we get

x (𝑌 ) = xr (𝑌 ) + / 𝜕 𝑈 1 xr (𝑌 ) • 𝐵 s (𝑌 ) + 𝑜 (𝐵 s (𝑌 )) + / 𝜕 𝑈 0 xr (𝑌 ) • 𝐵 s (𝑌 ) + 𝑜 (𝐵 s (𝑌 )) = xr (𝑌 ) + / 𝜕 𝑈 1 xr (𝑌 𝑐 ) • 𝐵 s (𝑌 ) + 𝑜 ((𝑌 𝑐 -𝑌 ) α-1 )
as 𝑌 → 𝑌 𝑐 . The function xr is analytic at 𝑌 𝑐 . It is not hard to see that xr

(𝑌 𝑐 ) = x (𝑌 𝑐 ) =𝑥 𝑐 and x r (𝑌 𝑐 ) = x (𝑌 𝑐 ) = 0. Also, we have 𝐵 s (𝑌 ) = -αC 𝐵 𝑌 𝑐 • 𝑆 α-1 . It follows that x (𝑌 ) = 𝑥 𝑐 -/ 𝜕 𝑈 1 x (𝑌 𝑐 ) • αC 𝐵 𝑌 𝑐 • 𝑆 α-1 + 𝑜 (𝑆 α-1 ) , ( ) that is, 1 -x (𝑌 ) 𝑥 𝑐 ∼ 𝜇 • 𝑆 α-1 with 𝜇 = αC 𝐵 𝑌 𝑐 • / 𝜕 𝑈 1 x (𝑌 𝑐 ) 𝑥 𝑐
. A similar computation shows that x (𝑌 ) 𝑥 𝑐 ∼ 𝛼-1 𝑌 𝑐 𝜇 • 𝑆 α-2 . All of the above asymptotics are valid when 𝑌 → 𝑌 𝑐 in 𝑌 𝑐 , or equivalently 𝑆 → 0 in 𝐾 𝛿 , thanks to the domain of validity of the expansions in Assumption ( * ). In the generic + phase, we have x (𝑌 𝑐 ) < 0 by Lemma . In the dilute -phase, one can check that the asymptotic positivity of the coe cients 𝑏 𝑙 = [𝑌 𝑛 ]𝐵(𝑌 ) implies that C 𝐵 < 0 when 2 < 𝛼 < 3. It follows that 𝜇 > 0 in both phases. This completes the proof of the lemma.

Lemma is the only place where we do calculations separately in the generic + phase and the dilute -phase. From now on, the two phases will be treated in a uni ed way (except for a technical proof in Section . which veri es the Δ-analyticity of Ŷ = x-1 ). Now we turn to the asymptotic expansions of F (𝑌 , 𝑦). Since Theorem contains both univariate and bivariate asymptotics of the coe cients of 𝐹 (𝑥, 𝑦), we need both univariate and bivariate asymptotic expansions of F (𝑌, 𝑦) to derive it. The univariate expansions are straightforward to compute, and it is not hard to seegiven our assumption ( * ) on 𝐵(𝑦) -that the dominant singular term must be of the classical power-law type. In the bivariate case, the classi cation of dominant singular terms is much less studied. For the particular example of F (𝑌, 𝑦), it seems that the correct generalization of power functions in the multivariate world is the following concept of homogenous functions:

De nition. We say that a function 𝐻 de ned on some domain Ω ⊆ C × C is homogenous of degree 𝛾 if for all 𝜎 ∈ C \ {0}, we have 𝐻 (𝜎𝑧, 𝜎𝑤) = 𝜎 𝛾 • 𝐻 (𝑧, 𝑤) whenever both (𝑧, 𝑤) and (𝜎𝑧, 𝜎𝑤) are in Ω.

In the formula of F (𝑌 , 𝑦), the only term where 𝑌 and 𝑦 cannot be easily separated is the square root of 𝑞(𝑌 , 𝑦). Lemma below gives the homogenous function 𝐻 𝛼 that is asymptotic equivalent to 𝑞 as (𝑌 , 𝑦) → (𝑌 𝑐 , 𝑌 𝑐 ). The next lemma (Lemma ) provides uniform bounds of 𝐻 𝛼 by a power function of the vector norm (𝑆, 𝑡) .

Notice that when 𝑥 is tied to 𝑌 by 𝑥 ≡ (1-𝑠)𝑥 𝑐 = x (𝑌 ), the asymptotics of x in Lemma becomes 𝑠 ∼ 𝜇 •𝑆 𝛼-1 . Therefore, at rst order approximation,

𝑥 → 𝑥 𝑐 in 𝑥 𝑐 is equivalent to 𝑆 → 0 in 𝐾 𝜃 𝛿 := {𝑧 𝜃 | 𝑧 ∈ 𝐾 𝛿 }, where 𝜃 = 1
𝛼-1 as de ned in Theorem . Hence the right domain for taking the limit (𝑆, 𝑡) → (0, 0) is 𝐾 𝜃 𝛿 × 𝐾 𝛿 . Lemma (Asymptotics of 𝑞(𝑌 , 𝑦)). When (𝑆, 𝑡) → (0, 0) in 𝐾 𝜃 𝛿 × 𝐾 𝛿 for some 𝛿 = 𝛿 (𝛼) > 0, we have

𝑞(𝑌 , 𝑦) ∼ C 𝑞 • 𝐻 𝛼 (𝑆, 𝑡) ( )
where the constant

C 𝑞 := 2/ 𝜕 𝑥 𝜙 (𝑌 𝑐 ) • 𝜇𝑥 𝑐
𝛼𝑌 𝑐 is positive, and 𝐻 𝛼 is the homogenous function of degree 𝛼 -2 de ned by

𝐻 𝛼 (𝑆, 𝑡) = 𝑡 𝛼 -𝑆 𝛼 -𝛼𝑆 𝛼-1 (𝑡 -𝑆) (𝑡 -𝑆) 2 . ( )
Proof. By Lemma ( ), we have / 𝜕 𝑥 𝜕 𝑦 𝑄 (𝑌 𝑐 , 𝑌 𝑐 ) = -2/ 𝜕 𝑥 𝜙 (𝑌 𝑐 ) ≠ 0. Hence Lemma implies that 𝜕 𝑌 𝜕 𝑦 𝑄 (𝑌 , 𝑦) = / 𝜕 𝑥 𝜕 𝑦 𝑄 (𝑌 , 𝑦) • x (𝑌 ) ∼ 𝑌 →𝑌 𝑐 -2/ 𝜕 𝑥 𝜙 (𝑌 𝑐 ) • 𝛼 -1 𝑌 𝑐 𝜇𝑥 𝑐 • 𝑆 𝛼-2 = -𝛼 (𝛼 -1)C 𝑞 • 𝑆 𝛼-2 . ( )
With the change of variables 𝑆 = 1 -𝑌 𝑌 𝑐 and 𝑡 = 1 -𝑦 𝑌 𝑐 , this gives

𝜕 𝑌 𝜕 𝑦 𝑄 𝑌 + 𝜆 2 (𝑦 -𝑌 ), 𝑌 + 𝜆 1 (𝑦 -𝑌 ) = -𝛼 (𝛼 -1)C 𝑞 • 1 - 𝑌 + 𝜆 2 (𝑦 -𝑌 ) 𝑌 𝑐 𝛼-2 • (1 + 𝑜 (1)) = -𝛼 (𝛼 -1)C 𝑞 • 𝑆 + 𝜆 2 (𝑡 -𝑆) 𝛼-2 + 𝑜 (𝑆, 𝑡) 𝛼-2
where • is any norm on the vector space C 2 , and the little-o is uniform over (𝜆 1 , 𝜆 2 ) ∈ [0, 1] 2 . Plug this into the integral formula of 𝑞(𝑌 , 𝑦) in Lemma ( ), we get

𝑞(𝑌 , 𝑦) = C 𝑞 • ˆ1 0 ˆ𝜆1 0 𝛼 (𝛼 -1) 𝑆 + 𝜆 2 (𝑡 -𝑆) 𝛼-2 d𝜆 2 d𝜆 1 + 𝑜 (𝑆, 𝑡) 𝛼-2 = C 𝑞 • ˆ1 0 𝛼 (𝑆 + 𝜆 1 (𝑡 -𝑆)) 𝛼-1 -𝑆 𝛼-1 𝑡 -𝑆 d𝜆 1 + 𝑜 (𝑆, 𝑡) 𝛼-2 = C 𝑞 • 𝑡 𝛼 -𝑆 𝛼 -𝛼𝑆 𝛼-1 (𝑡 -𝑆) (𝑡 -𝑆) 2 + 𝑜 (𝑆, 𝑡) 𝛼-2 .
-and also for some examples related to Ising-decorated planar maps, see [ ] -

Thanks to the lower bound of 𝐻 𝛼 (𝑡, 𝑆) in Lemma below, this implies the asymptotic equivalence ( ).

Lemma (" 𝐻 𝛼 (𝑆, 𝑡) (𝑆, 𝑡) 𝛼-2 "). For each 𝛼 ∈ (2, 3], there exist 𝛿 > 0 and 𝑐, 𝑐 > 0 such that

𝑐 • (𝑆, 𝑡) 𝛼-2 ≤ |𝐻 𝛼 (𝑆, 𝑡)| ≤ 𝑐 • (𝑆, 𝑡) 𝛼-2 ( )
for all (𝑆, 𝑡) ∈ 𝐾 𝜃 𝛿 × 𝐾 𝛿 . Proof. When 𝑆 = 0, we have 𝐻 𝛼 (0, 𝑡) = 𝑡 𝛼-2 and ( ) is obvious. When 𝑆 ≠ 0, because 𝐻 𝛼 is homogenous of degree 𝛼 -2, ( ) is equivalent to

𝑐 • (1 + |𝑧| 𝛼-2 ) ≤ |ℎ 𝛼 (𝑧)| ≤ 𝑐 • (1 + |𝑧| 𝛼-2 ) ( ) for all 𝑧 ∈ K := {𝑡/𝑆 : (𝑆, 𝑡) ∈ 𝐾 𝜃 𝛿 × 𝐾 𝛿 , 𝑆 ≠ 0}, where ℎ 𝛼 (𝑧) := 𝐻 𝛼 (1, 𝑧) = 𝑧 𝛼 -1-𝛼 (𝑧-1) (𝑧-1) 2
. Due to the 𝑧 𝛼 term, K should be viewed as a subdomain of the universal cover of C \ {0}, completed by a single point at 0. Notice that ℎ 𝛼 is continuous on this completed universal cover, because lim 𝑧→0 ℎ 𝛼 (𝑧) = 𝛼 and lim 𝑧→1 ℎ 𝛼 (𝑧) = 𝛼 (𝛼-1) 2 . Since ℎ 𝛼 (𝑧) ∼ 𝑧 𝛼-2 when 𝑧 → ∞, for any 𝑐 < 1 < 𝑐 , there exists 𝑅 > 0 such that ( ) holds for all |𝑧| > 𝑅.

On the other hand, the continuity of ℎ 𝛼 implies that it is bounded on the compact set {𝑧 ∈ K : |𝑧| ≤ 𝑅}. This proves the upper bound in ( ). For the lower bound, it su ces to show that ℎ 𝛼 have no zeros in K.

From its de nition, we see that

K = {𝑟𝑒 𝑖𝜏 | 𝑟 ≥ 0 , |𝜏 | ≤ (𝜃 + 1) ( 𝜋 2 + 𝛿)}. For all 𝛼 ∈ (2, 3], since 𝜃 = 1 𝛼-1 < 1, we can choose 𝛿 = 𝛿 (𝛼) > 0 such that (𝜃 + 1) ( 𝜋 2 + 𝛿) < 𝜋. Then K is contained in C \ (-∞, 0), the principal branch of the universal cover of C \ {0}.
Now let us show that ℎ 𝛼 has no zero in C\(-∞, 0) for all 𝛼 ∈ (2, 3]. This is clear for 𝛼 = 3, since ℎ 3 (𝑧) = 𝑧 + 2. Assume that ℎ 𝛼 has a zero on C \ (-∞, 0) for some 𝛼 ∈ (2, 3). Let 𝛼 * be the in mum of such 𝛼. By de nition, there exists a sequence of pairs (𝑧 𝑛 , 𝛼 𝑛 ) ∈ (C \ (-∞, 0)) × (2, 3) such that ℎ 𝛼 𝑛 (𝑧 𝑛 ) = 0 for all 𝑛, and 𝛼 𝑛 𝛼 * as 𝑛 → ∞. Using the equation ℎ 𝛼 𝑛 (𝑧 𝑛 ) = 0, it is not hard to see that the sequence (|𝑧 𝑛 |) 𝑛 ≥0 is bounded. Thus up to extracting a subsequence, we can assume that 𝑧 𝑛 → 𝑧 * as 𝑛 → ∞ for some 𝑧 * in the closure of C \ (-∞, 0). By the continuity of (𝑧, 𝛼) ↦ → ℎ 𝛼 (𝑧), we have ℎ 𝛼 * (𝑧 * ) = 0. However, we can check ℎ 𝛼 has no zero on the boundary of C \ (-∞, 0): we have ℎ 𝛼 (0) = 𝛼, and ℎ 𝛼 (𝑟𝑒 ±𝑖𝜋 ) = 𝑟 𝛼 𝑒 ±𝑖𝛼𝜋 -1+𝛼 (𝑟 +1)

(𝑟 +1) 2
≠ 0 for all 𝛼 ∈ (2, 3) and 𝑟 ∈ (0, ∞) because the imaginary part of the left hand side is nonzero. It follows that 𝑧 * ∈ C \ (-∞, 0). In addition, we have 𝛼 * > 2 because ℎ 2 (𝑧) ≡ 1. Since the mapping (𝑧, 𝛼) ↦ → ℎ 𝛼 (𝑧) is analytic in 𝑧, and jointly continuous in both variables, a version of the implicit function theorem (Lemma ) implies that there exists a continuous function ẑ : (𝛼 * -𝜀, 𝛼 * + 𝜀) → C \ (-∞, 0) such that ẑ (𝛼 * ) = 𝑧 * and ℎ 𝛼 ( ẑ (𝛼)) = 0 for all 𝛼. This contradicts the minimality of 𝛼 * . Therefore ℎ 𝛼 has no zero in C \ (-∞, 0) for all 𝛼 ∈ (2, 3], and this concludes the proof.

With the asymptotic expansions of x (𝑌 ) and 𝑞(𝑌 , 𝑦) in Lemmas and , we can now derive the desired asymptotic expansions of F (𝑌 , 𝑦) and 𝐹 (𝑥, 𝑦) by elementary calculations.

Lemma (Asymptotics of F (𝑌 , 𝑦)). Let C 𝐹 = √ C 𝑞 /2. When (𝑆, 𝑡) → (0, 0) in 𝐾 𝜃 𝛿 × 𝐾 𝛿 : F (𝑌 , 𝑦) -1 2 - 𝜙 (𝑌 ) 2𝑦 ∼ C 𝐹 • (𝑡 -𝑆) √ 𝐻 𝛼 (𝑆, 𝑡) . ( )
When 𝑆 → 0 in 𝐾 𝜃 𝛿 for xed 𝑦 ∈ 𝑌 𝑐 : / 𝜕 𝑥 F (𝑌 , 𝑦) -/ 𝜕 𝑥 F (𝑌 𝑐 , 𝑦) ∼ -𝑌 𝑐 • 𝜕 𝑌 / 𝜕 𝑥 F (𝑌 𝑐 , 𝑦) • 𝑆 . ( ) When 𝑡 → 0 in 𝐾 𝛿 : 𝑌 𝑐 • 𝜕 𝑌 / 𝜕 𝑥 F (𝑌 𝑐 , 𝑦) ∼ 𝛼 •C 𝐹 2𝜇𝑥 𝑐 •𝑡 -𝛼 2 and / 𝜕 𝑥 F (𝑌 𝑐 , 𝑦) ∼ 𝛼 •C 𝐹 2𝜇𝑥 𝑐 •𝑡 1-𝛼 2 . ( )
Proof. The asymptotic expansion ( ) follows directly from the de nition ( ) of F (𝑌 , 𝑦) and Lemma :

F (𝑌 , 𝑦) - 1 2 - 𝜙 (𝑌 ) 2𝑦 = 𝑌 -𝑦 2𝑦 √ 𝑞(𝑌 , 𝑦) ∼ 𝑡 -𝑆 2 √ C 𝑞 𝐻 𝛼 (𝑆, 𝑡) ( ) as (𝑆, 𝑡) → (0, 0) in 𝐾 𝜃 𝛿 × 𝐾 𝛿 .
/ 𝜕 𝑥 F (𝑌 , 𝑦) = 1 2𝑦 / 𝜕 𝑥 𝑄 (𝑌 , 𝑦) 2(𝑌 -𝑦) √ 𝑞(𝑌 , 𝑦) -/ 𝜕 𝑥 𝜙 (𝑌 ) .
( )

The same derivative formula implies that 𝜕 𝑌 gives ( ). The expression of the constant follows from the identity

1 (𝑌 -𝑦) √ 𝑞 (𝑌 ,𝑦) 𝑌 =𝑌 𝑐 = - / 𝜕 𝑥 𝑄 (𝑌 𝑐 ,𝑦) 2( (𝑌 𝑐 -𝑦)𝑞 (𝑌 𝑐 ,𝑦)) 3/2 • x (𝑌 𝑐 ) = 0.
/ 𝜕 𝑥 𝜙 (𝑌 𝑐 ) 2𝑌 𝑐 √ C 𝑞 = 𝛼 √ C 𝑞 4𝜇𝑥 𝑐 = 𝛼C 𝐹 2𝜇𝑥 𝑐 .
Recall the de nitions

𝛽 0 = 𝛼 𝛼-1 , 𝛽 1 = -𝛼 2 and 𝛾 0 = 𝛼 2 , 𝛾 1 = 1 -𝛼 2 from Theorem .
The following proposition translates the asymptotic expansions of F (𝑌 , 𝑦) in Lemma to asymptotic expansions of 𝐹 (𝑥, 𝑦).

Proposition (Asymptotics of 𝐹 (𝑥, 𝑦)). Let 𝐹 reg (𝑥, 𝑦) = 1 2 - 𝜙 ( Ŷ (𝑥)) 2𝑦
and 𝐹 hom (𝑆, 𝑡) = (𝑡 -𝑆) √ 𝐻 𝛼 (𝑆, 𝑡).

When (𝑥, 𝑦) → (𝑥 𝑐 , 𝑌 𝑐 ) in 𝑥 𝑐 × 𝑌 𝑐 : 𝐹 (𝑥, 𝑦) = 𝐹 reg (𝑥, 𝑦) + C 𝐹 • 𝐹 hom (𝑠/𝜇) 𝜃 , 𝑡 + 𝑜 (𝑠 𝜃 , 𝑡) 𝛾 0 . ( )
When 𝑥 → 𝑥 𝑐 in 𝑥 𝑐 for xed 𝑦 ∈ 𝑌 𝑐 : 𝐹 (𝑥, 𝑦) = 𝐹 (𝑥 𝑐 , 𝑦) -𝜕 𝑥 𝐹 (𝑥 𝑐 , 𝑦) (𝑥 𝑐 -𝑥) + 𝐺 (𝑦) • 𝑠 𝛽 0 + 𝑜 𝑠 𝛽 0 . ( )
When 𝑦 → 𝑌 𝑐 in 𝑌 𝑐 : 𝐺 (𝑦) ∼ 𝛼-1 2𝜇 𝛽 0 C 𝐹 • 𝑡 𝛽 1 and 𝜕 𝑥 𝐹 (𝑥 𝑐 , 𝑦) ∼ 𝛼 2𝜇𝑥 𝑐 C 𝐹 • 𝑡 𝛾 1 , ( )
where 𝐺 (𝑦) :=

𝜇𝑥 𝑐 𝛽 0 •𝜇 𝛽 0 • 𝜕 𝑌 / 𝜕 𝑥 F (𝑌 𝑐 , 𝑦).
Proof. Under the change of variable 𝑥 = x (𝑌 ), Equation ( ) in Lemma reads: 𝐹 (𝑥, 𝑦) -𝐹 reg (𝑥, 𝑦) ∼ C 𝐹 • 𝐹 hom (𝑆, 𝑡). To prove ( ), we just need to show that the error induced when replacing 𝐹 hom (𝑆, 𝑡) by 𝐹 hom ((𝑠/𝜇) 𝜃 , 𝑡) is of order 𝑜 ( (𝑠 𝜃 , 𝑡) 𝛾 0 ). Recall from Lemma that 𝑆 ∼ (𝑠/𝜇) 𝜃 when 𝑠 → 0. For general values 𝑆 1 , 𝑆 2 ∈ C, we have:

𝐹 hom (𝑆 1 , 𝑡) -𝐹 hom (𝑆 2 , 𝑡) ≤ |𝑆 1 -𝑆 2 | • sup 𝑆 ∈ [𝑆 1 ,𝑆 2 ] 𝜕 𝑆 𝐹 hom (𝑆, 𝑡) = |𝑆 1 -𝑆 2 | • sup 𝑆 ∈ [𝑆 1 ,𝑆 2 ] 𝛼 (𝛼 -1) 2 𝑆 𝛼-2 √ 𝐻 𝛼 (𝑆, 𝑡) ( ) When 𝑆 1 , 𝑆 2 → 0 and 𝑆 1 /𝑆 2 → 1, we have |𝑆 1 -𝑆 2 | = 𝑜 (𝑆 1 ), whereas the supremum on [𝑆 1 , 𝑆 2 ] is bounded by a constant times |𝑆 1 | 𝛼 -2
(𝑆 1 ,𝑡 ) 𝛼 /2-1 (the denominator is estimated using Lemma ). It follows that

𝐹 hom (𝑆 1 , 𝑡) -𝐹 hom (𝑆 2 , 𝑡) = 𝑜 (|𝑆 1 | 𝛼-1 ) (𝑆 1 , 𝑡) 𝛼/2-1 = 𝑜 ( (𝑆 1 , 𝑡) 𝛼/2 ) . ( )
Taking 𝑆 1 = (𝑠/𝜇) 𝜃 and 𝑆 2 = 𝑆 in the above formula gives the necessary estimate for proving ( ).

Under the change of variable 𝑥 = x (𝑌 ), the asymptotics ( ) in Lemma reads:

𝜕 𝑥 𝐹 (𝑥, 𝑦) -𝜕 𝑥 𝐹 (𝑥 𝑐 , 𝑦) = -𝑌 𝑐 • 𝜕 𝑌 / 𝜕 𝑥 F (𝑌 𝑐 , 𝑦) • (𝑠/𝜇) 𝜃 + 𝑜 (𝑠/𝜇) 𝜃 . ( )
Since 𝑠 = 1 -𝑥 𝑥 𝑐 and 𝛽 0 = 𝜃 + 1, by integrating the above equation from 𝑥 to 𝑥 𝑐 , we get

𝐹 (𝑥 𝑐 , 𝑦) -𝐹 (𝑥, 𝑦) -𝜕 𝑥 𝐹 (𝑥 𝑐 , 𝑦) • (𝑥 𝑐 -𝑥) = -𝑌 𝑐 • 𝜕 𝑌 / 𝜕 𝑥 F (𝑌 𝑐 , 𝑦) • 𝜇𝑥 𝑐 𝛽 0 𝑠 𝜇 𝛽 0 + 𝑜 𝑠 𝛽 0 . ( )
This is ( ) after rearrangement. Finally, ( ) is the direct translation of ( ) in Lemma .

Remark. In the bivariate expansion ( ) of 𝐹 (𝑥, 𝑦), the mapping 𝑦 ↦ → 𝐹 reg (𝑥, 𝑦) is analytic at 𝑌 𝑐 , and 𝐹 hom is a homogenous function of degree 𝛾 0 . These are the essential features of ( ) that will be used to prove the bivariate asymptotics ( ) of 𝐹 𝑛,𝑝 in Theorem .

Proof of Theorem

In this section, we prove the coe cient asymptotics stated in Theorem under the assumption that 𝐹 (𝑥, 𝑦) is Δ-analytic in both variables. More precisely, we assume that 𝐹 has an analytic continuation in some double Δ-domain 𝑥 𝑐 × 𝑌 𝑐 which is continuous on the boundary, and that 𝑦 ↦ → 𝐹 (𝑥 𝑐 , 𝑦) and 𝑦 ↦ → 𝜕 𝑥 𝐹 (𝑥 𝑐 , 𝑦) are analytic in 𝑌 𝑐 . Using ( ), it is not hard to see that these assumptions imply that 𝐺 (𝑦) is also analytic in 𝑌 𝑐 . We will verify the above Δ-analyticity assumptions in Section .

Proof of Theorem . Asymptotics of 𝐹 𝑝 (𝑥 𝑐 ) and 𝐹 𝑝 (𝑥 𝑐 ). When 𝑥 = 𝑥 𝑐 , the bivariate asymptotics ( ) reads

𝐹 (𝑥 𝑐 , 𝑦) = 𝐹 reg (𝑥 𝑐 , 𝑦) + C 𝐹 • 𝑡 𝛾 0 + 𝑜 (𝑡 𝛾 0 )
, where 𝑦 ↦ → 𝐹 reg (𝑥 𝑐 , 𝑦) is analytic at 𝑦 = 𝑌 𝑐 . Together with the second asymptotics in ( ), this gives the asymptotic expansion of 𝑦 ↦ → 𝐹 (𝑥 𝑐 , 𝑦) and 𝑦 ↦ → 𝜕 𝑥 𝐹 (𝑥 𝑐 , 𝑦) at their domiannt singularity 𝑌 𝑐 . By assumption, these functions are analytic in 𝑌 𝑐 . Thus, by the classical transfer theorem:

𝐹 𝑝 (𝑥 𝑐 ) ∼ 𝑝→∞ C 𝐹 Γ(-𝛾 0 ) • 𝑌 -𝑝 𝑐 • 𝑝 -𝛾 0 -1 and 𝐹 𝑝 (𝑥 𝑐 ) ∼ 𝑝→∞ 𝛼 2𝜇𝑥 𝑐 C 𝐹 Γ(-𝛾 1 ) • 𝑌 -𝑝 𝑐 • 𝑝 -𝛾 1 -1 . ( )
Asymptotics of 𝐹 𝑛,𝑝 as 𝑛 → ∞ for xed 𝑝, and then 𝑝 → ∞. For each 𝑦 ∈ 𝑌 𝑐 , ( ) and the Δ-analyticity of 𝑥 ↦ → 𝐹 (𝑥, 𝑦) imply that

𝐹 (𝑛) (𝑦) ∼ 𝑛→∞ 𝐺 (𝑦) Γ(-𝛽 0 ) • 𝑥 -𝑛 𝑐 • 𝑛 -𝛽 0 -1 , ( )
where 𝐹 (𝑛) (𝑦) := [𝑥 𝑛 ]𝐹 (𝑥, 𝑦) = ∞ 𝑝=0 𝐹 𝑛,𝑝 𝑦 𝑝 . Dividing the above asymptotics by its special case at 𝑦 = 𝑌 𝑐 gives

𝐹 (𝑛) (𝑦) 𝐹 (𝑛) (𝑌 𝑐 ) ----→ 𝑛→∞ 𝐺 (𝑦) 𝐺 (𝑌 𝑐 ) ( )
According to Vitali's theorem [ , p. ], the uniform convergence of a sequence of analytic functions in a neighborhood of zero implies the convergence of each coe cient in their Taylor expansions. Therefore

[𝑦 𝑝 ] 𝐹 (𝑛) (𝑦) 𝐹 (𝑛) (𝑌 𝑐 ) = 𝐹 𝑛,𝑝 𝐹 (𝑛) (𝑌 𝑐 ) ----→ 𝑛→∞ 𝐺 𝑝 𝐺 (𝑌 𝑐 ) ( )
for each xed 𝑝, where 𝐺 𝑝 = [𝑦 𝑝 ]𝐺 (𝑦). Multiply this by the asymptotics of 𝐹 (𝑛) (𝑌 𝑐 ), and we obtain

𝐹 𝑛,𝑝 ∼ 𝑛→∞ 𝐺 𝑝 Γ(-𝛽 0 ) • 𝑥 -𝑛 𝑐 • 𝑛 -𝛽 0 -1 ( )
for each xed 𝑝. And thanks to the rst asymptotics of ( ) and the Δ-analyticity of 𝐺 (𝑦), we have

𝐺 𝑝 ∼ 𝑝→∞ 𝛼 -1 2 𝜇 𝛽 0 C 𝐹 Γ(-𝛽 1 ) • 𝑌 -𝑝 𝑐 • 𝑝 -𝛽 1 -1 . ( )
Asymptotics of 𝐹 𝑛,𝑝 as 𝑛, 𝑝 → ∞ while 𝑛 ∼ 𝑣 • 𝑝 1/𝜃 . According to the Cauchy integral formula, we have

𝐹 𝑛,𝑝 = 1 2𝜋𝑖 2 ‹ 𝐹 (𝑥, 𝑦) 𝑥 𝑛+1 𝑦 𝑝+1 d𝑥 d𝑦 , ( )
where the integral is performed on the product of two small circles around the origin. Since 𝐹 is analytic in 𝑥 𝑐 × 𝑌 𝑐 and continuous on the boundary, we can deform the contour of integration to 𝜕 𝑥 𝑐 × 𝜕 𝑌 𝑐 . The contour 𝜕 1 can be decomposed into a circular part C := 𝜕 𝜖,𝛿 1 ∩ 𝜕D 1+𝜖 and a 𝑉 -shaped part V := 𝜕 1 \ C. For 𝑥 on the circular part 𝑥 𝑐 • C of its contour, we have

𝐹 (𝑥, 𝑦) 𝑥 𝑛+1 𝑦 𝑝+1 ≤ sup 𝜕 𝑥𝑐 ×𝜕 𝑌𝑐 |𝐹 | 𝑥 𝑛+1 𝑐 (1 + 𝜖) 𝑛+1 𝑌 𝑝+1 𝑐 = 𝑥 -𝑛 𝑐 𝑌 -𝑝 𝑐 • 𝑂 ((1 + 𝜖) -𝑛 ) . ( )
Similarly, when 𝑦 ∈ 𝑌 𝑐 • C, the integrand decays exponentially fast with respect to 𝑝 → ∞. It follows that

𝑥 𝑛 𝑐 𝑌 𝑝 𝑐 • 𝐹 𝑛,𝑝 = 1 2𝜋𝑖 2 ¨V×V 𝐹 (𝑥 𝑐 𝑢, 𝑌 𝑐 𝑣) 𝑢 𝑛+1 𝑣 𝑝+1 d𝑢 d𝑣 + 𝑂 ((1 + 𝜖) -𝑛 ) + 𝑂 (1 + 𝜖) -𝑝 ( )
when 𝑛, 𝑝 → ∞. Thanks to ( ), we have

1 2𝜋𝑖 2 ¨V×V 𝐹 (𝑥 𝑐 𝑢, 𝑌 𝑐 𝑣) 𝑢 𝑛+1 𝑣 𝑝+1 d𝑢 d𝑣 = 𝐼 reg + C 𝐹 • 𝐼 hom + 𝐼 rem , ( )
where 𝐼 reg , 𝐼 hom and 𝐼 rem are de ned by replacing 𝐹 (𝑥, 𝑦) in the integral on the left hand side by 𝐹 reg (𝑥, 𝑦), 𝐹 hom ((𝑠/𝜇) 𝜃 , 𝑡) and 𝑜 ( (𝑠 𝜃 , 𝑡) 𝛾 0 ), respectively. (Recall that 𝑠 := 1 -𝑥 𝑥 𝑐 and 𝑡 := 1 -𝑦 𝑌 𝑐 .) Since 𝑦 ↦ → 𝐹 reg (𝑥, 𝑦) is analytic in a neighborhood of 𝑌 𝑐 , one can deform the second component of the contour of integration of 𝐼 reg from V to C 𝑐 := 𝜕D 1+𝜖 \ C. Moreover, 𝐹 reg (𝑥 𝑐 𝑢, 𝑌 𝑐 𝑣) is bounded on V × C 𝑐 . So the same argument as before implies that 𝐼 reg = 𝑂 ((1 + 𝜖) -𝑝 ). We conclude that when 𝑛, 𝑝 → ∞ at any speed, we have

𝑥 𝑛 𝑐 𝑌 𝑝 𝑐 • 𝐹 𝑛,𝑝 = C 𝐹 • 𝐼 hom + 𝐼 rem + 𝑂 ((1 + 𝜖) -𝑛 ) + 𝑂 (1 + 𝜖) -𝑝 . ( ) Now assume 𝑛 ∼ 𝑣 • 𝑝 1/𝜃 . The change of variable 𝑢 = 1 -𝑠 maps V to {𝑠 ∈ 𝜕𝐾 𝛿 : |𝑠 | ≤ ε}, where ε = 𝑂 (𝜖). Therefore 𝐼 hom = 1 2𝜋𝑖 2 ¨(𝜕𝐾 𝛿 ) 2 𝐹 hom (𝑠/𝜇) 𝜃 , 𝑡 (1 -𝑠) 𝑛+1 (1 -𝑡) 𝑝+1 1 { |𝑠 | ≤ ε, |𝑡 | ≤ ε } d𝑠 d𝑡 . ( )
Using the fact that 𝐹 hom is homogenous of degree 𝛾 0 , we get after the rescaling 𝑠 ← 𝑠/𝑛 and 𝑡 ← 𝑡/𝑝 :

𝐼 hom = 1 𝑛 𝑝 1+𝛾 0 1 2𝜋𝑖 2 ¨(𝜕𝐾 𝛿 ) 2 𝐹 hom 𝑝 • (𝑠/(𝜇𝑛)) 𝜃 , 𝑡 (1 -𝑠/𝑛) 𝑛+1 (1 -𝑡/𝑝) 𝑝+1 1 { |𝑠 | ≤ ε𝑛, |𝑡 | ≤ ε𝑝 } d𝑠 d𝑡 . ( ) For 𝑠 ∈ 𝜕𝐾 𝛿 and |𝑠 | ≤ ε𝑛, we have -ℜ𝔢(𝑠) = |𝑠 | •sin 𝛿 ≤ ε sin 𝛿 •𝑛. Then, using the estimate log(1+𝑥) ≥ 𝑥 -𝑥 2 /2, one can show that | (1 -𝑠/𝑛) 𝑛+1 | ≥ |1 + -ℜ𝔢(𝑠) 𝑛 | 𝑛 ≥ exp(𝑐 1 • |𝑠 |) with 𝑐 1 = (1 -1 2 ε sin 𝛿) sin 𝛿.
The same bound holds for (1 -𝑡/𝑝) 𝑝+1 . Then it follows from the upper bound ( ) of 𝐻 𝛼 that there exists 𝑀 < ∞ such that

𝐹 hom 𝑝 • (𝑠/(𝜇𝑛)) 𝜃 , 𝑡 (1 -𝑠/𝑛) 𝑛+1 (1 -𝑡/𝑝) 𝑝+1 1 { |𝑠 | ≤ ε𝑛, |𝑡 | ≤ ε𝑝 } ≤ 𝑀 • |Λ𝑠 | 𝜃 + |𝑡 | 𝛾 0 𝑒 -𝑐 1 •( |𝑠 |+ |𝑡 |) ( )
for all 𝑛, 𝑝 such that

𝑝 1/𝜃 𝜇𝑛 ≤ Λ.
The right hand side of the abouve inequality is integrable on (𝜕𝐾 𝛿 ) 2 and independent of 𝑛, 𝑝. Thus by the dominanted convergence theorem, we have

𝑛 𝑝 1+𝛾 0 • 𝐼 hom 𝑛∼𝑣 •𝑝 1/𝜃 -------→ 𝑛,𝑝→∞ 1 2𝜋𝑖 2 ¨(𝜕𝐾 𝛿 ) 2 𝐹 hom 𝑠 𝜇𝑣 𝜃 , 𝑡 𝑒 𝑠+𝑡 d𝑠 d𝑡, ( )
which gives after simpli cation

𝐼 hom ∼ 𝜇 𝐼 𝛼 (𝜇𝑣) • 𝑝 -(𝛾 0 +1+1/𝜃 ) with 𝐼 𝛼 (𝜆) := 1 2𝜋𝑖 2 ¨(𝜕𝐾 𝛿 ) 2 𝐹 hom 𝑠 𝜃 , 𝑡 𝑒 𝜆𝑠+𝑡 d𝑠 d𝑡 .
( ) It is not hard to see that 𝐼 𝛼 (𝜆) is a well-de ned analytic function of 𝜆 for all 𝜆 > 0. (Recall that its expression is given without proof in the remark after Theorem .) By de nition, for all 𝑐 0 > 0, one can nd 𝜖 > 0 such that 𝑜 (𝑠 𝜃 , 𝑡) 𝛾 0 ≤ 𝑐 0 • (|𝑠 | 𝜃 + |𝑡 |) 𝛾 0 for all |𝑠 | ≤ ε and |𝑡 | ≤ ε. Then, using similar estimates as for 𝐼 hom , it is not hard to see that

|𝐼 rem | ≤ 𝑐 0 • 𝑀 ¨(𝜕𝐾 𝛿 ) 2 |𝑠 | 𝜃 + |𝑡 | 𝛾 0 𝑒 -𝑐 1 •( |𝑠 |+ |𝑡 |) d𝑠 d𝑡 • 𝑝 -(𝛾 0 +1+1/𝜃 ) . ( )
The integral is independent of 𝜖. The constant 𝑐 0 can be made arbitrarily small by taking smaller and smaller 𝜖.

It follows that 𝐼 rem = 𝑜 (𝑝 -(𝛾 0 +1+1/𝜃 ) ). Combining this estimate with ( ) and ( ), we obtain the bivariate asymptotics ( ) of 𝐹 𝑛,𝑝 when 𝑛, 𝑝 → ∞ and 𝑛 ∼ 𝑣 • 𝑝 1/𝜃 . This concludes the proof of Theorem .

Δ-analyticity of 𝐹 (𝑥, 𝑦)

The rest of this paper is devoted to the proof of the following Δ-analyticity result used in the proof Theorem .

Proposition . The function 𝐹 (𝑥, 𝑦) has an analytic continuation in some double Δ-domain 𝑥 𝑐 × 𝑌 𝑐 which is continuous on the boundary. And 𝑦 ↦ → 𝐹 (𝑥 𝑐 , 𝑦) and 𝑦 ↦ → 𝜕 𝑥 𝐹 (𝑥 𝑐 , 𝑦) are analytic in 𝑌 𝑐 .

We prove this result under the general assumptions speci ed in the introduction (in particular, we still restrict ourselves to the generic and the dilute phases). The proof comes in three steps, which are organized as follows: In Section . , we prove that 𝐹 (𝑥, 𝑦) is absolutely convergent on the double disk D 𝑥 𝑐 × D 𝑌 𝑐 , so in this sense (𝑥 𝑐 , 𝑌 𝑐 ) is indeed a dominant singularity of 𝐹 (𝑥, 𝑦). In Section . , we check that (𝑥 𝑐 , 𝑌 𝑐 ) is essentially the only dominant singularity, in the sense that 𝐹 (𝑥, 𝑦) is analytic everywhere on the boundary of D 𝑥 𝑐 × D 𝑌 𝑐 except when 𝑥 = 𝑥 𝑐 . This part relies crucially on Lemma which, in spite of its simple statement, has a quite long and technical proof. We postpone the proof of Lemma to Section . Section make use of some analysis results in Appendices A and B, which are organized separately because they are not speci c to the parking model, and is of independent interest. Finally, in Section . , we combine the conclusion of Section . with some asymptotic expansions from Section to construct the global analytic continuation of 𝐹 (𝑥, 𝑦) claimed in Proposition .

. Domain of convergence of 𝐹 (𝑥, 𝑦)

Recall from Section that Ŷ is a power series with nonnegative coe cients, and is the functional inverse of x. From the de nition of 𝑌 𝑐 , we see that Ŷ induces a homeomorphism from [0, 𝑥 𝑐 ] to [0, 𝑌 𝑐 ] that is analytic on [0, 𝑌 𝑐 ). In particular, the series Ŷ converges absolutely at 𝑥 𝑐 and Ŷ (𝑥 𝑐 ) = 𝑌 𝑐 .

Lemma (Domain of convergence of 𝐹 (𝑥, 𝑦)). The power series 𝐹 (𝑥, 𝑦) and 𝜕 𝑦 𝐹 (𝑥, 𝑦) are absolutely convergent on D 𝑥 𝑐 × D 𝑌 𝑐 , and 𝜕 𝑥 𝐹 (𝑥, 𝑦) is absolutely convergent on D 𝑥 𝑐 × D 𝑌 𝑐 .

Proof. Since 𝐹 (𝑥, 𝑦) has nonnegative coe cients, to prove the lemma it su ces to show that the series 𝜕 𝑦 𝐹 (𝑥, 𝑦) +𝑦 • 𝐹 (𝑥, 𝑦) converges absolutely at (𝑥 𝑐 , 𝑌 𝑐 ). Thanks to the parametrization of 𝐹 (𝑥, 𝑦) and Lemma ( ), we have

𝜕 𝑦 𝐹 ( x (𝑌 ), 𝑦) + 𝑦 • 𝐹 ( x (𝑌 ), 𝑦) = 𝜕 𝑦 F (𝑌 , 𝑦) + 𝑦 • F (𝑌 , 𝑦) = 1 2𝑦 • -𝜕 𝑦 𝑄 (𝑌 , 𝑦) 2(𝑌 -𝑦) √ 𝑞(𝑌 , 𝑦) . ( )
Using the identities in Lemma ( ), it is not hard to see that 𝑞(𝑌 , 𝑌 ) = 𝜙 (𝑌 ) and -

𝜕 𝑦 𝑄 (𝑌 ,𝑦) 2(𝑌 -𝑦) → 𝜙 (𝑌 ) as 𝑦 → 𝑌 . Therefore 𝜕 𝑦 𝐹 ( x (𝑌 ), 𝑌 ) + 𝑌 • 𝐹 ( x (𝑌 ), 𝑌 ) = 1 2𝑌 • √ 𝜙 (𝑌 ) . ( )
By taking 𝑌 = Ŷ (𝑥) in the above equation, we obtain that 𝑓 (𝑥)

:= 𝜕 𝑦 𝐹 (𝑥, Ŷ (𝑥)) + Ŷ (𝑥) • 𝐹 (𝑥, Ŷ (𝑥)) = √ 𝜙 ( Ŷ (𝑥))
2 Ŷ (𝑥) . It is clear that the series 𝑓 (𝑥) also has nonnegative coe cients.

By Lemma ( ), we have 𝜙 (𝑌 ) = (𝐵(𝑌 ) + 𝑌 𝐵 (𝑌 )) • x (𝑌 ), which is analytic and strictly positive on [0, 𝑌 𝑐 ). Therefore 𝑓 (𝑥) has an analytic continuation on [0, 𝑥 𝑐 ) with a nite limit at 𝑥 - 𝑐 . It follows that it converges absolutely at 𝑥 𝑐 . This implies that the double series 𝜕 𝑦 𝐹 (𝑥, 𝑦) + 𝑦 • 𝐹 (𝑥, 𝑦) converges absolutely at (𝑥 𝑐 , 𝑌 𝑐 ), and completes the proof of the lemma.

. Uniqueness of dominant singularity of 𝐹 (𝑥, 𝑦)

By convention, we say that a function is holomorphic (resp. meromorphic) on an arbitrary set 𝐷 ⊆ C 𝑛 if it is continuous on 𝐷 and holomorphic (resp. meromorphic) in the interior of 𝐷. A function is a conformal bijection from 𝐷 to 𝐷 if it is bijective and holomorphic on 𝐷, and its inverse is holomorphic on 𝐷 .

As the series Ŷ has nonnegative coe cients and converges at 𝑥 𝑐 , it de nes a holomorphic function on

D 𝑥 𝑐 . Let V = Ŷ (D 𝑥 𝑐 ) and V = Ŷ (D 𝑥 𝑐 ). It is a simple exercise to show that V is open and V is indeed the closure of V. π 2 θ ( π 2 +δ )θ δ Y c 0 δ x c 0 x Y , ∂D x c ∂ x c ∂D Y c ∂V ∂ ∂ Y c Ŷ x Figure :
The boundaries of various domains. The angles indicate the directions of their half tangents at 𝑥 𝑐 (for 𝑥 𝑐 ) or at 𝑌 𝑐 (for V, , 𝑌 𝑐 ). The function x induces a conformal bijection from V to D 𝑥 𝑐 , and a conformal bijection from ≡ 𝜖,𝛿 to 𝑥 𝑐 ≡ 𝜖,𝛿 𝑥 𝑐 . Its inverse is Ŷ .

Figure depicts the shape of V and its relation to various other domains, some of which will be de ned later.

The set V is a natural domain for the variable 𝑌 , in the following sense:

Lemma (Analyticity w.r.t. 𝑌 ∈ V). We have V ⊆ D 𝑌 𝑐 , and x induces a conformal bijection from V to D 𝑥 𝑐 . Moreover, the function 𝜙 is holomorphic on V, and 𝑞(𝑌 , 𝑦) is holomorphic on V × D 𝑌 𝑐 .

Proof. The series Ŷ has nonnegative coe cients.

Hence | Ŷ (𝑥)| ≤ Ŷ (𝑥 𝑐 ) = 𝑌 𝑐 for all 𝑥 ∈ D 𝑥 𝑐 , that is, V ⊆ D 𝑌 𝑐 .
By Assumption ( * ), 𝐵(𝑦) has an analytic continuation on D 𝑌 𝑐 that is 𝐶 2 -continuous at 𝑌 𝑐 . Since x (𝑌 ) is a rational function of 𝑌 , 𝐵(𝑌 ) and 𝐵 (𝑌 ), it is well-de ned and meromorphic in D 𝑌 𝑐 . We have x ( Ŷ (𝑥)) = 𝑥 for all 𝑥 in some neighborhood of 0. Thanks to the uniqueness of analytic continuation, the same identity holds for all 𝑥 ∈ D 𝑥 𝑐 . It follows that Ŷ is injective on D 𝑥 𝑐 , hence de nes a bijection from D 𝑥 𝑐 to V. This implies that its inverse x has no pole on V, and therefore induces a conformal bijection from V to D 𝑥 𝑐 .

Recall that x (𝑌 ) = 𝑌 𝐵 (𝑌 ) (𝐵 (𝑌 )+𝑌 𝐵 (𝑌 )) 2 and 𝜙 (𝑌 ) =𝑌 𝐵 (𝑌 )-𝑌 𝐵 (𝑌 ) 𝐵 (𝑌 )+𝑌 𝐵 (𝑌 ) . Like x, the function 𝜙 is also meromorphic on D 𝑌 𝑐 ⊃ V. Assume that it has a pole 𝑌 * ∈ V, that is, 𝐵(𝑌 * ) + 𝑌 * 𝐵 (𝑌 * ) = 0. Then 𝑌 * is at least a double zero of (𝐵(𝑌 ) + 𝑌 𝐵 (𝑌 )) 2 . Since x (𝑌 * ) is nite, we must have 𝐵(𝑌 * ) = 0. This implies that 𝑌 * ≠ 0 and therefore 𝑌 * is a zero of the same multiplicity of 𝐵(𝑌 ) -𝑌 𝐵 (𝑌 ) and of 𝐵(𝑌 ) + 𝑌 𝐵 (𝑌 ). It follows that 𝜙 is nite at 𝑌 * . This contradicts the assumption that 𝑌 * is a pole of 𝜙. Hence 𝜙 is holomorphic on V.

Recall that 𝑄 (𝑌 , 𝑦) = (𝜙 (𝑌 ) + 𝑦) 2 -4𝑦𝐵(𝑦) • x (𝑌 ) and 𝑞(𝑌 , 𝑦) = 𝑄 (𝑌 ,𝑦)

(𝑌 -𝑦) 2 for 𝑌 ≠ 𝑦. Since x and 𝜙 are holomorphic on V and 𝐵 is holomrphic on D 𝑌 𝑐 , the function 𝑞 is holomorphic on V × D 𝑌 𝑐 away from the diagonal. For (𝑦, 𝑦) ≠ (𝑌 𝑐 , 𝑌 𝑐 ) on the diagonal of V × D 𝑌 𝑐 , the function 𝑄 is analytic in a neighborhood of (𝑦, 𝑦). This is true even if (𝑦, 𝑦) is on the boundary of V × D 𝑌 𝑐 , because by assumption 𝐵 is analytic in a Δ-domain 𝑌 𝑐 . Then the integral formula of Lemma ( ) shows that 𝑞 is also analytic at (𝑦, 𝑦). Finally, by Lemma we have

𝑞(𝑌, 𝑦) → 𝑞(𝑌 𝑐 , 𝑌 𝑐 ) = 0 when (𝑌 , 𝑦) → (𝑌 𝑐 , 𝑌 𝑐 ) in V × D 𝑌 𝑐 . Hence 𝑞 is continuous at (𝑌 𝑐 , 𝑌 𝑐 ) in V × D 𝑌 𝑐 .
This shows that 𝑞(𝑌, 𝑦) is analytic in the interior, and continuous on the boundary of V × D 𝑌 𝑐 .

The proof of the following lemma is rather long and technical, and is deferred to Section .

Lemma . We have V \ {𝑌 𝑐 } ⊆ D 𝑌 𝑐 . Moreover, x (𝑌 ) ≠ 0 on V \ {𝑌 𝑐 } and 𝑞(𝑌 , 𝑦) ≠ 0 on V × D 𝑌 𝑐 \ {(𝑌 𝑐 , 𝑌 𝑐 )}.
It is not hard to deduce from Lemma that 𝐹 (𝑥, 𝑦) is analytic everywhere on the boundary of D 𝑥 𝑐 × D 𝑌 𝑐 except when 𝑥 = 𝑥 𝑐 . But we shall not insist on this here, since the next subsection will provide stronger results.

.

Analytic continuation of 𝐹 (𝑥, 𝑦) to a double Δ-domain

The rst part of Lemma tells us that x is analytic and locally invertible at every point of V except 𝑌 𝑐 . This allows us to analytically extend its inverse Ŷ to a neighborhood of each point on the circle 𝜕D 𝑥 𝑐 except 𝑥 𝑐 .

The following lemma says that we can also extend Ŷ analytically to a neighborhood of 𝑥 𝑐 in some Δ-domain 𝑥 𝑐 , and thus extend the conformal bijection x : V → D 𝑥 𝑐 to a conformal bijection onto 𝑥 𝑐 .

Lemma (De nition of 𝜖,𝛿 ). Ŷ extends to a holomorphic function on some closed Δ-domain 𝑥 𝑐 ≡ 𝜖,𝛿 𝑥 𝑐 . Let ≡ 𝜖,𝛿 = Ŷ ( 𝜖,𝛿 𝑥 𝑐 ). Then for all 𝜖, 𝛿 > 0 small enough, x induces a conformal bijection from to 𝑥 𝑐 .

Proof. When 𝑌 → 𝑌 𝑐 in D 𝑌 𝑐 , we have by Lemma :

𝑥 𝑐 -x (𝑌 ) ∼ 𝑐 • (𝑌 𝑐 -𝑌 ) 𝛼-1 and x (𝑌 ) ∼ 𝑐 • (𝛼 -1) • (𝑌 𝑐 -𝑌 ) 𝛼-2 ( )
for some 𝑐 > 0. Under the change of variables 𝑧 = (𝑌 𝑐 -𝑌 ) 𝛼-1 and ŵ (𝑧) = 𝑥 𝑐x (𝑌 ), the above asymptotics imply that ŵ (𝑧) → 0 and ŵ (𝑧) → 𝑐 as 𝑧 → 0 in the cone 𝐾 = {𝑟𝑒 𝑖𝜏 | 𝑟 > 0, |𝜏 | < 𝜏 0 }, for any 𝜏 0 < (𝛼 -1) 𝜋 2 . If ŵ were de ned in a neighborhood of 0, then the inverse function theorem would imply that it has a local inverse ẑ such that ẑ (𝑤) → 0 and ẑ (𝑤) → 𝑐 -1 when 𝑤 → 0. In Appendix B, we will show that this is still true when ŵ is only de ned in a cone. More precisely, Lemma implies that for any 𝐾 = {𝑟𝑒 𝑖𝜏 | 𝑟 > 0, |𝜏 | < 𝜏 0 } with 𝜏 0 < 𝜏 0 , there exist a neighborhood V of 0 and an analytic function ẑ : V∩𝐾 → 𝐾, such that ŵ ( ẑ (𝑤)) = 𝑤 for all 𝑤 ∈ V ∩ 𝐾 . By going back through the change of variables 𝑧 = (𝑌 𝑐 -𝑌 ) 𝛼-1 and 𝑤 = 𝑥 𝑐 -𝑥, we see that Ŷ

(𝑥) = 𝑌 𝑐 -( ẑ (𝑥 𝑐 -𝑥)) 1 𝛼 -1 is an analytic continuation of Ŷ on {𝑥 : 𝑥 𝑐 -𝑥 ∈ V ∩ 𝐾 }. Since 𝛼 > 2, we can choose 𝜏 0 > 𝜏 0 > 𝜋 2 .
Then the set {𝑥 : 𝑥 𝑐 -𝑥 ∈ V ∩ 𝐾 } can be written as U 𝑥 𝑐 ∩ 𝜖,𝛿 𝑥 𝑐 , where 𝜖 > 0 is arbitrary, 𝛿 = 𝜏 0 -𝜋 2 , and U 𝑥 𝑐 is a su ciently small neighborhood of 𝑥 𝑐 . We have constructed an analytic continuation of Ŷ on U 𝑥 𝑐 ∩ 𝜖,𝛿 𝑥 𝑐 . On the other hand, by the remark preceding Lemma , Ŷ also has an analytic continuation in a neighborhood U 𝑥 of each point 𝑥 ∈ D 𝑥 𝑐 \ {𝑥 𝑐 }. When 𝜖 > 0 is su ciently small, we have 𝜖,𝛿 𝑥 𝑐 ⊆ 𝑥 ∈D 𝑥𝑐 U 𝑥 . Then Ŷ has an analytic continuation on 𝜖,𝛿 𝑥 𝑐 . Moreover, Lemma used in the previous paragraph also ensures that ẑ (𝑤) → 0 as 𝑤 → 0 in 𝐾 , or equivalently Ŷ (𝑥) → 𝑌 𝑐 as 𝑥 → 𝑥 𝑐 in 𝜖,𝛿 𝑥 𝑐 . Thus by decreasing slightly both 𝜖 and 𝛿, we may assume that Ŷ is continuous on the boundary of 𝜖,𝛿 𝑥 𝑐 , hence holomorphic on 𝜖,𝛿 𝑥 𝑐 in our terminology. By the uniqueness of analytic continuation, we have x ( Ŷ (𝑥)) = 𝑥 for all 𝑥 ∈ 𝜖,𝛿 𝑥 𝑐 . It follows that x is injective on 𝜖,𝛿 := Ŷ ( 𝜖,𝛿 𝑥 𝑐 ) and induces a conformal bijection from 𝜖,𝛿 to 𝜖,𝛿 𝑥 𝑐 .

In the proof of Lemma , we deduced the holomorphicity of 𝜙 and 𝑞(𝑌 , 𝑦) on their respective domains from the holomorphicity of x on V. Now we know that x is holomorphic on the larger domain . The exact same argument can be used to show the following corollary. We leave the reader to check the details.

Corollary . For 𝜖, 𝛿 > 0 small enough, 𝜙 is holomorphic on and 𝑞(𝑌 , 𝑦) is holomorphic on × 𝑌 𝑐 .

The second part of Lemma asserts that 𝑞(𝑌 , 𝑦) has no zero on V × D 𝑌 𝑐 except (𝑌 𝑐 , 𝑌 𝑐 ). By continuity, for any neighborhood U of (𝑌 𝑐 , 𝑌 𝑐 ), there exists 𝜖 > 0 such that 𝑞(𝑌 , 𝑦) has no zero on × 𝑌 𝑐 \ U neither. The following lemma states that for 𝜖, 𝛿 > 0 small enough, (𝑌 𝑐 , 𝑌 𝑐 ) is actually the only zero.

Lemma (Analyticity of √ 𝑞(𝑌 , 𝑦)). For 𝜖, 𝛿 > 0 small enough, (𝑌 𝑐 , 𝑌 𝑐 ) is the only zero of 𝑞(𝑌 , 𝑦) on × 𝑌 𝑐 .

Proof. Recall from Section that 𝐾 𝛿 is the cone

{𝑧 : | arg(𝑧)| < 𝜋 2 + 𝛿 } and 𝜃 = 1 𝛼-1 .
As shown in Figure , the boundaries of 𝑌 𝑐 and each have two half tangents at 𝑌 𝑐 forming an angle of 2(𝜋 + 𝛿) and 2(𝜋 + 𝛿)𝜃 , respectively. It follows that for any 𝛿 > 𝛿, there exists a neighborhood U 0 of (𝑌 𝑐 , 𝑌 𝑐 ) such that

× 𝑌 𝑐 ∩ U 0 ⊆ (𝑌 , 𝑦) 1 - 𝑌 𝑌 𝑐 , 1 - 𝑦 𝑌 𝑐 ∈ 𝐾 𝜃 𝛿 × 𝐾 𝛿 . ( )
Therefore by Lemma , for 𝛿 and 𝛿 small enough, we have 𝑞(𝑌 , 𝑦)

∼ C 𝑞 •𝐻 𝛼 (1-𝑌 𝑌 𝑐 , 1- 𝑦 𝑌 𝑐 ) when (𝑌 , 𝑦) → (𝑌 𝑐 , 𝑌 𝑐 ) in × 𝑌 𝑐 , where C 𝑞 > 0 and 𝐻 𝛼 (𝑆, 𝑡) = 𝑡 𝛼 -𝑆 𝛼 -𝛼𝑆 𝛼 -1 (𝑡 -𝑆) (𝑡 -𝑆) 2
. The lower bound in Lemma implies that (𝑆, 𝑡) = (0, 0) is the only zero of 𝐻 𝛼 (𝑆, 𝑡) in 𝐾 𝜃 𝛿 × 𝐾 𝛿 . It follows that there exists a neighborhood U of (𝑌 𝑐 , 𝑌 𝑐 ), such that (𝑌 𝑐 , 𝑌 𝑐 ) is the only zero of 𝑞 in × 𝑌 𝑐 ∩ U. On the other hand, by the remark preceding Lemma , there exists 𝜖 > 0 such that 𝑞 has no zero on × 𝑌 𝑐 ∩ U. It follows that for 𝜖, 𝛿 > 0 small enough, (𝑌 𝑐 , 𝑌 𝑐 ) is the only zero of 𝑞 in × 𝑌 𝑐 .

Proof of Proposition . By Corollary and Lemma , the function F (𝑌 , 𝑦)

= 1 2 + 1 2𝑦 (𝑌 -𝑦) √ 𝑞(𝑌 , 𝑦) -𝜙 (𝑌 )
is holomorphic on × 𝑌 𝑐 . (The factor 𝑦 in the denominator is not a problem, since we know that F (𝑌 , 𝑦) de nes a formal power series in 𝑦.) And Ŷ , the inverse of x, induces a holomorphic function from 𝑥 𝑐 onto according to Lemma . It follows that 𝐹 (𝑥, 𝑦) = F ( Ŷ (𝑥), 𝑦) is holomorphic on (i.e. analytic in the interior and continuous on the boundary of) 𝑥 𝑐 × 𝑌 𝑐 .

When 𝑥 = 𝑥 𝑐 , we have 𝐹 (𝑥 𝑐 , 𝑦) = F (𝑌 𝑐 , 𝑦) and 𝜕 𝑥 𝐹 (𝑥 𝑐 , 𝑦) = / 𝜕 𝑥 F (𝑌 𝑐 , 𝑦), where / 𝜕 𝑥 F (𝑌 , 𝑦) is given by ( ). Then, thanks to the Δ-analyticity of 𝐵(𝑦) and the fact that 𝑞(𝑌 𝑐 , 𝑦) ≠ 0 for all 𝑦 ∈ 𝑌 𝑐 , both 𝑦 ↦ → 𝐹 (𝑥 𝑐 , 𝑦) and 𝑦 ↦ → 𝜕 𝑥 𝐹 (𝑥 𝑐 , 𝑦) are analytic on 𝑌 𝑐 .

Proof of Lemma

. The inclusion V \ {𝑌 𝑐 } ⊆ D 𝑌 𝑐 Recall that | Ŷ (𝑥)| ≤ Ŷ (𝑥 𝑐 ) = 𝑌 𝑐
for all 𝑥 ∈ D 𝑥 𝑐 because the series Ŷ has nonnegative coe cients. To prove the inclusion V \ {𝑌 𝑐 } ⊆ D 𝑌 𝑐 , it su ces to show that the inequality is strict for all |𝑥 | ≤ 𝑥 𝑐 di erent from 𝑥 𝑐 . With a bit of thought, one sees that this is true if and only if the series Ŷ is aperiodic.

Recall that supp 𝐵 = {𝑙 ∈ N : 𝑏 𝑙 ≠ 0} denotes the support of the coe cients of 𝐵. Since 𝑏 0 ≠ 0, the series 𝐵 is aperiodic if and only if supp 𝐵 𝑚Z for all 𝑚 ≥ 2. Similarly, since Ŷ (0) = 0 and Ŷ (0) = 1

x (0) ≠ 0, the series Ŷ is aperiodic if and only if supp (𝑥 -1 Ŷ ) 𝑚Z for all 𝑚 ≥ 2. The following simple lemma provides a method to relate the (a)periodicity of one formal power series to another. We will use it to deduce the aperiodicity of Ŷ (𝑥) from that of 𝐵(𝑌 ).

Lemma

(Heredity of periodicity).

If Φ : Ω ⊆ C[[𝑥]] → C[[𝑥]
] is a mapping between formal power series such that Φ(𝑆 (𝜔𝑥)) = Φ(𝑆) (𝜔𝑥) for all roots of unity 𝜔 ∈ {𝑒 𝑖2𝜋𝑞 | 𝑞 ∈ Q}, then for all integer 𝑚 ≥ 1, supp 𝑆 ⊆ 𝑚Z implies supp Φ(𝑆) ⊆ 𝑚Z.

Proof. Observe that supp 𝑆 ⊆ 𝑚Z if and only if 𝑆 (𝑒 𝑖 2𝜋 𝑚 𝑥) = 𝑆 (𝑥). The property of the mapping Φ ensures that if

𝑆 (𝑒 𝑖 2𝜋 𝑚 𝑥) = 𝑆 (𝑥), then Φ(𝑆) (𝑒 𝑖 2𝜋 𝑚 𝑥) = Φ(𝑆 (𝑒 𝑖 2𝜋 𝑚 𝑥)) = Φ(𝑆) (𝑥).
Hence supp 𝑆 ⊆𝑚Z implies supp Φ(𝑆) ⊆𝑚Z.

Lemma The power series Ŷ (𝑥) is aperiodic, and therefore V \ {𝑌 𝑐 } ⊆ D 𝑌 𝑐 .

Proof. The fact that x (𝑌 ) and Ŷ (𝑥) are inverse of each other can be written as Ŷ (𝑥) = 𝑥 • 𝑊 ( Ŷ (𝑥)), where

𝑊 (𝑌 ) = 𝑌 x (𝑌 ) = 𝐵(𝑌 ) • 1 + 𝑌 𝐵 (𝑌 ) 𝐵 (𝑌 )

2

. The Lagrange inversion formula states that [𝑥 𝑛 ] Ŷ (𝑥) = 1 𝑛 [𝑌 𝑛-1 ]𝑊 (𝑌 ) for all 𝑛 ≥ 1. Therefore the series Ŷ (𝑥) is aperiodic if and only if 𝑊 (𝑌 ) is. Moreover, since 𝑊 (0) = 𝐵(0) = 𝑏 0 is nonzero by assumption, the series 𝑊 (𝑌 ) is aperiodic if and only if supp𝑊 𝑚Z for all 𝑚 ≥ 2.

We know that 𝐵(𝑌 ) is aperiodic. In particular, supp 𝐵 𝑚Z for all 𝑚 ≥ 2. Hence to prove that 𝑊 (𝑌 ) is also aperiodic, it su ces to show that Φ : 𝑊 (𝑌 ) ↦ → 𝐵(𝑌 ) is a well-de ned mapping that satis es the assumption of Lemma . The mapping Φ is well-de ned if the relation between 𝐵(𝑌 ) and 𝑊 (𝑌 ), which can be written as

𝐵(𝑌 ) = 𝑊 (𝑌 ) • 1 + 𝑌 𝐵 (𝑌 ) 𝐵(𝑌 ) -2 , ( )
uniquely determines the coe cients of 𝐵(𝑌 ) for any given 𝑊 (𝑌 ). Let us prove this by induction: Write 𝑊 (𝑌 ) = 𝑛 ≥0 𝑤 𝑛 𝑌 𝑛 and 𝐵(𝑌 ) = 𝑛 ≥0 𝑏 𝑛 𝑌 𝑛 . We see easily that 𝑏 0 = 𝑤 0 . For 𝑛 ≥ 1, Equation ( ) gives:

𝑏 𝑛 = 𝑛 ∑ 𝑚=0 𝑤 𝑛-𝑚 • [𝑌 𝑚 ] 1 + 𝑌 𝐵 (𝑌 ) 𝐵(𝑌 ) -2 = 𝑛 ∑ 𝑚=0 𝑤 𝑛-𝑚 ∞ ∑ 𝑘=0 (-1) 𝑘 (𝑘 + 1) • [𝑌 𝑚 ] 𝑌 𝐵 (𝑌 ) 𝐵(𝑌 ) 𝑘 . ( )
Since 𝑌 𝐵 (𝑌 ) 𝐵 (𝑌 ) = 𝑌 𝑏 1 +•••+𝑛𝑏 𝑛 𝑌 𝑛-1 +••• 𝑏 0 +𝑏 1 𝑌 +•••+ 𝑏 𝑛 𝑌 𝑛 +••• and 𝑏 0 ≠ 0, it is not hard to see that the coe cient [𝑌 𝑚 ] 𝑌 𝐵 (𝑌 ) 𝐵 (𝑌 )

𝑘

in the double sum is a function of (𝑏 0 , . . . , 𝑏 𝑛-1 ) unless 𝑚 = 𝑛 and 𝑘 = 1. It follows that 𝑏 𝑛 can be written as

𝑏 𝑛 = 𝑓 (𝑏 0 , . . . , 𝑏 𝑛-1 ; 𝑤 0 , . . . , 𝑤 𝑛 ) -2𝑤 0 • [𝑌 𝑛 ] 𝑌 𝐵 (𝑌 ) 𝐵(𝑌 ) . ( )
Since [𝑌 𝑛 ] 𝑌 𝐵 (𝑌 ) 𝐵 (𝑌 ) = 𝑛𝑏 𝑛 𝑏 0 and 𝑤 0 = 𝑏 0 , the above formula gives (1 + 2𝑛)𝑏 𝑛 = 𝑓 (𝑏 0 , . . . , 𝑏 𝑛-1 ; 𝑤 0 , . . . , 𝑤 𝑛 ). By induction, this implies that all the coe cients 𝑏 𝑛 are determined by 𝑊 (𝑌 ), i.e. the mapping Φ is well-de ned. By repalcing 𝐵(𝑌 ) with 𝐵(𝜔𝑊 ) in the de nition of 𝑊 (𝑌 ), we see that Φ(𝑊 (𝜔𝑌 )) = 𝐵(𝜔𝑌 ) = Φ(𝑊 ) (𝜔𝑥) for any 𝜔 ∈ C \ {0}. This shows that Φ satis es the assumption of Lemma . It follows that the series 𝑊 , and hence Ŷ , is aperiodic. As explained at the beginning of Section . , this implies that V \ {𝑌 𝑐 } ⊆ D 𝑌 𝑐 .

.

x has no critical point in V \ {𝑌 𝑐 }

We have seen in Lemma that x induces a conformal bijection from V to D 𝑥 𝑐 , so it has no critical point in the interior V. In this subsection, we check that the same is true on the boundary 𝜕V except at 𝑌 𝑐 . We will use a variational method which provides additional equations on the critical points of x on 𝜕V by considering perturbations of the parameters 𝑏 𝑘 = [𝑦 𝑘 ]𝐵(𝑦). This variational method uses very little information on the speci c function x and applies in a much more general setting in analytic combinatorics. For this reason, we will discuss it in full detail in Appendix A. The method itself is summarized as Proposition .

We highlight the fact that this variational method can provide an additional equation by perturbing 𝑏 𝑘 only if 𝑏 𝑘 > 0. When |supp 𝐵| = ∞, we obtain an in nite sequence of equations. It turns out that the asymptotics of these equations as 𝑘 → ∞ is quite simple, and the proof of Lemma below make use of this asymptotics. When |supp 𝐵| < ∞, our method provides only nitely many equation. While there is still in theory enough equations for eliminating the critical points of x on 𝜕V \ {𝑌 𝑐 } (see Remark for a detailed count), we did not nd a proof that works in general (we veri ed that Lemma remains true when supp 𝐵 = {0, 1, 2}, {0, 1, 3} and {0, 2, 3}). This is why we assumed in Assumption ( * ) that |supp 𝐵| = ∞.

Lemma . x (𝑌 ) ≠ 0 for all 𝑌 ∈ V \ {𝑌 𝑐 }.

Proof. Assume that x has a critical point 𝑌 * ∈ V \ {𝑌 𝑐 }. Fix an integer 𝑘 such that 𝑏 𝑘 ≡ [𝑦 𝑘 ]𝐵(𝑦) > 0, and consider a perturbation 𝜀 to the weight 𝑏 𝑘 . The perturbed model has a weight generating function 𝐵(𝑦, 𝜀) = 𝐵(𝑦) + 𝜀𝑦 𝑘 . Let x (𝑌 , 𝜀), 𝑥 𝑐 (𝜀), 𝑌 𝑐 (𝜀) and V(𝜀) denote the perturbed versions of x (𝑌 ), 𝑥 𝑐 , 𝑌 𝑐 and V, respectively.

For all 𝜀 ∈ I := (-𝑏 𝑘 , 𝑏 𝑘 ), the perturbed weight sequence (𝑏 𝑙 + 𝜀𝛿 𝑘,𝑙 ) 𝑙 ≥0 remains nonnegative and satis es the same assumptions (in particular, Assumption ( * )) as the non-perturbed one. Hence we can apply Lemma to conclude that x ( • , 𝜀) induces a conformal bijection from V(𝜀) to D 𝑥 𝑐 (𝜀) . Notice that x (𝑌 , 𝜀) is meromorphic in 𝑌 ∈ D 𝑥 𝑐 , rational in 𝜀, adn nite at (𝑌 , 𝜀) = (𝑌 * , 0). Hence it is analytic in an open neighborhood U ⊆ C × I of (𝑌 * , 0). Without loss of generality, assume that {(𝑌 , 𝜀) | 𝑌 ∈ V(𝜀), 𝜀 ∈ I} ⊆ U. Then, one can check that x (𝑌, 𝜀), 𝑥 𝑐 (𝜀) and V(𝜀) satisfy the conditions of Proposition (see also Remark ), provided that 𝑥 𝑐 (𝜀) is di erentiable at 0.

Let us show that 𝑥 𝑐 (𝜀) is indeed di erentiable at 0: By de nition, 𝑥 𝑐 (𝜀) = x (𝑌 𝑐 (𝜀), 𝜀), where 𝑌 𝑐 (𝜀) < 𝜌 is a critical point of x ( • , 𝜀) in the generic phase, and 𝑌 𝑐 (𝜀) = 𝜌 in the non-generic phase. Recall the characterization of the phases from Proposition .

• If the weight generating function 𝐵( • , 0) is in the generic phase, then so is 𝐵( • , 𝜀) for all 𝜀 close to zero.

In this case, 𝑌 𝑐 (𝜀) is a critical point of x ( • , 𝜀), which is analytic in a neighborhood of 𝑌 𝑐 (𝜀). Hence we can apply Lemma , which implies that 𝑥 𝑐 (𝜀) = x (𝑌 𝑐 (𝜀), 𝜀) is di erentiable at 0, with 𝑥 𝑐 (0) = 𝜕 𝜀 x (𝑌 𝑐 (0), 0).

• If 𝐵( • , 0) is in the non-generic dilute phase, then x ( • , 0) is not analytic at 𝑌 𝑐 (0) = 𝜌, so Lemma no longer applies. But the proof of Lemma can be adapted as follows: The functions x (𝑌 , 𝜀) and 𝜕 𝑌 x (𝑌 , 𝜀), though not analytic, are still 𝐶 1 at (𝑌 𝑐 (0), 0), in particular, we have

x (𝑌 , 𝜀) = x (𝑌 , 0) + 𝜕 𝜀 x (𝑌 𝑐 (0), 0) • 𝜀 + 𝑜 (𝜀) and 𝜕 𝑌 x (𝑌 , 𝜀) = 𝜕 𝑌 x (𝑌 , 0) + 𝑂 (𝜀) ( ) as (𝑌, 𝜀) → (𝑌 𝑐 (0), 0). Let I 0 be the set of values of 𝜀 for which the perturbed model is in the generic phase.

For 𝜀 ∈ I \ I 0 , the value 𝑌 𝑐 (𝜀) = 𝜌 is independent of 𝜀. It follows that x (𝑌 𝑐 (𝜀),𝜀)x (𝑌 𝑐 (0),0)

𝜀 → 𝜕 𝜀 x (𝑌 𝑐 (0), 0)
when 𝜀 → 0 in I\I 0 . For 𝜀 ∈ I 0 , we have 𝜕 𝑌 x (𝑌 𝑐 (𝜀), 𝜀) = 0, hence the second expansion in ( ) implies that 𝜕 𝑌 x (𝑌 𝑐 (𝜀), 0) = 𝑂 (𝜀). But from the asymptotic expansion of x (𝑌 , 0) and 𝜕 𝑌 x (𝑌 , 0) in Lemma , we can see that x (𝑌 𝑐 (0), 0)x (𝑌 , 0) = 𝑜 (𝜕 𝑌 x (𝑌 , 0)). Therefore we have x (𝑌 𝑐 (0), 0)x (𝑌 , 0) = 𝑜 (𝜀). Plugging this into the rst expansion in ( ), we obtain that x (𝑌 𝑐 (𝜀), 𝜀) = x (𝑌 𝑐 (0), 0) + 𝜕 𝜀 x (𝑌 𝑐 (0), 0) • 𝜀 + 𝑜 (𝜀), that is, x (𝑌 𝑐 (𝜀),𝜀)x (𝑌 𝑐 (0),0) 𝜀 → 𝜕 𝜀 x (𝑌 𝑐 (0), 0) when 𝜀 → 0 in I 0 as well. It follows that 𝑥 𝑐 (𝜀) = x (𝑌 𝑐 (𝜀), 𝜀) is di erentiable at 0 and we have 𝑥 𝑐 (0) = 𝜕 𝜀 x (𝑌 𝑐 (0), 0) as well.

We conclude that 𝑥 𝑐 (𝜀) is indeed di erentiable at 0, and we always have 𝑥 𝑐 (0) = 𝜕 𝜀 x (𝑌 𝑐 (0), 0). (This is also obviously true in the dense phase, though we do not need this fact here.) Then, Proposition states that 𝑌 * must satisfy ℜ𝔢 𝜕 𝜀 x (𝑌 * ,0)

x (𝑌 * ,0) = 𝜕 𝜀 x (𝑌 𝑐 ,0)

x (𝑌 𝑐 ,0) for every 𝑘 such that 𝑏 𝑘 > 0. A straightforward computation gives the explicit equation

ℜ𝔢 𝑌 𝑘 * 𝐵(𝑌 * ) 2𝑘 + 1 -𝜓 (𝑌 * ) 1 + 𝜓 (𝑌 * ) = 𝑌 𝑘 𝑐 𝐵(𝑌 𝑐 ) 2𝑘 + 1 -𝜓 (𝑌 𝑐 ) 1 + 𝜓 (𝑌 𝑐 )
, where 𝜓 (𝑌 ) := 𝑌 𝐵 (𝑌 ) 𝐵(𝑌 )

.

( )

In particular, we have that

𝑌 𝑘 * 𝐵(𝑌 * ) 2𝑘 + 1 -𝜓 (𝑌 * ) 1 + 𝜓 (𝑌 * ) ≥ 𝑌 𝑘 𝑐 𝐵(𝑌 𝑐 ) 2𝑘 + 1 -𝜓 (𝑌 𝑐 ) 1 + 𝜓 (𝑌 𝑐 ) ( )
By Assumption ( * ), there are in nitely many 𝑘 such that 𝑏 𝑘 > 0. Hence we can take the limit 𝑘 → ∞ in the above inequality, which implies that |𝑌 * | ≥ 𝑌 𝑐 . But according to the previous subsection, we have |𝑌 * | < 𝑌 𝑐 for all 𝑌 * ∈ V \ {𝑌 𝑐 }. Therefore x cannot have a critical point in V \ {𝑌 𝑐 }.

Before moving on, let us register a useful fact whose proof uses a similar variational argument as Lemma .

Lemma . 𝐵(𝑌 ) ≠ 0 and / 𝜕 𝑥 𝜙 (𝑌 ) ≡ 𝐵(𝑌 ) + 𝑌 𝐵 (𝑌 ) ≠ 0 for all 𝑌 ∈ V.

Proof. Assume that 𝐵(𝑌 * ) = 0 for some 𝑌 * ∈ V. Since 𝐵(0) > 0, 𝑌 * ≠ 0. Using the fact that x (𝑌 ) = 𝑌 𝐵 (𝑌 ) (𝐵 (𝑌 )+𝑌 𝐵 (𝑌 )) 2 is bounded and nonzero on V \ {0}, it is not hard to see that 𝑌 * must be a zero of 𝐵 of multiplicity exactly . Now let us show that this is impossible using the variational method:

Consider a perturbation 𝐵(𝑦, 𝜀) = 𝐵(𝑦) + 𝜀 to the constant term of the weight generating function, and denote by x (𝑌, 𝜀), 𝑥 𝑐 (𝜀) and V(𝜀) the perturbed versions of x (𝑌 ), 𝑥 𝑐 and V. (This is the special case 𝑘 = 0 of the perturbation considered in the proof of Lemma .) When 𝜀 > -𝐵(0), we can still apply the argument of the rst paragraph to the perturbed model. It follows that the zeros of 𝐵( • , 𝜀) in V(𝜀) are all double zeros. But the critical points of 𝐵( • , 𝜀) do not depend on 𝜀, while its zeros do. More precisely, since 𝑌 * is a double zero of 𝐵(𝑌 ), the equation 𝐵(𝑌 , 𝜀) = 𝐵(𝑌 ) + 𝜀 = 0 has two solutions 𝑌 + * (𝜀) and 𝑌 - * (𝜀) such that 𝑌 ± * (𝜀) -𝑌 * ∼ ±𝑐 • 𝜀 1/2 as 𝜀 → 0, where 𝑐 = √ 2 𝐵 (𝑌 * ) . It follows that 𝑌 ± * (𝜀) are both simple zeros of 𝐵( • , 𝜀), and hence 𝑌 ± * (𝜀) ∉ V(𝜀) for all 𝜀 ≠ 0 close to zero.

By continuity, 𝑌 * must be on the boundary of V. It is clear that 𝑌 * ≠ 𝑌 𝑐 . By Lemma , we have x (𝑌 * ) ≠ 0 and thus x is locally injective at 𝑌 * . It follows that in a small neighborhood of 𝑌 * , the preimage x-1 (D 𝑥 𝑐 ) coincides with V, that is, 𝑌 ∈ V if and only if | x (𝑌 )| ≤ 𝑥 𝑐 . By continuity, the same is true in the perturbed model when 𝜀 is small enough. Hence 𝑌 ± * (𝜀) ∉ V(𝜀) implies that x (𝑌 ± * (𝜀), 𝜀) > 𝑥 𝑐 (𝜀) for all 𝜀 ≠ 0 close to 0. However, the asymptotics

𝑌 ± * (𝜀) -𝑌 * ∼ ±𝑐 • 𝜀 1/2 implies that x (𝑌 ± * (𝜀), 𝜀) = x (𝑌 * ) + x (𝑌 * ) • (𝑌 ± * (𝜀) -𝑌 * ) + 𝑂 (𝜀) = x (𝑌 * ) ± 𝑐 x (𝑌 * ) • 𝜀 1/2 + 𝑜 (𝜀 1/2 ). ( ) Since 𝑌 * ∈ 𝜕V, we have | x (𝑌 * )| = 𝑥 𝑐 . It follows that x (𝑌 ± * (𝜀), 𝜀) = 𝑥 𝑐 • 1 ± c • 𝜀 1/2 + 𝑜 (𝜀 1/2 ) = 𝑥 𝑐 • 1 ± ℜ𝔢( c • 𝜀 1/2 ) + 𝑜 (𝜀 1/2 ) , ( )
where c = 𝑐 x (𝑌 * )

x (𝑌 * ) ≠ 0. In the proof of Lemma , we have shown that 𝑥 𝑐 (𝜀) is di erentiable at 𝜀 = 0. Hence the asymptotic expansion of x (𝑌 ± * (𝜀), 𝜀) and the inequality x (𝑌 ± * (𝜀), 𝜀) > 𝑥 𝑐 (𝜀) implies that ℜ𝔢( c • 𝜀 1/2 ) = 0 for all 𝜀 ≠ 0 close to 0. But this is impossible, because arg( c • 𝜀 1/2 ) changes by 𝜋/2 when 𝜀 changes sign. We conclude by contradiction that 𝐵(𝑌 ) ≠ 0 for all 𝑌 ∈ V. Since x (𝑌 ) = 𝑌 𝐵 (𝑌 ) (𝐵 (𝑌 )+𝑌 𝐵 (𝑌 )) 2 is bounded on V, we also have 𝐵(𝑌 ) + 𝑌 𝐵 (𝑌 ) ≠ 0 for all 𝑌 ∈ V. . Since 𝑌 * ≠ 𝑦 * , the zero sets of 𝑄 and 𝑞 coincide near (𝑌 * , 𝑦 * ). We have seen that Z does not contain points in V × D 𝑌 𝑐 . It follows that the graphs of the functions Ỹ𝑗 do not intersect V × D 𝑌 𝑐 in a neighborhood of (𝑌 * , 𝑦 * ), or equivalently, V ∩ Ỹ𝑗 (D 𝑌 𝑐 ) = ∅ locally near 𝑌 * . When 𝑦 → 𝑦 * along a half-line, Ỹ𝑗 (𝑦) → 𝑌 * also along a half-line. In this asymptotic regime, we have

𝑌 ∈ V ⇔ x (𝑌 ) x (𝑌 * ) < 1 ⇔ ℜ𝔢 x (𝑌 ) x (𝑌 * ) -1 < 0 and 𝑦 ∈ D 𝑌 𝑐 ⇔ 𝑦 𝑦 * < 1 ⇔ ℜ𝔢 𝑦 𝑦 * -1 < 0
We cannot have Ỹ 𝑗 (𝑦 * ) = 0 because otherwise Ỹ𝑗 (D 𝑌 𝑐 ) would contain a cone of angle arbitrarily close to 2𝜋 near 𝑌 * , which would intersect V. It follows that when 𝑦 → 𝑦 * , x ( Ỹ𝑗 (𝑦))

x (𝑌 * )

-1 ∼ x (𝑌 * ) x (𝑌 * ) ( Ỹ𝑗 (𝑦) -𝑌 * ) ∼ x (𝑌 * ) x (𝑌 * ) 𝑦 * Ỹ 𝑗 (𝑦 * ) • 𝑦 -𝑦 * 𝑦 * . ( )
It

A Variational method for nding the dominant singularities of an inverse

In this appendix, we discuss the variational method used in the proof of Lemma to nd additional constraints on the critical points of x on the boundary of V. We will describe the method in a general context: Proposition states the result of the variational method under the minimal conditions for its application, and we discuss in Remarks and how the setting of Proposition arises naturally in analytic combinatorics.

Proposition . Let 𝑥 𝑐 : I → R >0 be a continuous function on an open interval I that is di erentiable at 0 ∈ I. Let x : U → C be a 𝐶 1 function on an open domain U ⊆ C × I that is analytic in its rst variable. For each 𝜀 ∈ I, let V(𝜀) be a connected component of the (lower) level set 𝐿 𝑥 𝑐 (𝜀) := {𝑌 ∈ U 𝜀 : | x (𝑌 , 𝜀)| < 𝑥 𝑐 (𝜀)} that does not contain any critical point of x ( • , 𝜀), where U 𝜀 denotes the set {𝑌 ∈ C : (𝑌 , 𝜀) ∈ U}. In addition, we assume that the family (V(𝜀)) 𝜀 ∈I contains a continuous function 𝑌 0 : I → C in the sense that 𝑌 0 (𝜀) ∈ V(𝜀) for all 𝜀 ∈ I.

Under the above conditions, if 𝜕 𝑌 x (𝑌 * , 0) = 0 for some 𝑌 * ∈ U 0 on the boundary of V(0), then we have As explained in the proof of Lemma , the domain V of the parking model and its perturbation V(𝜀) satisfy the global version of the assumptions (hence also the local one).

ℜ𝔢 𝜕 𝜀 x (𝑌 * , 0) x (𝑌 * , 0) = 𝑥 𝑐 (0) 𝑥 𝑐 (0) . ( )
Remark

Remark

(Applications in analytic combinatorics). The situation addressed in this appendix has also appeared in the enumeration of Ising-decorated triangulations in [ ]. In the proof of [ , Lemma ], the authors used a simpli ed version of the variational method discussed here to nd one extra equation satis ed by the critical points of the function x𝑅 (the counterpart of x ( • , 𝜀) in [ ]) on the boundary of the domain H 0 (𝑅) (the counterpart of V(𝜀)). The main simpli cation in [ ] comes from the fact that the non-trivial critical points of x𝑅 are known to be simple.

More generally, whenever we have a power series Ŷ of radius of convergence 𝑥 𝑐 whose inverse x := Ŷ -1 has an analytic continuation on V := Ŷ (D 𝑥 𝑐 ), the function x will induce a conformal bijection from V to D 𝑥 𝑐 . In the context of analytic combinatorics, a natural question is to ask where are the singularities of Ŷ on its circle of convergence 𝜕D 𝑥 𝑐 . If 𝑥 * ∈ 𝜕D 𝑥 𝑐 is a point such that 𝑌 * = Ŷ (𝑥 * ) ∈ 𝜕V is well-de ned and that x has an analytic continuation in a neighborhood U of 𝑌 * , then Ŷ has a singularity at 𝑥 * if and only if 𝑌 * is a critical point of x. The fact that 𝑌 * is a critical point of x ( • , 0) with a critical value in 𝜕D 𝑥 𝑐 implies the equations

𝜕 𝑌 x (𝑌 * , 0) = 0 and | x (𝑌 * , 0)| = 𝑥 𝑐 (0) . ( )
This is a system of three real equations on two real variables ℜ𝔢(𝑌 * ) and ℑ𝔪(𝑌 * ). So generically, this system should be able to eliminate all the "unexpected" singularities of Ŷ on 𝜕D 𝑥 𝑐 . However, if the function x (thus also Ŷ and V) depends on one or more extra (real) parameters 𝜀, then the system ( ) would generically have "unexpected" solutions for some subset of 𝜀 of codimension one. Proposition provides a solution to this problem when the depence of x on the extra parameters 𝜀 is 𝐶 1 . More precisely, it provides one additional (real) equation on 𝑌 * for each (real) parameter 𝜀, which is generically enough for eliminating all the "unexpected" solutions.

The basic idea behind Proposition is the following: Since x ( • , 𝜀) has no critical point in V(𝜀) for any 𝜀, if x ( • , 0) has a critical point 𝑌 * on the boundary of V(0), then the perturbation 𝜀 must "move 𝑌 * away from V(0)" for both positve and negative values of 𝜀, and this gives a stationarity equation that 𝑌 * must satisfy.

The implementation of the above idea is complicated by the fact that both x ( • , 𝜀) and the domain V(𝜀) change with the perturbation. For this we need to understand how the critical points and the level sets of x ( • , 𝜀) depends on 𝜀, and the interplay between the two. This is the subject of the two lemmas below. More precisely, Lemma de nes the branches of the critical points of x ( • , 𝜀) near (𝑌 * , 0), and computes the derivative of x ( • , 𝜀) along these branches. Lemma establishes the connectedness of the level set of x ( • , 𝜀) near 𝑌 * , when the level is higher than all the critical values of x ( • , 𝜀).

Lemma . If 𝑌 * is a critical point of x ( • , 0) of multiplicity 𝑛 ≥ 1, then there exists a neighborhood V×J of (𝑌 * , 0) in which the critical points of x ( • , 𝜀) in V × J are parametrized by 𝑛 (not necessarily distinct) continuous functions, that is, there exist 𝑛 continuous functions 𝑌 (𝑘) * : J → V such that

𝑌 (1) * (0) = • • • = 𝑌 (𝑛) * (0) = 𝑌 * and 𝑌 ∈ V 𝜕 𝑌 x (𝑌 , 𝜀) = 0 = 𝑌 (1) * (𝜀), • • • , 𝑌 (𝑛) * (𝜀) ( )
for all 𝜀 ∈ J. Moreover, for all 1 ≤ 𝑘 ≤ 𝑛, we have d d𝜀 x (𝑌 (𝑘) * (𝜀), 𝜀)

𝜀=0 = 𝜕 𝜀 x (𝑌 * , 0).
Proof. The generalization of implicit function theorem given in Lemma ensures that the zero set of 𝜕 𝑌 x de nes 𝑛 continuous functions 𝑌 (1) * , . . . , 𝑌 (𝑛) * satisfying ( ). By Cauchy's integral formula, 𝜕 𝑌 x (𝑌 , 𝜀) = 1 2𝜋𝑖 ¸x (𝜂,𝜀) (𝑌 -𝜂) 2 d𝜂. So the 𝐶 1 -continuity of x implies that of 𝜕 𝑌 x. Since both x and 𝜕 𝑌 x are 𝐶 1 with respect to (𝑌 , 𝜀), and 𝑌 * is a zero of 𝑌 ↦ → 𝜕 𝑌 x (𝑌 , 0) of multiplicity 𝑛, we have

x (𝑌 , 𝜀) = x (𝑌 , 0) + 𝜕 𝜀 x (𝑌 * , 0) • 𝜀 + 𝑜 (𝜀) = x (𝑌 * , 0) + 𝑂 ((𝑌 -𝑌 * ) 𝑛+1 ) + 𝜕 𝜀 x (𝑌 * , 0) • 𝜀 + 𝑜 (𝜀) and 𝜕 𝑌 x (𝑌 , 𝜀) = 𝜕 𝑌 x (𝑌 , 0) + 𝜕 𝜀 𝜕 𝑌 x (𝑌 * , 0) • 𝜀 + 𝑜 (𝜀) = 𝑐 • (𝑌 -𝑌 * ) 𝑛 + 𝑂 ((𝑌 -𝑌 * ) 𝑛+1 ) + 𝜕 𝜀 𝜕 𝑌 x (𝑌 * , 0) • 𝜀 + 𝑜 (𝜀)
as (𝑌, 𝜀) → (𝑌 * , 0), where 𝑐 = 1 𝑛! 𝜕 𝑛+1 𝑌 x (𝑌 * , 0) ≠ 0. Applying the above expansion of 𝜕 𝑌 x to the equation 𝜕 𝑌 x (𝑌 (𝑘) * (𝜀), 𝜀) = 0 shows that 𝑌 (𝑘) * (𝜀) -𝑌 * = 𝑂 (𝜀 1/𝑛 ) as 𝜀 → 0. Plugging this into the expansion of x then gives x (𝑌 (𝑘) * (𝜀), 𝜀) = x (𝑌 * , 0) + 𝜕 𝜀 x (𝑌 * , 0) • 𝜀 + 𝑜 (𝜀), that is, the function 𝜀 ↦ → x (𝑌 (𝑘) * (𝜀), 𝜀) is di erentiable at 0, with a derivative equal to 𝜕 𝜀 x (𝑌 * , 0).

Lemma . Assume that x (𝑌 * , 0) ≠ 0. Then the neighborhood V × J in Lemma can be chosen in such a way that for all 𝜀 ∈ J and ℎ > max 𝑘=1,...,𝑛

x (𝑌 (𝑘) * (𝜀), 𝜀) , the local level set 𝐿

V ℎ (𝜀) := {𝑌 ∈ V : | x (𝑌 , 𝜀)| < ℎ} is connected.
Proof. Without loss of generality, we assume that 𝑌 * = 0 and x (0, 0) = 1. We choose V to be the closed ball of radius 𝑟 centered at 0 and J = (-𝜀 0 , 𝜀 0 ), for some 𝑟 > 0 and 𝜀 0 > 0 to be speci ed later. We assume that 𝑟 and 𝜀 0 are small enough so that by continuity, x does not vanish on V × J. In the rest of the proof, unless otherwise mentioned, we x an 𝜀 ∈ J and drop it from the notations. Let 𝐻 (𝑌 ) = | x (𝑌 )| and ℎ 𝑐 = max 1≤𝑘 ≤𝑛 𝐻 (𝑌 (𝑘) * ). Then we have 𝐿 V ℎ = {𝑌 ∈ V : 𝐻 (𝑌 ) < ℎ}, and the lemma claims that 𝐿 V ℎ is connected for all ℎ > ℎ 𝑐 . For technical reasons, we will prove the claim for the closed level set 𝐿 V ℎ := {𝑌 ∈ V : 𝐻 (𝑌 ) ≤ ℎ} instead of 𝐿 V ℎ . This is clearly equivalent, since we have 𝐿 V ℎ ⊆ 𝐿 V ℎ ⊆ 𝐿 V ℎ for all ℎ < ℎ. Notice that the maximum ℎ 0 = max 𝐻 (V) is nite, and for all ℎ ≥ ℎ 0 , we have 𝐿 V ℎ = V, which is connected. For the other values of ℎ, we will construct a continuous mapping 𝛷 : V × (ℎ 𝑐 , ℎ 0 ] → V with the property: 𝛷 (𝑌 , ℎ) = 𝑌 when ℎ ≥ 𝐻 (𝑌 ) and 𝐻 (𝛷 (𝑌 , ℎ)) = ℎ when ℎ ≤ 𝐻 (𝑌 ). ( ) Since ℎ 0 is the maximum of 𝐻 on V, the above property dictates that 𝛷 ( • , ℎ 0 ) : V → V is the identity map. For general ℎ ∈ (ℎ 𝑐 , ℎ 0 ], it says that 𝛷 ( • , ℎ) is equal to the identity on 𝐿 V ℎ , while projects the complement of 𝐿 V ℎ to the level line {𝑌 ∈ V : 𝐻 (𝑌 ) = ℎ} ⊆ 𝐿 V ℎ . These facts ensure that for each ℎ ∈ (ℎ 𝑐 , ℎ 0 ), the restriction 𝛷 | V×[ℎ,ℎ 0 ] de nes a deformation retraction from V to 𝐿 V ℎ . We refer to [ ] for the de nition and properties of deformation retractions. In particular, it implies that 𝐿 V ℎ is homotopy equivalent to V, therefore also connected. We construct 𝛷 by de ning its marginals Ỹ ≡ 𝛷 (𝑌 , • ) : (ℎ 𝑐 , ℎ 0 ] → V using the following backward ODE: for all ℎ ≥ 𝐻 (𝑌 ), let Ỹ (ℎ) = 𝑌 (this is the initial condition), and for ℎ ∈ (ℎ 𝑐 , 𝐻 (𝑌 )], let Ỹ (ℎ) satisfy

d Ỹ dℎ = F( Ỹ ) :=        1 t•∇𝐻 ( Ỹ ) t if Ỹ ∈ 𝜕V and Ỹ • ∇𝐻 ( Ỹ ) < 0, 1 ∇𝐻 ( Ỹ ) 2 ∇𝐻 ( Ỹ ) otherwise.
( ) Here we identify Ỹ with a vector in R 2 and use the notations of real vector analysis: ∇𝐻 ( Ỹ ) is the gradient of the scalar function 𝐻 ( Ỹ ), t is any nonzero vector orthogonal to Ỹ (i.e. a tangent vector of the circle 𝜕V), a • b stands for the inner product of two vectors a and b, and • denotes the Euclidean norm on R 2 .

Intuitively, the above ODE describes how a point 𝑌 ∈ V should move when we lower the height ℎ from ℎ 0 to ℎ 𝑐 , and force 𝑌 to remain in the level set 𝐿 V ℎ . For large values of ℎ, the point 𝑌 is already in 𝐿 V ℎ , so it does not have to move, that is, Ỹ (ℎ) = 𝑌 for ℎ ≥ 𝐻 (𝑌 ) (the rst half of property ( )). When ℎ decreases below 𝐻 (𝑌 ), we move the point 𝑌 to new positions Ỹ (ℎ) by gradient descent: In general, Ỹ moves in the direction of -∇𝐻 ( Ỹ ) as ℎ decreases (the second case in ( )). But when Ỹ is on the boundary of the disk V and -∇𝐻 ( Ỹ ) points to the exterior of V, we project the vector -∇𝐻 ( Ỹ ) onto the tangent of 𝜕V, and move Ỹ in that direction instead (the rst case in ( )). In both cases, the movement speed is adjusted so that d dℎ 𝐻 ( Ỹ (ℎ)) = ∇𝐻 ( Ỹ ) • d Ỹ dℎ ≡ 1, which implies 𝐻 ( Ỹ (ℎ)) = ℎ for all ℎ ≤ 𝐻 (𝑌 ) (the second half of property ( )). Due to the identity 𝐻 ( Ỹ (ℎ)) = ℎ for ℎ ≤ 𝐻 (𝑌 ), only the vector eld F on V \ 𝐿 V ℎ 𝑐 is involved in determining the solution of the ODE for ℎ ∈ (ℎ 𝑐 , ℎ 0 ]. Let us show that the vector eld F is locally bounded on V \ 𝐿 V ℎ 𝑐 . More precisely, let us show that when 𝑟 > 0 and 𝜀 0 > 0 are chosen appropriately, F is bounded on V \ 𝐿 V ℎ for all ℎ > ℎ 𝑐 and 𝜀 ∈ J: Since ∇𝐻 is continuous on the closed set V \ 𝐿 V ℎ , it is not hard to see from ( ) that F is bounded on V \ 𝐿 V ℎ if and only if ∇𝐻 ( Ỹ ) ≠ 0 for all Ỹ ∈ V \ 𝐿 V ℎ , and ∇𝐻 ( Ỹ ) ∉ R <0 • Ỹ for all Ỹ ∈ 𝜕V \ 𝐿 V ℎ .

( )

Recall that 𝐻 = | x |. It is a simple exercise to check that under the canonical identi cation of C = R 2 , we have

∇𝐻 = x | x | • (𝜕 𝑌 x) * , ( )
where 𝑧 * denotes the complex conjugate of 𝑧. By assumption, x do not vanish on V, for all 𝜀 ∈ J. Moreover, ℎ > ℎ 𝑐 = max 1≤𝑘 ≤𝑛 𝐻 (𝑌 (𝑘) * ) implies that the 𝑛 critical points 𝑌 (1) * , • • • , 𝑌 (𝑛) * of x are all in 𝐿 V ℎ , hence 𝜕 𝑌 x does not vanish on V\ 𝐿 V ℎ . It follows that ∇𝐻 ( Ỹ ) ≠ 0 for all Ỹ ∈ V\ 𝐿 V ℎ , i.e. the rst half of the condition ( ) is true. On the other hand, by ( ) and the de nition of 𝐿 V ℎ , the second half of ( ) is true if and only if Ỹ 𝜕 𝑌 x ( Ỹ )

x ( Ỹ ) ∉ R <0 for all Ỹ ∈ 𝜕V such that | x ( Ỹ )| ≥ ℎ .

( )

Taking the limit ℎ ℎ 𝑐 , we see that the above condition holds for all ℎ > ℎ 𝑐 if and only if

Ỹ 𝜕 𝑌 x ( Ỹ ) x ( Ỹ ) , x ( Ỹ ) -ℎ 𝑐 ∉ R <0 × R >0 for all Ỹ ∈ 𝜕V. ( )
To nd 𝑟 and 𝜀 0 such that the above condition is satis ed for all 𝜀 ∈ J = (-𝜀 0 , 𝜀 0 ), we would like to obtain a non-trivial uniform limit of the pair in ( ) when (𝑟, 𝜀 0 ) → (0, 0). A bit of thought reveals that it is convenient to take the limit 𝑟 → 0 after 𝜀 0 → 0, and we should renormalize both components of the pair by 1/𝑟 𝑛+1 . Indeed, since x, 𝜕 x and ℎ 𝑐 are all (uniformly) continuous in 𝜀 and ℎ 𝑐 (𝜀 = 0) = x (𝑌 * , 0) = 1, we have uniformly in 𝜏 ∈ [0, 2𝜋) and 𝜀 ∈ (-𝜀 0 , 𝜀 0 ). It is not hard to see that for any xed 𝑐 ∈ C \ {0} and 𝑛 ≥ 1, the set (𝑐𝑒 𝑖 (𝑛+1)𝜏 , 1 𝑛+1 ℜ𝔢(𝑐𝑒 𝑖 (𝑛+1)𝜏 )) 𝜏 ∈ [0, 2𝜋) is bounded away from R <0 × R >0 in C × R. Then the uniform convergence ( ) implies that there exist 𝑟 > 0 and 𝜀 0 > 0 such that ( ) is true for all 𝜀 ∈ (-𝜀 0 , 𝜀 0 ). With this choice of 𝑟 and 𝜀 0 , the second half of ( ) is also true for all 𝜀 ∈ (-𝜀 0 , 𝜀 0 ). We conclude that the vector eld F is bounded on V \ 𝐿 V ℎ for each ℎ > ℎ 𝑐 . Due to the di erence between the cases on the right hand side of ( ), the vector eld F is not continuous. But the discontinuity only occurs on the circle 𝜕V, and can be avoided using the following regularization: For 𝜎 ∈ (0, 𝑟 ), let V 𝜎 denote the closed disk of radius 𝑟 -𝜎 centered at 0. De ne F • 𝜎 : V → R 2 by F • 𝜎 ( Ỹ ) = F( Ỹ ) for all Ỹ ∈ V 𝜎 ∪𝜕V, and by linear interpolation on the segment {𝑟 𝑒 𝑖𝜏 | 𝑟 ∈ [𝑟 -𝜎, 𝑟 ]} for each 𝜏 ∈ [0, 2𝜋). Recall that the vector eld F has the property that F( Ỹ ) • ∇𝐻 ( Ỹ ) = 1 for all Ỹ ∈ V (except at its singularities), which ensures that every solution of the backward ODE ( ) must satisfy 𝐻 ( Ỹ (ℎ)) = ℎ for all ℎ ∈ (ℎ 𝑐 , 𝐻 (𝑌 )]. The linear interpolation in the de nition of F • 𝜎 breaks this property. To restore it, we de ne

F 𝜎 = 1 F • 𝜎 •∇𝐻 F • 𝜎 . One can check that F 𝜎 = F • 𝜎 = F on V 𝜎 ∪ 𝜕V.
In particular, F 𝜎 converges to F pointwise on V as 𝜎 → 0. With a close look at the proof in the previous paragraph, it is not hard to see that for 𝜎 > 0 small enough, the vector eld F 𝜎 is bounded and Lipschitz continuous on V \ 𝐿 V ℎ for all ℎ > ℎ 𝑐 . Then, by the Cauchy-Lipschitz theorem (a.k.a. Picard-Lindelöf theorem, see e.g. [ , Theorem . ]), the backward ODE d Ỹ dℎ = F 𝜎 ( Ỹ ) with the initial condition Ỹ (ℎ) = 𝑌 for ℎ ∈ [𝐻 (𝑌 ), ℎ 0 ] has a unique solution Ỹ𝜎 : (ℎ 𝑐 , ℎ 0 ] → V, such that 𝛷 𝜎 (𝑌 , ℎ) = Ỹ𝜎 (ℎ) de nes a continuous function 𝛷 𝜎 : V × (ℎ 𝑐 , ℎ 0 ] → V. By construction, the vector eld F 𝜎 satis es F 𝜎 • ∇𝐻 ≡ 1 on V. Hence we have 𝐻 ( Ỹ𝜎 (ℎ)) = ℎ for all ℎ ∈ (ℎ 𝑐 , 𝐻 (𝑌 )] and 𝑌 ∈ V. It follows that 𝛷 𝜎 satis es the condition ( ). As discussed in the second paragraph of the proof, this implies the conclusion of the lemma.

Remark. In the above proof, we expect Ỹ𝜎 to converge to a solution of the backward ODE ( ) when 𝜎 → 0. However this is not needed for proving Lemma .

This proof is an adaptation of the proof of a classical theorem [ , Theorem . ] of Morse theory in di erential topology, which states that if 𝐻 : V → R is a smooth function on a manifold V (without boundary), and 𝑎 < 𝑏 are such that 𝐻 -1 ( [𝑎, 𝑏]) is compact and contains no critical point of 𝐻 , then there exists a deformation retraction from 𝐻 -1 ((-∞, 𝑏]) to 𝐻 -1 ((-∞, 𝑎]). The main di culty in adapting the classical proof to Lemma is that now V has a boundary.

2 2𝜇𝛼 1 +

 21 The non-universal constants 𝜇 and C 𝐹 in the expansions ( )-( ) are given by 𝑐 ) 𝑥 𝑐 in the generic + phase2 α C 𝐵 𝐵 (𝑌 𝑐 )+𝑌 𝑐 𝐵 (𝑌 𝑐 )in the dilute -phase andC 𝐹 = 1 𝑌 𝑐 𝐵 (𝑌 𝑐 )𝐵 (𝑌 𝑐 )

  Random fully packed trees and equivalent weight sequences. When 𝐹 (𝑥, 𝑦; b) < ∞, we can de ne a probability measure on FT by P 𝑥,𝑦 b (𝔱, ℓ) = 𝑤 b (𝔱, ℓ) • 𝑥 |𝑉 (𝔱) | 𝑦 s(𝔱,ℓ) 𝐹 (𝑥, 𝑦; b) . ( ) The de nition of 𝑤 b and the relation ( ) imply that for any 𝜆, 𝑟 > 0, the weight sequence b𝑙 = 𝜆𝑟 𝑙 • 𝑏 𝑙 satis es 𝐹 (𝑥, 𝑦; b) = 𝐹 (𝜆𝑟 • 𝑥, 𝑟 • 𝑦; b) and P

  𝜕 𝑌 𝑄 (𝑌 +𝜆 1 (𝑦 -𝑌( ( ( ( ( ( ( (

(

  Global version of the assumptions on (V(𝜀)) 𝜀 ∈I ). Proposition states the local version of the assumptions on the family (V(𝜀)) 𝜀 ∈I . A stronger global version goes as follows: we assume that for each 𝜀 ∈ I, x ( • , 𝜀) induces a conformal bijection from V(𝜀) to the disk D 𝑥 𝑐 (𝜀) such that the preimage of 0, characterized by x (𝑌 0 (𝜀), 𝜀) = 0 and 𝑌 0 (𝜀) ∈ V(𝜀), is a continuous function of 𝜀. It is clear that the global version of the assumptions implies the local one: if x ( • , 𝜀) induces a conformal bijection from V(𝜀) to D 𝑥 𝑐 (𝜀) , then V(𝜀) is a connected component of the lower level set 𝐿 𝑥 𝑐 (𝜀) = {𝑌 ∈ U 𝜀 : | x (𝑌 , 𝜀)| < 𝑥 𝑐 (𝜀)} that does not contain any critical point of x ( • , 𝜀).

ỸỸ

  𝜕 𝑌 x ( Ỹ , 𝜀) x ( Ỹ , 𝜀) , x ( Ỹ , 𝜀)ℎ 𝑐 (𝜀) ----→ 𝜀 0 →0 1 𝑟 𝑛+1 Ỹ 𝜕 𝑌 x ( Ỹ , 0) x ( Ỹ , 0) , x ( Ỹ , 0) -1 ( ) uniformly in 𝜀 ∈ (-𝜀 0 , 𝜀 0 ). Next, since 𝑌 * = 0 is a zero of multiplicity 𝑛 of 𝜕 𝑌 x ( • , 0), we have 𝜕 𝑌 x ( Ỹ , 0) ∼ 𝑐 • Ỹ 𝑛 and x ( Ỹ, 0) = 1 + 𝑐 𝑛+1 Ỹ 𝑛+1 + 𝑜 ( Ỹ 𝑛+1) for some 𝑐 ∈ C \ {0} when Ỹ → 0. We parametrize the point Ỹ ∈ 𝜕V by Ỹ = 𝑟𝑒 𝑖𝜏 with 𝜏 ∈ [0, 2𝜋). Then taking the limit 𝑟 → 0 of the previous display gives lim 𝜕 𝑌 x ( Ỹ , 𝜀)x ( Ỹ , 𝜀) , x ( Ỹ , 𝜀)ℎ 𝑐 (𝜀) = 𝑐 • 𝑒 𝑖 (𝑛+1)𝜏 , 1 𝑛 + 1 ℜ𝔢(𝑐 • 𝑒 𝑖 (𝑛+1)𝜏 ) ( )

  Hence the 𝑌 -derivative of / 𝜕 𝑥 𝐹 (𝑌 , 𝑦) simpli es to 𝜕 𝑌 / 𝜕 𝑥 F (𝑌 𝑐 , 𝑦) = 𝜕 𝑥 𝑄 (𝑌 𝑐 , 𝑦) 2(𝑌 𝑐 -𝑦) √ 𝑞(𝑌 𝑐 , 𝑦) -𝜕 𝑌 / 𝜕 𝑥 𝜙 (𝑌 𝑐 ) . ( ) On the one hand, Lemma ( ) tells us that / 𝜕 𝑥 𝑄 (𝑌 𝑐 , 𝑦) ∼ -𝜕 𝑦 / 𝜕 𝑥 𝑄 (𝑌 𝑐 , 𝑌 𝑐 ) • (𝑌 𝑐 -𝑦) = 2𝜕 𝑥 𝜙 (𝑌 𝑐 )𝑌 𝑐 • 𝑡 as 𝑦 → 𝑌 𝑐 , and 𝜕 𝑌 / 𝜕 𝑥 𝑄 (𝑌 𝑐 , 𝑌 𝑐 ) = 2𝜕 𝑥 𝜙 (𝑌 𝑐 ) ≠ 0. On the other hand, Lemma implies (𝑌 𝑐 -𝑦) √ 𝑞(𝑌 𝑐 , 𝑦) ∼ 𝑌 𝑐 √ C 𝑞 • 𝑡 𝛼/2 in the special case where 𝑆 = 0. Plugging these asymptotics into the expressions of / 𝜕 𝑥 F (𝑌 𝑐 , 𝑦) and 𝜕 𝑌 / 𝜕 𝑥 F (𝑌 𝑐 , 𝑦)

	1	𝜕 𝑌 /
	2𝑦

  graphs of two analytic functions Ỹ1 , Ỹ2 such that Ỹ1 (𝑦 * ) = Ỹ2 (𝑦 * ) = 𝑌 * and Ỹ 1 (𝑦 * ), Ỹ 2 (𝑦 * ) are the two roots of the polynomial 𝜕 2 𝑌 𝑄 (𝑌 * , 𝑦 * ) • 𝑟 2 + 2𝜕 𝑌 𝜕 𝑦 𝑄 (𝑌 * , 𝑦 * ) • 𝑟 + 𝜕 2 𝑦 𝑄 (𝑌 * , 𝑦 * ) . 2𝜕 𝑌 𝜕 𝑦 𝑄 (𝑌 * ,𝑦 * ) 𝜕 2 𝑌 𝑄 (𝑌 * ,𝑦 * ) = -2 𝜙 (𝑌 * )

	( )
	In particular, Ỹ 1 (𝑦 * ) + Ỹ 2 (𝑦 * ) = -

  is not hard to see that the condition V ∩ Ỹ𝑗 (D 𝑌 𝑐 ) = ∅ constraints the coe cient x (𝑌 𝑌 𝑐 , this contradicts the result that 𝑦 * ∉ D 𝑌 𝑐 . Therefore Z must be empty.

* )

x (𝑌 * ) 𝑦 * Ỹ 𝑗 (𝑦 * ) to be negative. It follows that x (𝑌 * )

x

(𝑌 * ) 𝑦 * • ( Ỹ 1 (𝑦 * ) + Ỹ 2 (𝑦 * )) < 0. Using Ỹ 1 (𝑦 * ) + Ỹ 2 (𝑦 * ) = -2 𝜙 (

𝑌 * ) and 𝑦 * = -𝜙 (𝑌 * ), one can check that the inequality simpli es to 𝜓 * := 𝑌 * 𝐵 (𝑌 * ) 𝐵 (𝑌 * ) > 1. It follows that 𝑦 * = -𝜙 (𝑌 * ) = 𝑌 * 𝜓 * -1 𝜓 * +1 ∈ (0, 𝑌 * ). But since 𝑌 * ∈ V ⊆ D

Acknowledgement. I would like to thank Nicolas Curien and Olivier Hénard for introducing the parking

. 𝑞(𝑌 , 𝑦) does not vanish on V × D 𝑌 𝑐 except at (𝑌 𝑐 , 𝑌 𝑐 )

Recall that 𝑞 is a holomorphic function on V × D 𝑌 𝑐 such that 𝑞(𝑌 , 𝑌 ) = 𝜙 (𝑌 ) and 𝑞(𝑌 , 𝑦) = 𝑄 (𝑌 ,𝑦) (𝑌 -𝑦) 2 when 𝑌 ≠ 𝑦. We start by showing that a zero of the function 𝑄 in V × D 𝑌 𝑐 must also be a zero of both / 𝜕 𝑥 𝑄 and 𝜕 𝑦 𝑄, using a variant of the quadratic method. The proof is complicated by the fact that 𝜕 𝑥 𝐹 (𝑥, 𝑦) is absolutely convergent only on D 𝑥 𝑐 × D 𝑌 𝑐 and not on 𝜕D 𝑥 𝑐 × D 𝑌 𝑐 (c.f. Lemma ). We solve this problem with a continuity argument by studying the local geometry of the zero set {(𝑌 , 𝑦) : 𝑄 (𝑌 , 𝑦) = 0}.

Lemma . If 𝑄 (𝑌 , 𝑦) = 0 for some (𝑌 , 𝑦) ∈ V × D 𝑌 𝑐 , then 𝜕 𝑦 𝑄 (𝑌 , 𝑦) = / 𝜕 𝑥 𝑄 (𝑌 , 𝑦) = 0 as well.

Proof. Let 𝑇 (𝑌, 𝑦) = 2𝑦 • 𝐹 ( x (𝑌 ), 𝑦) -1 2 +𝜙 (𝑌 ). A simple rearrangement of ( ) shows that 𝑄 (𝑌 , 𝑦) = 𝑇 (𝑌 , 𝑦) 2 . By Lemmas and , the power series 𝐹 (𝑥, 𝑦) and 𝜕 𝑦 𝐹 (𝑥, 𝑦) are absolutely convergent on D 𝑥 𝑐 ×D 𝑌 𝑐 , and x maps V continuously to D 𝑥 𝑐 . It follows that 𝜕 𝑦 𝑇 (𝑌 , 𝑦) = 2𝑦 • 𝜕 𝑦 𝐹 ( x (𝑌 ), 𝑦) + 2 𝐹 ( x (𝑌 ), 𝑦) -1 2 is bounded on V × D 𝑌 𝑐 . Hence 𝑄 (𝑌, 𝑦) = 𝑇 (𝑌 , 𝑦) 2 = 0 implies 𝜕 𝑦 𝑄 (𝑌 , 𝑦) = 2𝑇 (𝑌 , 𝑦) • 𝜕 𝑦 𝑇 (𝑌 , 𝑦) = 0, for all (𝑌 , 𝑦) ∈ V × D 𝑌 𝑐 . Similarly, since 𝜕 𝑥 𝐹 (𝑥, 𝑦) is absolutely convergent on D 𝑥 𝑐 × D 𝑌 𝑐 , the function / 𝜕 𝑥 𝑇 (𝑌 , 𝑦) = 2𝑦 • 𝜕 𝑥 𝐹 ( x (𝑌 ), 𝑦) + / 𝜕 𝑥 𝜙 (𝑌 ) takes nite values on V × D 𝑌 𝑐 . Therefore 𝑄 (𝑌 , 𝑦) = 𝑇 (𝑌 , 𝑦) 2 = 0 implies / 𝜕 𝑥 𝑄 (𝑌 , 𝑦) = 2𝑇 (𝑌 , 𝑦) • / 𝜕 𝑥 𝑇 (𝑌 , 𝑦) = 0 for all (𝑌, 𝑦) ∈ V × D 𝑌 𝑐 .

It remains to show that 𝑄 (𝑌 , 𝑦) = 0 also implies / 𝜕 𝑥 𝑄 (𝑌 , 𝑦) = 0 for (𝑌 , 𝑦) ∈ 𝜕V × D 𝑌 𝑐 . When 𝑌 * = 𝑦 * = 𝑌 𝑐 , we have / 𝜕 𝑥 𝑄 (𝑌 𝑐 , 𝑌 𝑐 ) = 0 directly by Lemma ( ). When (𝑌 * , 𝑦 * ) ∈ 𝜕V × D 𝑌 𝑐 , the mapping 𝑦 ↦ → 𝑄 (𝑌 , 𝑦) is analytic in a neighborhood of 𝑦 * for all 𝑌 ∈ V. By the generalization of the implicit function theorem in Lemma , there exists a continuous function ỹ : V → C such that ỹ (𝑌 * ) = 𝑦 * and 𝑄 (𝑌, ỹ (𝑌 )) = 0 for all 𝑌 ∈ V close enough to 𝑌 * . Since 𝑦 * is in the interior of D 𝑌 𝑐 , the graph of this function ỹ contains a sequence (𝑌 𝑗 , 𝑦 𝑗 ) ∈ V × D 𝑌 𝑐 that converges to (𝑌 * , 𝑦 * ). But the rst paragraph of the proof ensures that 𝑄 (𝑌 𝑗 , 𝑦 𝑗 ) = / 𝜕 𝑥 𝑄 (𝑌 𝑗 , 𝑦 𝑗 ) = 0 for all 𝑗. Therefore / 𝜕 𝑥 𝑄 (𝑌 * , 𝑦 * ) = 0 by continuity. It remains the case where (𝑌 * , 𝑦 * ) ∈ 𝜕V × 𝜕D 𝑌 𝑐 and (𝑌 * , 𝑦 * ) ≠ (𝑌 𝑐 , 𝑌 𝑐 ). Thanks to the Δ-analyticity of 𝐵(𝑦), the function 𝑄 (𝑌 , 𝑦) is analytic in 𝑌 when 𝑌 ∈ 𝜕V \ {𝑌 𝑐 }, and analytic in 𝑦 when 𝑦 ∈ 𝜕D 𝑌 𝑐 \ {𝑌 𝑐 }. In both cases, we can apply Lemma to express locally the zero set of 𝑄 as the graphs of some functions. We will use the asymptotics of these functions provided in Lemma to show that their graphs contain a sequence (𝑌 𝑗 , 𝑦 𝑗 ) ∈ V × D 𝑌 𝑐 that converges to (𝑌 * , 𝑦 * ). As in the previous paragraph, this implies / 𝜕 𝑥 𝑄 (𝑌 * , 𝑦 * ) = 0 by continuity. Actually, we will proceed by contradiction: Assume that 𝑄 (𝑌 * , 𝑦 * ) = 0 and / 𝜕 𝑥 𝑄 (𝑌 * , 𝑦 * ) ≠ 0 for some (𝑌 * , 𝑦 * ) ∈ 𝜕V × 𝜕D 𝑌 𝑐 \ {(𝑌 𝑐 , 𝑌 𝑐 )}. We have two cases:

When 𝑌 * ≠ 𝑌 𝑐 , the mapping 𝑌 ↦ → 𝑄 (𝑌 , 𝑦) is analytic at 𝑌 * for all 𝑦 ∈ D 𝑌 𝑐 . Moreover: 𝜕 𝑌 𝑄 (𝑌 * ,𝑦 * ) as 𝑦 → 𝑦 * . The last asymptotics implies that the tangent of the disk D 𝑌 𝑐 at 𝑦 * is mapped by Ỹ to an angle of size 𝛾𝜋 ≥ 2𝜋 at 𝑌 * . This means that the image Ỹ (D 𝑌 𝑐 ) contains a neighborhood of 𝑌 * , possibly with a cone of arbitrarily small angle removed. In particular, V ∩ Ỹ (D 𝑌 𝑐 ) contains a cone of positive angle at 𝑌 * . It follows that there exists a sequence (𝑦 𝑗 ) 𝑗 ≥0 such that ( Ỹ (𝑦 𝑗 ), 𝑦 𝑗 ) ∈ V × D 𝑌 𝑐 for all 𝑗 and ( Ỹ (𝑦 𝑗 ), 𝑦 𝑗 ) → (𝑌 * , 𝑦 * ) as 𝑗 → ∞.

When 𝑌 * = 𝑌 𝑐 and 𝑦 * ≠ 𝑌 𝑐 , the mapping 𝑦 ↦ → 𝑄 (𝑌 , 𝑦) is analytic at 𝑦 * for all 𝑌 ∈ V. Moreover:

• 𝑄 (𝑌 𝑐 , 𝑦 * ) = 𝜕 𝑦 𝑄 (𝑌 𝑐 , 𝑦 * ) = 0, that is, 𝑦 * is a zero of 𝑦 ↦ → 𝑄 (𝑌 𝑐 , 𝑦) of some multiplicity 𝑛 ≥ 2.

• 𝜕 𝑘 𝑦 𝑄 (𝑌, 𝑦 * ) = 𝑂 ((𝑌 𝑐 -𝑌 ) 𝛼-1 ) for all 0 ≤ 𝑘 < 𝑛 and 𝑄 (𝑌 , 𝑦 * ) ∼ 𝑐 • (𝑌 𝑐 -𝑌 ) 𝛼-1 as 𝑌 → 𝑌 𝑐 in V. Indeed, 𝜕 𝑘 𝑦 𝑄 (𝑌 , 𝑦 * ) is a rational function of (𝑌 , 𝐵 (𝑌 ), 𝐵(𝑌 )). Due to Assumption ( * ), it is 𝐶 1 -continuous in a neighborhood of 𝑌 𝑐 in V. By the de nition of 𝑛, we have 𝜕 𝑘 𝑦 𝑄 (𝑌 𝑐 , 𝑦 * ) = 0 for all 0 ≤ 𝑘 < 𝑛. On the other hand, we have

) and therefore after integration, 𝜕 𝑘 𝑦 𝑄 (𝑌, 𝑦 * ) = 𝑂 ((𝑌 𝑐 -𝑌 ) 𝛼-1 ) for all 0 ≤ 𝑘 < 𝑛. When 𝑘 = 0, since / 𝜕 𝑥 𝑄 (𝑌 𝑐 , 𝑦 * ) ≠ 0 by assumption, we have

𝑛 as 𝑌 → 𝑌 𝑐 , where 𝑐 ≠ 0 and 𝜔 1 , . . . , 𝜔 𝑛 are all the 𝑛-th roots of unity. The asymptotics of ỹ𝑘 implies that, in a neighborhood of 𝑦 * , the union 𝑛 𝑘=1 ỹ𝑘 (V) contains a cone of positive angle at 𝑦 * , and all of its images under the rotations 𝑧 ↦ → 𝜔 𝑘 𝑧 (𝑘 = 1, . . . , 𝑛). Since 𝑛 ≥ 2, there is at least one 𝑘 for which D 𝑌 𝑐 ∩ ỹ𝑘 (V) contains a cone of positive angle at 𝑦 * . It follows that there exists a sequence

In both cases, the set V×D 𝑌 𝑐 contains a sequence of zeros of 𝑄 that converges to (𝑌 * , 𝑦 * ). As discussed before, this implies / 𝜕 𝑥 𝑄 (𝑌 * , 𝑦 * ) = 0. This completes the proof by contradiction for

Lemma . 𝑞(𝑌 , 𝑦) does not vanish on V × D 𝑌 𝑐 \ {(𝑌 𝑐 , 𝑌 𝑐 )}.

Proof. We prove that the set

First, we derive from Lemma that all points in Z satisfy 𝜙 (𝑌 ) + 𝑦 = 𝐵(𝑦) = 𝐵 (𝑦) = 0, hence Z is a discrete set. Then, we show that a solution of the system 𝜙 (𝑌 ) + 𝑦 = 𝐵(𝑦) = 𝐵 (𝑦) = 0 cannot be an isolated point in Z, hence Z must be empty. Consider (𝑌 * , 𝑦 * ) ∈ Z. We have seen in Lemma ( ) that 𝑞(𝑌 , 𝑌 ) = 𝜙 (𝑌 ) = / 𝜕 𝑥 𝜙 (𝑌 ) • x (𝑌 ), which does not vanish on V \ {𝑌 𝑐 } by Lemmas and . Since 𝑌 * ∈ V and (𝑌 𝑐 , 𝑌 𝑐 ) ∉ Z, we have 𝑌 * ≠ 𝑦 * . One can check that 𝑞(0, 𝑦) = 1 for all 𝑦, so 𝑌 * ≠ 0 as well. On the other hand, we have 𝑄 (𝑌 * , 𝑦 * ) = (𝑌 * -𝑦 * ) 2 𝑞(𝑌 * , 𝑦 * ) = 0 and hence / 𝜕 𝑥 𝑄 (𝑌 * , 𝑦 * ) = 𝜕 𝑦 𝑄 (𝑌 * , 𝑦 * ) = 0 by Lemma . Explicitly, We have seen that 𝜙 does not vanish on V \ {𝑌 𝑐 }. Therefore 𝑌 * is a double zero of 𝑌 ↦ → 𝑄 (𝑌 , 𝑦 * ). According to the Newton-Puiseux theorem (see e.g. [ ]), the zero set of 𝑄 coincides in a neighborhood of (𝑌 * , 𝑦 * ) with the Proof of Proposition . Let 𝑌 * ∈ U be a critical point of x ( • , 0) on the boundary of V(0), and let (𝑌 (𝑘) * (𝜀)) 1≤𝑘 ≤𝑛 be the critical points of x ( • , 𝜀) in a neighborhood of 𝑌 * as de ned in Lemma . We prove Proposition in two steps: First, we show that for each 𝜀 close enough to 0, there exists 𝑘 ∈ {1, . . . , 𝑛} such that x (𝑌 (𝑘) * (𝜀), 𝜀) ≥ 𝑥 𝑐 (𝜀). Then, we derive ( ) from the previous inequality. The rst step is topological in nature and makes crucial use of the connectedness result of Lemma , while the second step basically calculates the derivative of the ratio | x (𝑌 (𝑘) * (𝜀), 𝜀)|/𝑥 𝑐 (𝜀) at 𝜀 = 0. We start by showing that for each neighborhood V of 𝑌 * , we have V ∩ V(𝜀) ≠ ∅ for all 𝜀 close enough to 0: Since 𝑌 * ∈ 𝜕V(0), we have V ∩ V(0) ≠ 0. Recall that for all 𝜀 ∈ I, V(𝜀) is a connected component of the level set 𝐿 𝑥 𝑐 (𝜀) = {𝑌 ∈ U : | x (𝑌 , 𝜀)| < 𝑥 𝑐 (𝜀)}. In particular, V(𝜀) is open and connected, thus path connected. Let Γ ⊆ V(0) be a path that connects 𝑌 0 (0) to an arbitrary point 𝑌 1 ∈ V ∩ V(0) (recall that 𝑌 0 : I → U is a continuous function such that 𝑌 0 (𝜀) ∈ V(𝜀) for all 𝜀), and let 𝐾 ⊆ V(0) be a compact neighborhood of 𝑌 0 (0). By construction, Γ ∪ 𝐾 is a compact subset of 𝐿 𝑥 𝑐 (0). Then the continuity of x and 𝑥 𝑐 implies that Γ ∪ 𝐾 ⊆ 𝐿 𝑥 𝑐 (𝜖) for all |𝜀 | < 𝜀 0 . Up to decreasing 𝜀 0 , we also have 𝑌 0 (𝜀) ∈ 𝐾 for all |𝜀 | < 𝜀 0 . Since 𝑌 0 (𝜀) ∈ V(𝜀) and Γ ∪ 𝐾 is connected, it implies Γ ∪ 𝐾 ⊆ V(𝜀). In particular, we have 𝑌 1 ∈ V(𝜀) and therefore V ∩ V(𝜀) ≠ ∅ for all |𝜀 | < 𝜀 0 .

Let V × J be a neighborhood of (𝑌 * , 0) having the properties stated in Lemma . Without loss of generality, we assume that J ⊆ (-𝜀 0 , 𝜀 0 ). Now x 𝜀 ∈ J. By the previous paragraph, (𝑘) * (𝜀), 𝜀)| for all 𝑘 ∈ {1, . . . , 𝑛}, then on the one hand, Lemma states that 𝐿 V 𝑥 𝑐 (𝜀) is connected, which implies 𝐿 V 𝑥 𝑐 (𝜀) ⊆ V(𝜀), and on the other hand, 𝑌 (𝑘) * (𝜀) ∈ 𝐿 V 𝑥 𝑐 (𝜀) by the de nition of the level set. It follows that V(𝜀) must contain all the 𝑌 (𝑘) * (𝜀). This contradicts the assumption that V(𝜀) does not contain any critical point of x ( • , 𝜀). Therefore we must have 𝑥 𝑐 (𝜀) ≤ | x (𝑌 (𝑘) * (𝜀), 𝜀)|, or equivalently

for at least one 𝑘 ≡ 𝑘 (𝜀) ∈ {1, . . . , 𝑛}.

Since 𝑌 * is on the boundary of V(0), we have | x (𝑌 * , 0)| = 𝑥 𝑐 (0), that is, the above inequality becomes an equality at 𝜀 = 0. So the derivative of the left hand side at 𝜀 = 0, if exists, is equal to zero. But by Lemma , we have d d𝜀

x (𝑌 (𝑘) * (𝜀), 𝜀)

regardless of the choice of 𝑘 ≡ 𝑘 (𝜀) ∈ {1, . . . , 𝑛}. It follows that the left hand side of ( ) is indeed di erentiable at 𝜀 = 0, and the vanishing of the derivative gives Equation ( ).

B Modi ed inverse/implicit function theorem

In this appendix we prove two analytic lemmas used in this paper. They can be viewed as modi cations of the inverse function theorem and of the implicit function theorem, respectively.

Lemma (Inverse function theorem in a cone). Let ŵ : 𝐾 → C be a holomorphic function such that ŵ (𝑧) → 0 and ŵ (𝑧) → 𝑐 when 𝑧 → 0 in 𝐾, where 𝑐 ∈ C \ {0}, and 𝐾 = {𝑟𝑒 𝑖𝜏 | 𝑟 ∈ (0, 𝑟 0 ), 𝜏 ∈ (𝜏 1 , 𝜏 2 )} is a truncated cone with an angle 𝜏 2 -𝜏 1 ∈ (0, 2𝜋). Then there is a neighborhood U of 0 such that ŵ | 𝐾∩U is injective and its inverse ẑ : ŵ (𝐾 ∩ U) → 𝐾 ∩ U is an analytic function such that ẑ (𝑤) → 0 and ẑ (𝑤) → 𝑐 -1 when 𝑤 → 0 in ŵ (𝐾 ∩ U).

Moreover, for all 𝐾 ⊆ 𝐾 of the form

Proof. Since the cone 𝐾 has an angle strictly smaller than 2𝜋, one can nd a constant 𝑀 < ∞ such that for all 𝑧 1 , 𝑧 2 ∈ 𝐾, there exists a smooth path

The above bound implies that ŵ is injective on 𝐾 ∩ U. By the classical inverse function theorem, the inverse ẑ of ŵ | 𝐾∩U is analytic on ŵ (𝐾 ∩U). Moreover, taking 𝑧 1 = 𝑧 and 𝑧 2 → 0 in the above display gives | ŵ (𝑧) -𝑐𝑧| < |𝑐𝑧|, which implies that ŵ (𝑧) → 0 if and only if 𝑧 → 0. It follows that ẑ (𝑤) → 0 and ẑ (𝑤)

. The limits ŵ (𝑧) → 0 and ŵ (𝑧) → 𝑐 imply that ŵ (𝑧)

𝑐𝑧 → 1 when 𝑧 → 0 in 𝐾. So there exists 𝑟 𝛿 > 0 such that ŵ (𝑧) 𝑐𝑧 -1 < 𝛿 for all 𝑧 ∈ 𝐾 ∩ D 𝑟 𝛿 . Without loss of generality, we assume that

𝑐𝑧 -1 < 𝛿 for 𝑧 ∈ 𝐾 ∩D 𝑟 𝛿 , the boundary of K mapped by ŵ to a curve that encloses the truncated cone 𝑐

Lemma

(Modi ed implicit function theorem). Let 𝑆 be a topological space containing 0 that is locally connected at 0, and let U be a neighborhood of (0, 0) in C × 𝑆. Assume that 𝑓 : U → C is a continuous function which is analytic in its rst variable, such that 𝑧 ↦ → 𝑓 (𝑧, 0) has a zero of multiplicity 𝑛 ≥ 1 at 𝑧 = 0. Then the zeros set of 𝑓 can be parametrized by 𝑛 continuous functions near (0, 0), that is, there exist a neighborhood V × S of (0, 0) and 𝑛 continuous functions 𝑧 𝑘 : S → V such that for each 𝑠 ∈ S, the function 𝑧 ↦ → 𝑓 (𝑧, 𝑠) has exactly 𝑛 zeros in V (counted with multiplicity), given by 𝑧 1 (𝑠), • • • , 𝑧 𝑛 (𝑠).

In addition, if there is a continuous function ℎ : 𝑆 → C such that for all 0 ≤ 𝑚 ≤ 𝑛, the limit Notice that, since the logarithm is well-de ned modulo 2𝜋𝑖Z, the exponential is well-de ned for all 𝑧 ∈ D 𝑟 . The above expression shows that 𝑝 (𝑧, 𝑠) is continuous in (𝑧, 𝑠). Thanks to Cauchy's di erentiation formula, all of its 𝑧-derivatives are also continuous with respect to (𝑧, 𝑠). In particular, 𝑎 𝑘 (𝑠) = 1 𝑘! 𝜕 𝑘 𝑧 𝑝 (0, 𝑠) is a continuous function of 𝑠 ∈ S, for all 0 ≤ 𝑘 < 𝑛. It is well-known that the zeros of polynomial are continuous functions of its coe cients. It follows that, up to adjusting the ordering of the roots 𝑧 1 (𝑠), • • • , 𝑧 𝑛 (𝑠) for each 𝑠 ∈ S, the functions 𝑧 𝑘 : S → D 𝑟 are continuous. This proves the rst part of the lemma with V = D 𝑟 .

Notice that since 𝑝 (𝑧, 𝑠) and 𝑓 (𝑧, 𝑠) have the same set of zeros with multiplicity on V × S, their quotient 𝑐 (𝑧, 𝑠) = 𝑓 (𝑧,𝑠) 𝑝 (𝑧,𝑠) is continuous on V × S and does not vanish there (the joint continuity in (𝑧, 𝑠) follows from Cauchy's integral formula). Now assume that 𝑓 satis es ( ). When 𝑚 = 𝑛, the formula gives 𝑐 𝑛 = 1 𝑛! 𝜕 𝑛 𝑧 𝑓 (0, 0). Since 𝑧 = 0 is a zero of multiplicity 𝑛 for 𝑓 ( • , 0), this implies 𝑐 𝑛 ≠ 0. If ℎ(0) ≠ 0, then we have 𝑐 𝑚 = 1 𝑚! 𝜕 𝑚 𝑧 𝑓 (0,0) ℎ (0) 𝑛-𝑚 = 0 for all 𝑚 < 𝑛 and 𝑟 𝑘 = 𝑧 𝑠 (0) ℎ (0) = 0 for all 1 ≤ 𝑘 ≤ 𝑛, so the claims of the lemma are trivially true. When ℎ(0) = 0, the limits ( ) imply that 𝜕 𝑗 𝑧 𝑓 (0, 𝑠) = 𝑂 (ℎ(𝑠) 𝑛-𝑗 ) = 𝑜 (ℎ(𝑠) 𝑛-𝑚 ) for all 𝑗 < 𝑚 ≤ 𝑛. Hence the ratio 𝑧 𝑘 (𝑠) ℎ (𝑠) must stay bounded when 𝑠 → 0. If 𝑧 𝑘 (𝑠) ℎ (𝑠) → 𝑟 ∈ C as 𝑠 → 0 along some subsequence, then ( ) implies that 𝑛 𝑚=0 𝑐 𝑚 𝑟 𝑚 = 0. In other words, all limit points of the function 𝑧 𝑘 (𝑠) ℎ (𝑠) as 𝑠 → 0 are roots of the polynomial 𝑐 0 + • • • + 𝑐 𝑛 𝑟 𝑛 . Since 𝑧 𝑘 ℎ is continuous in a neighborhood of 0 ∈ 𝑆 and 𝑆 is locally connected at 0, it is a simple exercise to check that the limit point must be unique, i. where the functions 𝑐 and 𝑎 𝑚 are continuous, 𝑧 ↦ → 𝑐 (𝑧, 𝑠) is analytic, 𝑎 𝑚 (0) = 0 and 𝑐 (0, 0) ≠ 0. This is a version of the Weierstrass preparation theorem. The classical version usually assumes that 𝑠 ∈ C 𝑘 and 𝑓 (𝑧, 𝑠) is analytic in both variables. Our proof can be easily amended to show that if 𝑓 is 𝐶 𝑛 -continuous or analytic with respect to (𝑧, 𝑠), then so are the function 𝑐 and 𝑎 𝑚 .