Trees with power-like height dependent weight

We consider planar rooted random trees whose distribution is even for ﬁxed height h and size N and whose height dependence is given by a power function h α . Deﬁning the total weight for such trees of ﬁxed size to be Z N , a detailed analysis of the analyticity properties of the corresponding generating function is provided. Based on this, we determine the asymptotic form of Z N and show that the local limit at large size is identical to the Uniform Inﬁnite Planar Tree, independent of the exponent α of the height distribution function


Introduction
The study of random geometric objects has been actively pursued in recent years.In particular, deep results have been obtained for various models of random planar maps (or surfaces) concerning local limits and scaling limits.Thus, the Uniform Infinite Planar Triangulation was constructed in [4] and the Uniform Infinite Planar Quadrangulation appeared in [8], see also [23] and [11,29], while initial studies of the scaling limit of planar maps in [10] have been continued in numerous papers leading to proofs of existence of the limiting measure as well as establishing important properties of the limit, see e.g.[27,24,25].While the subject is of natural interest in its own right within probability theory, important motivations and applications also arise from problems in theoretical physics, in particular statistical mechanics and quantum gravity, see e.g.[3] and references therein.
A more classical topic in the same realm, and equally relevant from a physical point of view, is that of random trees, see [13] and references therein.Local limits and scaling limits have been constructed in this case as well and detailed information about the limiting measures can frequently be obtained using the highly developed theory of branching processes.We shall in this paper concentrate on local limit results, in particular relating to the Uniform Infinite Planar Tree (UIPT) that can be obtained as a local (or weak) limit of uniformly distributed finite rooted planar trees of fixed size tending to infinity [14].Indeed, this limit can be viewed as a particular instance of a more general local limit result for critical Bienaymé-Galton-Watson (BGW) trees conditioned on size, namely when the off-spring distribution is a geometric sequence, a result first established in [21] and [2].Earlier results on local limits of critical (and subcritical) BGW trees conditioned on height go back to H. Kesten [22] and turn out to yield the same limit distribution supported on one-ended trees.More recent results allowing more general conditionings can be found in [20] and [1].Properties of the limiting measures, e.g.relating to their Hausdorff dimension and spectral dimension, have been established in [6] and [15].
While the results just described depend heavily on the fact that the weights of individual trees are local, in the sense of being products of weights associated with the vertices of the tree, the case of non-local weight functions has been much less addressed in the literature.Thus, while the average height of various ensembles of planar trees has been of interest in many works, such as [12,28,18,9,30], it seems that properties of random planar trees with height-dependent weights have not been extensively explored, an exception being [26], where a model of depth weighted random recursive trees is studied, with branching probability depending on vertex height.
In the present work we investigate a rather different case where the weight function f (h) depends on the height h of the tree and otherwise is constant for fixed size (see section 2 for a precise definition).It is easy to see that different limits can be obtained by a judicious choice of f .A detailed study of such cases will be given in [17].Here, we consider the case where f is a power function, f (h) = h α , making use of transfer theorems from analytic combinatorics.The main result, stated in Theorem 4.2, is that the local limit in this case equals the UIPT, independently of the value of the exponent α.
The paper is organised as follows.Section 2 contains a precise definition of the metric and associated Borel algebra on the space of rooted planar trees to be considered.Furthermore, the finite size distributions, whose limits we aim at calculating, are defined.In section 3, the analytic behaviour of the generating function for the total weights of trees as a function of the size N is examined for fixed α, based on well known results for the corresponding generating functions for trees of fixed height.Some technical details of the calculations involved are deferred to an appendix.Applying transfer theorems, the asymptotic behaviour of the coefficients at large N is determined.These results are used in section 4 to prove existence of the weak limit and to identify it as the UIPT.Finally, section 5 contains some concluding remarks.

Preliminaries
In the following, T N will denote the set of rooted planar trees of size |T | = N with root vertex of degree 1.Here N ∈ N := {1, 2, 3, . . .} or N = ∞ (in which case T is assumed to be locally finite) and we set For T ∈ T the notation h(T ) will be used for the height of T , i.e. the maximal length of a simple path in T originating from the root.For fixed α ∈ R we consider the probability measures µ N , N ∈ N, on T given by µ where Z N is a normalisation factor, also called the finite size partition function, given by Thus, µ N is supported on T N and our goal is to study the weak limit of µ N as N → ∞ as a probability measure on T .Here T is considered as a metric space whose metric dist is defined as follows.Denoting the root of T ∈ T by v 0 , the ball B r (T ) of radius r in T around v 0 is by definition the subgraph of T spanned by vertices at distance at most r from v 0 , i.e.
where d T designates the graph distance on T and V (G) denotes the vertex set of any given graph G.For T, T ′ ∈ T we then set It is easily verified that dist is a metric on T .In fact, it is an ultrametric in the sense that for any triple T, T ′ , T ′′ of trees in T we have The ball of radius s > 0 around a tree T will be denoted by B s (T ).If s = 1 r , where r ∈ N, it is seen that where T 0 = B r (T ).For additional properties of the metric space (T , dist), including the fact that it is a separable and complete metric space, see e.g.[14].
By definition, a sequence of probability measures (ν N ) N ∈N on T is weakly convergent to a probability measure ν on T T f dν N → T f dν as N → ∞ for all bounded continuous functions f on T .We refer to [7] for a detailed account of weak convergence of probability measures.
As mentioned earlier, the main result of this paper is a proof that the weak limit of the sequence (µ N ) N ∈N exists and equals the UIPT, which will be described in more detail in section 4. A basic ingredient in the proof is the generating function W α for the Z N , defined by where g is a real or complex variable.It is known, and will also be shown below, that the sum is convergent for |g| < 1 4 and divergent for |g| > 1 4 .It will be convenient for the following discussion to define It is a known fact, and easy to verify, that X m fulfills the recursion relation , m ≥ 1, and X 1 (g) = g .
This equation can be rewritten in linear form and hence solved explicitly.The result is given by (see e.g.[13]) Defining X 0 = 0 and z = 1 − 4g , this gives Noting that for z = x + iy ∈ C we have x > 0, and hence, by (6), X m is an analytic function of z away from the imaginary axis for all m.Note also that for fixed such z the right-hand side of (8) decays exponentially with m and hence the series in (5) converges.Taking z to be real we get, in particular, that W α is well-defined for 0 ≤ g < 1  4 .On the other hand, considering the denominator in (6) we see that it is an odd polynomial in z of degree m + 1 if m is odd and otherwise of degree m, and it is straightforward to see that its roots are given by Hence, it follows from (6) that these are precisely the (simple) poles of X m apart from z = 0 corresponding to p = 0 where both numerator and denominator have simple zeroes.Therefore, X m is analytic for |z| < |z 1 | and hence for |g| < g m where with a singularity at g = g m .Since g m → 1 4 as m → ∞ it follows from (5) that the sum defining W α (g) is divergent for g > 1  4 , as claimed above.
3 Generating functions

Singular behaviour
In order to establish the existence of the limit of (µ N ) we shall determine the asymptotic behaviour of Z N as N → ∞.This will be done by analysing the singularity of W α at g = 1 4 in more detail and using so-called transfer theorems [19].When considering W α as a function of z we will use the notation W α (z) for W α (g).
Before stating the main result of this section, we note a few elementary facts that will be needed, formulated in the following three lemmas.Lemma 3.1.For |z| < 1 and m ≥ 1 we have where the coefficients for all k.Moreover, Proof.First, we note that the term of order k in the series expansion is identical to the one in the polynomial From this expression we see that the coefficient of (2mz) k is given by where the first inequality follows by replacing the prefactors by 1 and lifting the restriction r 1 + 2r 2 + ... + (k − 1)r k−1 = k − r, while the second inequality results from estimating the finite sum in parenthesis by the corresponding infinite sum.This proves the first part of the lemma.
To show (11), we rewrite (12) for k ≤ m in the form Here, the latter sum can be estimated as above by while expanding the first term yields an expression of the form where the coefficient c m k is easily seen to fulfill the bound Combining the two estimates yields the claimed bound.
The next lemma concerns properties of the denominator function appearing in (8), for small values of z.We use the notation A k for the Taylor coefficients of as well as where K and L are positive constants.b) For |z| ≤ tan( π m+1 ) we have where the coefficients and there exist positive numbers B k , k = 1, 2, . . ., independent of m, such that Proof.With notation as in Lemma 3.1 we have where it has been used that a m 1 = 1 and a m 2 = 1 2 .Hence, and ( 14) follows immediately from (10) with K = 2e 2 .Similarly, the bound (15) follows easily from (10) and (11) with L = 2(e 3 + 1).This proves part a) of the lemma.
For the second part we note that z 2 Dm(z) is a meromorphic function of z which is analytic for |z| < tan( π m+1 ) as shown in section 2. The power series for z 2 /D m (z) in this disc is obtained by inverting that of Dm(z)  z 2 , i.e. the coefficients c m 2k are determined by where c m 0 ≡ 1 .Using the the bound ( 14) just proven, we obtain which implies ( 16) by a simple induction argument.Using that Using the bounds of part a), this gives Since c 0 = A 0 = 1, the inequality ( 17) now follows by induction with B k defined recursively as B 0 = 0 and This concludes the proof.
For fixed a > 0 we denote in the remainder of this paper by V a the wedge in the right half-plane given by Lemma 3.3.Let a > 0 and 0 < ǫ < 1 be given.Then there exist positive constants K 0 , K 1 , m 0 depending on a, and δ 0 depending on a and ǫ, such that the following statements hold.
b) For z = x + iy ∈ V a , |z| ≤ δ 0 , and m ≥ m 0 , Proof.a) For any z ∈ C we have On the other hand, for z ∈ V a we have Combining these two estimates yields (19) with where O(z) is analytic and fulfills for z in a suitably small disc around 0, where c is some constant independent of m.Therefore, choosing δ 0 small enough, it follows that 2mz(1 Introducing the shorthand we get Consider first the last expression in the case where mx ′ ≥ 1.In the second term inside square brackets, the factor multiplying |z| is numerically bounded by a constant depending only on a.Hence, choosing δ 0 sufficiently small, the term in square brackets is bounded from below, say by1 2 , for |z| < δ 0 .On the other hand, if mx ′ ≤ 1 we observe that the factor inside round brackets is bounded while the prefactor |z| sinh(mx ′ ) can be estimated as follows.First, choosing δ 0 sufficiently small such that Then, choosing m 0 sufficiently large, the term in square brackets in ( 22) is bounded from below by 1 2 if m ≥ m 0 .Thus, we have shown that for m 0 sufficiently large and δ 0 > 0 small enough it holds that Given ǫ, choose δ 0 small enough such that |zO(z)| < |z| 1+a ǫ for |z| < δ 0 .For such z in V a we then have where from which the lower bound in (20) follows in view of (23).By a slight modification of the arguments above, the upper bound follows similarly.
We are now ready to prove the first main result of this section.
for z small in V a .
Proof.We claim that the polynomial W α is given by replacing the summand in the definition (5) of W α by its Taylor polynomial of degree 2n, i.e.
and that c α is given by where L n (t) stands for the Laurent polynomial of order 2(n − 1) for Setting we start by rewriting where in the last line we converted the integral over the positive real axis to a line integral along the half line ℓ z : t → tz inside the wedge V a , by using Cauchy's theorem and the fact that α + 2k − 2 < −1 for k ∈ {0, 1, .., n} implies that the integrand decays fast enough and uniformly to 0 at infinity inside V a , such that the integral along the circular part of the contour at infinity vanishes.
It is convenient to split the sum and integral in the last expression into three regions given by 3) r −1 < m|z| , r −1 < t|z| , respectively 2 .Here, r will ultimately be chosen as a suitable function of |z| tending to 0 as z tends to 0, but for the moment we merely assume 0 < r < 1. Denoting the corresponding contributions to the series and integral in (28) by S 1 , S 2 , S 3 and I 1 , I 2 , I 3 , respectively, we have and we shall proceed by successively estimating the numerical values of S 1 , I 1 first, then S 2 − I 2 , and finally of S 3 , I 3 .In the remainder of this proof, "cst" will denote a generic positive constant independent of z and r within the stated ranges, and likewise O(w) will denote a generic function of order w for |w| small, i.e. |O(w)| ≤ cst • |w|.
2 More precisely, t is restricted according to 0 < t ≤ r |z| , r |z| < t ≤ 1 r|z| and 1 r|z| < t, respectively, where ⌊x⌋ denotes the integer part of x ∈ R. For the sake of simplicity, we shall in the following apply the notation in (29) and omit writing explicitly the relevant integer parts.
S 1 and I 1 : It follows from Lemma 3.2 that which we henceforth assume to hold.It follows that For I 1 , on the other hand, we get S 2 − I 2 : We further decompose this expression as and proceed to establish bounds on A, B and C. In order to deal with A, we note that with the notation z ′ introduced earlier in (21) we have By Lemma 3.3 with, say, ǫ = 1 2 the denominator fulfills To estimate the numerator, we note the inequalities valid if m|z| 2 is bounded, which is the case if (39) holds and m|z| < 1 r .For m|z| ≤ 1 we hence get cosh(2mz and together with (37) and (38) this implies provided (39) holds, which we assume in the following.
On the other hand, in the range 1 < m|z| < r −1 the inequalities (40) imply Combining this with the bound which follows from Lemma 3.3 a) and b) with ǫ = 1 2 , we get Thus, using (41) and (42), we obtain the bound valid for any (fixed) value of α ∈ R.
To derive an upper bound for |B| it is useful to rewrite It is straight-forward to estimate the integrand using (19) and recalling that α < 1.One finds which yields the bound In order to estimate |C|, we first use the mean value theorem to write where and recalling (17), which holds for all m in the summation range provided z fulfills the condition we see that (46) implies for such values of z.
S 3 and I 3 : Using Lemma 3.3 a) we have and similarly Lemma 3.3 b) yields Furthermore, using (16) we have Taking into account also the obvious bound it follows that Combining the estimates (32), ( 33), ( 49) and (54), we conclude that provided z, r and n fulfill conditions (31), ( 39) and (47).Choosing r = |z| β where 0 < β < 1, these conditions are evidently satisfied for |z| small enough.Noting that 1 − α − 2n > 0, it follows from (55) that if β is chosen such that This concludes the proof of Theorem 3.4.
The next theorem is similar to the previous one and covers the case α > 1.
Theorem 3.5.Assume α > 1 and let a > 0. Then W α is analytic in the right half-plane C + and there exists ∆ > 0 such that for z ∈ V a small, where Proof.Applying Cauchy's theorem as in (28) gives As previously, we split the sum and integral in (57) into three regions as in (29) and denote the corresponding contributions by S 1 , S 2 , S 3 and I 1 , I 2 , I 3 , respectively.The relevant estimates can then be obtained by suitably modifying the arguments of the previous proof as follows.
The result for the remaining values of α, i.e. α = −(2n − 1), n ∈ N 0 , is stated in the following theorem whose detailed proof is deferred to Appendix A. Theorem 3.6.Assume α = −(2n − 1), n ∈ N 0 , and let a > 0. Then W α (z) is analytic in the right half-plane and there exists a polynomial P n (z) of degree 2n and a constant ∆ > 0 such that for z small in V a , where

Coefficient asymptotics
Recalling that z = √ 1 − 4g, it follows from Theorems 3.4, 3.5 and 3.6 that W α is an analytic function of g in the slit-plane C \ [ 1 4 , +∞).The asymptotic behaviour of the coefficients Z N in its power series expansion (4) around g = 1  4 , that will be needed in the next section, can be deduced by applying transfer theorems yielding the following result.Proposition 3.7.For fixed α ∈ R it holds for large N that where the constant C α is given by Proof.Consider first the case α = −(2n − 1) , n ∈ N 0 .With notation as in section VI.3 of [19], it follows from Theorems 3.4 and 3.5 with a > 1 that there exists a ∆-domain where φ a = π − 2 tan −1 (a) < π 2 and η > 0 can be chosen arbitrarily.Applying Corollary VI.1 in [19] then gives the result.
For α = −2n + 1 we recall the well known fact (see e.g.section VI.2 in [19]) that This immediately implies that Applying Theorem VI.3 of [19] to the remainder term O(|z| 2n+∆ ) in (65) then us to conclude that as N → ∞, as well as that (−1) n A n = |A n |, since Z N by definition is positive.This completes the proof of the theorem.
In the particular case, α = 0, the partition function can be calculated in closed form (see e.g.[14] for details) and is given by Moreover, its Taylor coefficients Z N are given by the Catalan numbers, For later use, we also note that Finally, we shall also need the asymptotic behaviour for large N of the Taylor coefficients Z N,M of the function i.e. the contribution to W α from trees of height at most M .Since each X m is a rational function of g by ( 6), the same holds for W α,M , and it has a unique pole closest to g = 0 which is simple and located at g M given by (9).Denoting its residue by r M , it follows that for N large.
Clearly, the sets A M i , i = 1, . . ., R, are disjoint subsets of A, if N > |T 0 | + RM , and hence Moreover, for such where the modified partition function ZK,M is given by ZK,M = Here, the last inequality is a consequence of (73), assuming that M > M 0 .With notation as in section 3.2, the last sum in (76) equals Z K − Z K,M , and according to (66) and (69) the bound holds for fixed M and for N large enough, where we have also taken into account that g M ≤ 1 and , the right-hand side of (77) decays exponentially with N .It thus follows from (66), for fixed M and fixed N i , i = i 0 , fulfilling the summation constraints of (75), that Using this together with (76) and (74), we get for any M > M 0 and N sufficiently large that which is equivalent to (72).
For ǫ > 0 and M > M 0 , let us denote the large-N limit of the right hand side of (72) by Λ(T 0 , M, ǫ), i.e.

Λ(T
and let Λ(T 0 ) := lim Lemma 4.2.For any r ∈ N it holds that Proof.We use an inductive argument.For r = 1 the statement trivially holds, so let r ≥ 2 be arbitrary and assume (80) holds for r−1.Recall that the R-factor in (79) originates from summing over the position i 0 of the long branch out of R branches.This branch has an ancestor j in T 0 at height r − 1, which is the root of the branch T i 0 .Letting R ′ denote the number of vertices at height r − 1, it follows that summing over trees T 0 of height r with T ′ 0 := B r−1 (T 0 ) fixed and with a marked root edge of the large branch amounts to summing over all possible choices of the remaining R − 1 edges at maximal height.Since the number of such choices equals Since the last expression in (81) equals Λ(T ′ 0 ), this completes the proof.
Corollary 4.3.The sequence (µ N ) N ∈N of measures on T given by (1) is tight.
Proof.It is easy to verify (see e.g.[14] for details) that sets of the form where (K r ) r∈N is any sequence of positive numbers, are compact.In order to establish (71) for sets of this form, it is sufficient to show that for every δ > 0 and r ∈ N there exists K r > 0, such that Indeed, choosing δ to be r-dependent of the form δ r = ε 2 r and defining C by (82) for the corresponding values of K r determined by (83), we obtain (84)

Concluding remarks
We note that, despite the widely varying singular behaviour of the generating function W α (g): it is finite at the critical point for α < 1, has a logarithmic divergence when α = 1 and a power-like divergence for α > 1, we have found that the local limit of the distributions (1) is independent of the exponent α ∈ R. Whether more general BGW trees respond in a similar way to a powerlike height coupling, we do not know, since our approach relies on knowing the explicit form of the generating function for fixedheight partition functions, given in (8), the analogue of which is not generally available.It is, however, conceivable that more general techniques based on recursion relations alone could be developed.Another way of extending the results of this paper would be allowing different forms of height couplings.In [17], we consider weights of exponential form, k h , at fixed size, partly motivated by findings in [16], where an analysis of certain statistical mechanical models of loops on random so-called causal triangulations is carried out.Via a bijective correspondence between causal triangulations and rooted planar trees, it turns out that some of those models can be related to models of planar random trees with exponential height coupling with k > 1.As shown in [17], it turns out that the local limits exhibit qualitatively different behaviours, depending on whether 0 < k < 1, k = 1, or k > 1.
Recalling the definition (25) of W Here, the first term inside round parenthesis is harmonic and yields the contribution Next, we proceed to estimate |S − I| as before by splitting the summation and integration domains corresponding to r < m|z| ≤ 1 r and m|z| > 1 r and similarly for s.Calling the corresponding sums and integrals S 2 , S 3 and I 2 , I 3 , respectively, we first rewrite

1
is defined by(27).Consider first the contribution S 1 to this sum from m ≤ r |z| , where r satisfies (31).With notation as in Lemma 3.2 b) we rewrite