Analyticity for rapidly determined properties of Poisson Galton--Watson trees

Let $T_\lambda$ be a Galton--Watson tree with Poisson($\lambda$) offspring, and let $A$ be a tree property. In this paper, are concerned with the regularity of the function $\mathbb{P}_\lambda(A):= \mathbb{P}(T_\lambda \vdash A)$. We show that if a property $A$ can be uniformly approximated by a sequence of properties $A_k$, depending only on the first $k$ vertices in the breadth first exploration of the tree, with a bound in probability of $\mathbb{P}_\lambda(A\triangle A_k) \le Ce^{-ck}$ over an interval $I = (\lambda_0, \lambda_1)$, then $\mathbb{P}_\lambda(A)$ is real analytic in $\lambda$ for $\lambda \in I$. We also present some applications of our results, particularly to properties that are not expressible in the first order language of trees.


Introduction
Let X 1 , X 2 , . . . be a sequence of independent Poisson random variables of parameter λ. Set X = (X 1 , X 2 , X 3 , . . . ) and construct a tree T so that node i has X i children, labelling the nodes from top to bottom and left to right, i.e., breadth first ordering (see Figure 1). We call the sequence X the seed of the Poisson Galton-Watson tree T with parameter λ. Note that if the tree has a finite number n of vertices then the values X j for j > n are irrelevant.
Although the offspring distribution completely determines the law of T , it does not provide an immediate sense of the tree's structure. A more transparent structural description of T is provided by tree property probabilities, i.e., for a given tree property A, what is the probability that T has this property? For convenience, we will identify this event T ⊢ A with the property A itself, defining where we write P λ (·) to indicate that the parameter of the Poisson distribution is λ. In this paper we are interested in the regularity of f λ (A) as a function of λ for certain choices of the tree property A.
In essence, this is a question about phase transitions: loss of regularity in P λ (A) at a particular value of λ is interpreted as phase transition in structure of T λ , as 'seen by' property A. We illustrate this idea as follows. Consider the two events As is well known (see, e.g., Prop. 5.4 in [3]), the probability f A 1 (λ) that T λ is infinite satisfies where W 0 (x) is the principle branch of the Lambert W function studied in [2], the unique real solution to is real analytic on I 1 = (0, 1) and on I 2 = (1, ∞), but has a branch cut singularity at λ = 1 and so is not real analytic on any interval containing this point: the interpretation is that the size of a Poisson Galton-Watson tree undergoes a phase transition at λ = 1. On the other hand, the probability that the root node has exactly one child is which is a real analytic function over the entire domain I = (0, ∞). From the perspective of A 2 , there is no phase transition.
Recently, Podder and Spencer [6,7] studied this question in the context of first order properties on the tree. Informally speaking, a first order property can be expressed as a sentence in first order logic, which contains an infinite number of variables, the equality "=" relation, the binary parent relation π(x, y) which is true if y is the parent of x, the root symbol R, universal and existential quantifiers and the usual Boolean connectives.
In [7], Podder and Spencer used the Ehrenfeucht game for rooted trees and a contraction mapping theorem to prove the following: Our main result, Theorem 1.5, is an extension of this to a larger class of properties. We also improve the smoothness. Before stating our result, we introduce some notation and definitions.
The k-truncated seed X (k) = (X 1 , . . . , X k ) is given by the first k elements of the seed X.
(1.7) Definition 1.3. Let 0 ≤ λ 0 < λ 1 ≤ ∞ and let I = (λ 0 , λ 1 ) be an interval. An event A is called rapidly determined over I, if for every λ ∈ I there exist positive constants c and C, k 0 ∈ N, and a sequence of k-tautologically determined events A k such that for all k ≥ k 0 . Every first order property is rapidly determined over (0, ∞).
We can now state our main result.
Theorem 1.5. Let 0 ≤ λ 0 < λ 1 ≤ ∞ and let I = (λ 0 , λ 1 ) be an interval. Suppose that the property A is rapidly determined over the interval I. Then f A (λ) is a real analytic function on I.
The conclusion of the theorem means that for every λ ∈ I, there exists δ > 0 so that the function f A (λ) can be extended to a complex analytic function f A (z) on the disc D δ (λ) = {z | |z − λ| ≤ δ}. Theorem 1.5 improves on Theorem 1.1 in two ways. Firstly, we broaden the scope of applicability to the larger class of rapidly determined properties, and secondly, we improve the regularity from C ∞ to real analytic. The collection of first order properties is countable, since every first order property is specified by a finite sequence from a countable alphabet. On the other hand, Proposition 3.4 in Section 3 describes uncountably many rapidly determined properties. Unlike Podder and Spencer, our methods are not model theoretic in nature. Instead, we take a more direct, complex analytic approach. It is similar in spirit to the route taken in [4,5], where the regularity of Lyapunov exponents for products of discrete random matrices was studied.

Analyticity for rapidly determined properties
In this section we prove Theorem 1.5. We begin with a preliminary result. it follows that P λ (A) may be analytically continued to an entire function P z (A).
Proof of Theorem 1.5. Let λ ∈ I. Since A is rapidly determined over the interval I, there exist constants c and C, k 0 ∈ N and a sequence of k-tautologically determined events (A k ) so that for all k ≥ k 0 From this it then follows that From Lemma 2.1 we get that f A k (λ) = P λ (A k ) can be extended to a complex analytic function over C that we denote f A k (z). In order to establish that f A (λ) can also be extended to an analytic function in some neighbourhood of λ ∈ I, it suffices to show that for every λ ∈ I there exist positive constants c 1 and c 2 and δ > 0 such that for all z ∈ D δ (λ) = {z ∈ C : |z − λ| ≤ δ} we have Indeed, this will then imply that f An (z) converges uniformly to a function denoted f A (z), which will also be analytic on D δ (λ).
Using this for all k terms of the product appearing in (2.6) now proves (2.5).
Since the event A k \ A k−1 is k-tautologically determined, we let B ⊆ N k be such that We can now apply (2.5) and get for ε and δ as above (2.7) Since k i=1 X i has the Poisson distribution with parameter kλ, it follows that there exists a positive constant c 1 so that for ℓ > [3kλ] From (2.3) we get that there exist positive constants c 2 and c 3 so that for all k Taking ε sufficiently small and using the two bounds above into (2.7) we obtain for positive constants c 4 and c 5 |f A k \A k−1 (z)| ≤ c 4 e −c 5 k and this concludes the proof of (2.4) and also the proof of the theorem.

Examples of rapidly determined properties
In this section we provide some examples of rapidly determined properties to demonstrate the applicability of Theorem 1.5.
We start by showing that when the tree is subcritical, every property is rapidly determined. For a tree T we write |T | for the total number of vertices of T .
Then every property A is rapidly determined on the interval I = (λ 0 , λ 1 ).
Proof. Let A k = A ∩ {|T | < k}. Since A k is a k-tautologically determined event, it suffices to show that for every λ < 1 there exist positive constants c and C so that for all k P λ (A△A k ) ≤ Ce −ck . (3.1) We now have Using that k i=1 X i has the Poisson distribution with parameter kλ and λ < 1 proves (3.1) (See, e.g., Appendix A in [1]), and this concludes the proof. Remark 3.2. One interpretation of the proposition above is that Poisson Galton-Watson trees do not exhibit a phase transition in any property over the interval I = (0, 1). Lemma 3.3. Let E k be the set of nodes amongst the first k which lie on an even level. Then for every λ ∈ (0, ∞) there exists a positive constant c so that Proof. Let E k (O k ) be the set of nodes amongst the first k which lie on an even (odd) level. On the event {|T | ≥ k} all the first k nodes exist, and hence Set n = ⌊k/(2λ + 1)⌋. There exists a positive constant c so that and this concludes the proof.
Next we prove a more general statement than the one given in Proposition 1.7. As noted in the Introduction, this statement implies that there are uncountably many rapidly determined properties.
In the following, if F ⊂ N is a set of levels, we say a node lies on an F -level if the level of the node is contained in F . Proof. Let E k and F k be the sets of nodes among the first k which lie on an even/F -level, respectively. As in the proof of Lemma 3.3, let Y 1 , . . . be i.i.d. Poisson(λ) random variables, representing the number of children that are attached to the i-th vertex on an even level. We now define the event A k (resp. B k ) that in E k (F k ) there exists a node with exactly one (two) child(ren), i.e., Set n = ⌊k/(2λ + 1)⌋. We now have for positive constants c, c 1 and c 2 , where in the last inequality we have used Lemma 3.3.