Sesqui-type branching processes

TL;DR

This paper analyzes a special two-type branching process where only one type reproduces, providing key estimates for survival and total particles, which are crucial for understanding certain random graph processes.

Contribution

It introduces and analyzes a two-type branching process with a barren type, offering new estimates for survival probability and total particles, relevant for bounded-size Achlioptas processes.

Findings

Derived survival probability estimates

Established tail bounds for total particles

Linked results to bounded-size Achlioptas processes

Abstract

We consider branching processes consisting of particles (individuals) of two types (type L and type S) in which only particles of type L have offspring, proving estimates for the survival probability and the (tail of) the distribution of the total number of particles. Such processes are in some sense closer to single- than to multi-type branching processes. Nonetheless, the second, barren, type complicates the analysis significantly. The results proved here (about point and survival probabilities) are a key ingredient in the analysis of bounded-size Achlioptas processes in a recent paper by the last two authors.

Equations

$$g_{Y,Z}(y,z):=E(y^Y{z}^Z)=\sum _{k,l⩾0}P(Y=k,Z=l)y^k{z}^l,$$

$$f_{Y,Z}(y,z):=g_{Y,Z}\bigl{(}e^{y},e^{z}\bigr{)}=\operatorname{\mathbb{E}{}}\bigl{(}e^{yY+zZ}\bigr{)}.$$

$$D^{2}f(z)[u,v]=(uv)D^{2}f(z)\left(\begin{array}{c}u\\ v\end{array}\right).$$

$$A⩾cI⟺v^{\mathrm{tr}}Av⩾c|v{|}^2\text{ for all v∈Rd}.$$

$$c_1⩽c_8⩽c_7⩽c_6⩽c_5\text{and}{c}_4⩽c_2$$

$$E{R}^{Y+Z}$$

$$EY$$

$${\pi }_{k_1,k_2}⩾\delta ,{\pi }_{k_1+1,k_2}⩾\delta ,{\pi }_{k_1,k_2+1}⩾\delta .$$

$$EZ⩾P(Z=k_2+1)⩾\delta ,$$

$$\operatorname{\mathbb{P}{}}(|\mathfrak{X}|=N)=N^{-3/2}e^{-N\xi}\bigl{(}\theta+O(N^{-1})\bigr{)},$$

$$\xi ={\xi }_{Y,Z}:=-\Psi (x^{\ast })⩾0,$$

$$\theta ={\theta }_{Y^0,Z^0,Y,Z}:=\sqrt{2\pi /|{\Psi }^{\prime \prime }(x^{\ast })|}\Phi (x^{\ast })=\Theta (1),$$

$$\xi =\Theta (|EY-1{|}^2).$$

$$\begin{array}{r}{p}_{n,m}:=P(|{\mathfrak{X}}^L|=n,|{\mathfrak{X}}^S|=m).\end{array}$$

$$\operatorname{\mathbb{P}{}}\bigl{(}|\mathfrak{X}^{L}|=n,\,|\mathfrak{X}^{S}|=m\bigm{|}Y^{0}=n_{0},\,Z^{0}=m_{0}\bigr{)}=\frac{n_{0}}{n}\operatorname{\mathbb{P}{}}\biggl{(}n_{0}+\sum_{1\leqslant j\leqslant n}Y_{j}=n,\ m_{0}+\sum_{1\leqslant j\leqslant n}Z_{j}=m\biggr{)}.$$

$$\operatorname{\mathbb{P}{}}\bigl{(}|\mathfrak{X}^{L}|=n,\,|\mathfrak{X}^{S}|=m\bigm{|}Y^{0}=n_{0},\,Z^{0}=m_{0}\bigr{)}\\ =\operatorname{\mathbb{P}{}}\biggl{(}n_{0}+\min_{0\leqslant n^{\prime}<n}\sum_{1\leqslant j\leqslant n^{\prime}}(Y_{j}-1)>0,\ n_{0}+\sum_{1\leqslant j\leqslant n}(Y_{j}-1)=0,\ m_{0}+\sum_{1\leqslant j\leqslant n}Z_{j}=m\biggr{)}.$$

$$[y^{n-n_{0}}z^{m-m_{0}}]\bigl{(}g(y,z)^{n}\bigr{)}=[y^{n}z^{m}]\bigl{(}y^{n_{0}}z^{m_{0}}g(y,z)^{n}\bigr{)}.$$

$$\begin{split}np_{n,m}&=\sum_{n_{0},m_{0}\geqslant 0}\operatorname{\mathbb{P}{}}(Y^{0}=n_{0},Z^{0}=m_{0})n_{0}[y^{n}z^{m}]\bigl{(}y^{n_{0}}z^{m_{0}}g(y,z)^{n}\bigr{)}\\ &=[y^{n}z^{m}]\bigl{(}\tilde{g}_{0}(y,z)g(y,z)^{n}\bigr{)},\end{split}$$

$$\begin{array}{r}{\tilde{g}}_0(y,z):=\sum _{n_0,m_0⩾0}P(Y^0=n_0,Z^0=m_0)n_0{y}^{n_0}{z}^{m_0}=y\frac{\partial }{\partial y}{g}_{Y^0,Z^0}(y,z).\end{array}$$

$$\tilde{f}_{0}(y,z):=\tilde{g}_{0}(e^{y},e^{z})=\frac{\partial}{\partial y}f_{Y^{0},Z^{0}}(y,z)=\operatorname{\mathbb{E}{}}\bigl{(}Y^{0}e^{yY^{0}+zZ^{0}}\bigr{)}.$$

$$\begin{array}{r}n{p}_{n,m}=\frac{1}{(2\pi \mathrm{i}{)}^2}\oint \oint y^{-n}{z}^{-m}{\tilde{g}}_0(y,z)g(y,z{)}^n\frac{\mathrm{d}y}{y}\frac{\mathrm{d}z}{z},\end{array}$$

$$n{p}_{n,m}=\frac{1}{4{\pi }^2}{\int }_{-\pi }^{\pi }{\int }_{-\pi }^{\pi }{e}^{-n(\alpha +\mathrm{i}u)-m(\beta +\mathrm{i}v)}{\tilde{f}}_0(\alpha +\mathrm{i}u,\beta +\mathrm{i}v)f(\alpha +\mathrm{i}u,\beta +\mathrm{i}v{)}^n\mathrm{d}u\mathrm{d}v.$$

$$\phi (y,z)={\phi }_{Y,Z}(y,z):=\log f_{Y,Z}(y,z)=\log f(y,z),$$

$$\bigl{|}f(\alpha+\mathrm{i}u,\beta+\mathrm{i}v)-1\bigr{|}\leqslant(|\alpha+\mathrm{i}u|+|\beta+\mathrm{i}v|)C\leqslant 4c_{2}C\leqslant 1/2,$$

$$\bigl{|}f(\alpha+\mathrm{i}u,\beta+\mathrm{i}v)\bigr{|}\leqslant f(\alpha,\beta)e^{-c_{3}(u^{2}+v^{2})}.$$

$$f(\alpha +\mathrm{i}u,\beta +\mathrm{i}v)=\sum _{k,l⩾0}{\pi }_{k,l}{e}^{k(\alpha +\mathrm{i}u)+l(\beta +\mathrm{i}v)},$$

$$f(\alpha,\beta)^{2}-|f(\alpha+\mathrm{i}u,\beta+\mathrm{i}v)|^{2}=\sum_{k,l,m,n}\pi_{k,l}\pi_{m,n}e^{(k+m)\alpha+(l+n)\beta}\Bigl{(}1-\operatorname{Re}e^{\mathrm{i}(k-m)u+\mathrm{i}(l-n)v}\Bigr{)}.$$

$$\begin{array}{rl}f(\alpha ,\beta {)}^2-|f(\alpha +\mathrm{i}u,\beta +\mathrm{i}v){|}^2& ⩾{\delta }^2{e}^{(2k_1+1)\alpha +2k_2\beta }(1-\cos u)+{\delta }^2{e}^{2k_1\alpha +(2k_2+1)\beta }(1-\cos v)\\ & =\Omega (u^2+v^2),\end{array}$$

$$1-|f(\alpha +\mathrm{i}u,\beta +\mathrm{i}v){|}^2/f(\alpha ,\beta {)}^2⩾2c_3(u^2+v^2)$$

$$|f(\alpha +\mathrm{i}u,\beta +\mathrm{i}v){|}^2/f(\alpha ,\beta {)}^2⩽1-2c_3(u^2+v^2)⩽e^{-2c_3(u^2+v^2)},$$

$$D^2\phi (\alpha ,\beta )[u,v]⩾c_3(u^2+v^2),u,v\in \mathbb{R}.$$

Full text

1 Introduction

Throughout the paper we consider branching processes in which every particle is of one of two types, called (for compatibility with the notation in [22]), ‘type $L$’ and ‘type $S$’. Particles of type $S$ may be thought of as barren: they have no children. Each particle of type $L$ will have some random number of children of each type; as usual, we have independence between the children of different particles, but the numbers $Y$ and $Z$ of type-$L$ and type-$S$ children of one particle need not be independent. The formal definition is as follows.

Definition 1.1**.**

Let $(Y,Z)$ and $(Y^{0},Z^{0})$ be probability distributions on $\mathbb{N}^{2}$. We write $\mathfrak{X}^{1}=\mathfrak{X}^{1}_{Y,Z}$ for the Galton–Watson branching process started with a single particle of type $L$, in which each particle of type $L$ has $Y$ children of type $L$ and $Z$ of type $S$. Particles of type $S$ have no children, and the children of different particles are independent. We write $\mathfrak{X}=\mathfrak{X}_{Y,Z,Y^{0},Z^{0}}$ for the branching process defined as follows: start in generation one with $Y^{0}$ particles of type $L$ and $Z^{0}$ of type $S$. Those of type $L$ have children according to $\mathfrak{X}^{1}_{Y,Z}$, independently of each other and of the first generation. Those of type $S$ have no children. We write $|\mathfrak{X}|$ ($|\mathfrak{X}^{1}|$) for the total number particles in $\mathfrak{X}$ $(\mathfrak{X}^{1})$.

These branching processes are in some sense essentially single-type: one could first generate the tree of type-$L$ particles as a classical single-type Galton–Watson process, and then consider particles of type $S$. However, since the numbers of type-$S$ and type-$L$ children are not necessarily independent, this two-stage description does not seem particularly easy to work with.

The motivation for considering such processes (and in particular for allowing a different rule for the first generation) comes from the application to studying the phase transition in Achlioptas processes in [22]. Achlioptas processes are evolving random graph models that have received considerable attention (see, e.g., [1; 19; 4; 24; 14; 20; 15; 3; 21] and the references therein). We shall say nothing further about these random graph processes here, aiming to keep the paper self-contained, and purely about branching processes.

We shall prove two main results. Firstly, in Section 2, we consider an individual branching process of the type above, giving an asymptotic formula for the point probability $p_{N}=\operatorname{\mathbb{P}{}}(|\mathfrak{X}|=N)$ under certain conditions on the distributions $(Y,Z)$ and $(Y^{0},Z^{0})$. This formula is proved in Sections 2.1–2.3, which are the heart of the paper. Then, in Section 3, we consider families of processes where the offspring distribution varies analytically in an additional parameter $t$. Roughly speaking, we show that the key quantities in the formula in Section 2 then vary analytically in $t$. This result (which in particular implies properties of the near-critical case) is needed in [22]. Finally, in Section 4, we prove corresponding results for the survival probability $\operatorname{\mathbb{P}{}}(|\mathfrak{X}|=\infty)$. Here the barren type plays no role, so the results effectively concern single-type processes and are much simpler.

Remark 1.2**.**

Although the definition of sesqui-type branching processes is adapted to the application in [22], the results here are applicable, at least in principle, to a more general class of branching processes. Consider a finite-type Galton–Watson process in which there is one special type (type $L$), and all other types are ‘doomed’ (lead to finite trees of descendants a.s.). Such a process may be transformed into a sesqui-type process in a natural way: for each type-$L$ particle replace its children of all doomed types, and their (necessarily doomed) descendents, by type-$S$ children (keeping the same total number of particles). For our results to apply to the transformed process we need further conditions, roughly speaking that the ‘doomed’ subtrees are not too close to critical; but in outline, all processes with (at most) one type that can potentially survive are covered. Branching processes of this type (with one doomed type) have been studied by several authors, giving various results different from ours; see for example [23; 25; 7].

1.1 Some notation and conventions

Throughout we write $\mathbb{N}:=\{0,1,2,\dots\}$ for the non-negative integers.

Given a two-dimensional random variable $(Y,Z)$ taking values in $\mathbb{N}^{2}$, we denote its bivariate probability generating function by

[TABLE]

for all complex $y$ and $z$ such that the expectation (or sum) converges absolutely. We will also consider the bivariate moment generating function

[TABLE]

When considering a particular branching process as in Definition 1.1, we often write $g=g_{Y,Z}$ and $f=f_{Y,Z}$ for brevity.

We denote the coefficient of $y^{k}z^{l}$ in a power series $G(y,z)$ by $[y^{k}z^{l}]G(y,z)$.

We say that a function $f$ defined on $I\subseteq\mathbb{C}$ is analytic if for every $x_{0}\in I$ there is an $r>0$ and a power series $g(x)=\sum_{j\geqslant 0}a_{j}(x-x_{0})^{j}$ with radius of convergence at least $r$ such that $f$ and $g$ coincide on $(x_{0}-r,x_{0}+r)\cap I$. A function $f$ defined on some domain including $I$ is analytic on $I$ if $f|_{I}$ is analytic. The definitions for functions of several real or complex variables are analogous.

If $f$ is an analytic function of $d$ variables, defined in an open set $U\subseteq\mathbb{C}^{d}$, we denote its derivative by $Df$, and its $m$th derivative by $D^{m}f$. Note that $D^{m}f$ is an analytic function from $U$ to the linear space of all (symmetric) $m$-linear forms $\mathbb{C}^{d}\to\mathbb{C}$. In particular, for each $z\in U$, $Df(z)$ is a linear form, which can also be regarded as a vector (the usual gradient); we write $D_{i}f:=\frac{\partial}{\partial x_{i}}f$, so Df(z)=\bigl{(}D_{1}f(z),\dots,D_{d}f(z)\bigr{)}. Similarly, $D^{2}f(z)$ is a bilinear form, which may be regarded as a $d\times d$ matrix with entries $D_{ij}f(z)$, where $D_{ij}=D_{i}D_{j}$. We denote its determinant by $\operatorname{Det}(D^{2}f(z))$. (This is known as the Hessian of $f$.)

For a vector $x\in\mathbb{C}^{d}$, let $D^{m}f(z)[x]$ denote $D^{m}f(z)(x,\dots,x)$, where the vector $x$ is repeated $m$ times. When using coordinates $x=(u,v)$ in the case $d=2$, we write $[u,v]$ for $[(u,v)]$, so, regarding $D^{2}f$ as a matrix and $x$ as a (column) vector, we have

[TABLE]

We denote the usual Euclidean norm of vectors by $|\cdot|$. For operators and the multilinear forms $D^{m}f$ we use $\|\cdot\|$ for the usual norm (any other norm would do as well).

For real symmetric matrices, $A\leqslant B$ means that $B-A$ is positive definite, i.e., that $v^{\mathrm{tr}}(B-A)v\geqslant 0$ for all real vectors $v$. In particular, if $A$ is a $d\times d$ symmetric matrix and $c\in\mathbb{R}$, then

[TABLE]

Remark 1.3**.**

We adopt the following notational convention regarding constants. $c$ and $C$ are used ‘locally’ (within a single proof), while numbered constants $c_{1}$, $C_{1}$ etc retain their meaning throughout the paper. The constants $c_{i}$, which are numbered in the order they are introduced, obey the inequalities

[TABLE]

We write $y$, $z$, $w$ for complex variables, and $u$, $v$, $\alpha$, $\beta$ for real variables. All constants $c_{i}$, $C_{i}$ etc are positive.

2 Point probabilities of a single branching process

In this section we study the point probabilities $\operatorname{\mathbb{P}{}}(|\mathfrak{X}|=N)$ of the branching process $\mathfrak{X}=\mathfrak{X}_{Y,Z,Y^{0},Z^{0}}$ from Definition 1.1. To formulate our main result we need some further definitions (which encapsulate fairly mild and natural conditions for the offspring distributions).

Definition 2.1**.**

Suppose that $R>1$, $M<\infty$, $k_{1},k_{2}\in\mathbb{N}$ and $\delta>0$.

1. (i)

Let $\mathcal{K}^{0}=\mathcal{K}^{0}(R,M,\delta)$ be the set of probability distributions $\nu$ on $\mathbb{N}^{2}$ such that if $(Y,Z)\sim\nu$, then

[TABLE] 2. (ii)

Let $\mathcal{K}^{1}=\mathcal{K}^{1}(k_{1},k_{2},\delta)$ be the set of probability distributions $\nu=(\pi_{i,j})_{i,j\geqslant 0}$ on $\mathbb{N}^{2}$ such that

[TABLE] 3. (iii)

Let $\mathcal{K}=\mathcal{K}(R,M,k_{1},k_{2},\delta):=\mathcal{K}^{0}(R,M,\delta)\cap\mathcal{K}^{1}(k_{1},k_{2},\delta)$.

We write $(Y,Z)\in\mathcal{K}^{0}$ if the distribution of $(Y,Z)$ is in $\mathcal{K}^{0}$, and similarly for $\mathcal{K}^{1}$ and $\mathcal{K}$. The key condition here is the (uniform) bound (2.1) on the probability generating functions. The condition (2.3) is needed, roughly speaking, to ensure that $(Y,Z)$ is not essentially supported on a sublattice of $\mathbb{N}^{2}$. Note that $(Y,Z)\in\mathcal{K}^{1}$ trivially implies

[TABLE]

and similarly $\operatorname{\mathbb{E}{}}Y\geqslant\delta$.

The following theorem gives the qualitative behaviour of the size–$N$ point probabilities of the branching process $\mathfrak{X}=\mathfrak{X}_{Y,Z,Y^{0},Z^{0}}$ from Definition 1.1. The statement of Theorem 2.2 is not self contained since the parameters $\Psi$, $\Phi$ and $x^{*}$ are defined (in a rather involved way) from the generating functions of $(Y,Z)$ and $(Y^{0},Z^{0})$, see (2.43)–(2.44) and Lemma 2.15 in Section 2.3. A key feature of the result is that the estimates and error-terms are uniform over all distributions $(Y^{0},Z^{0})\in\mathcal{K}^{0}$ and $(Y,Z)\in\mathcal{K}$, i.e., the explicit and implicit constants depend only on $R,M,k_{1},k_{2}$ and $\delta$. Note that, from (2.8) below, $\xi=0$ if and only if $\operatorname{\mathbb{E}{}}Y=1$, and that $\operatorname{\mathbb{P}{}}(|\mathfrak{X}|=N)$ decays exponentially in $\Theta(\varepsilon^{2}N)$ in the near-critical case $\operatorname{\mathbb{E}{}}Y=1\pm\varepsilon$.

Theorem 2.2** (Point probabilities of $\mathfrak{X}$).**

Suppose that $R>1$, $M<\infty$, $k_{1},k_{2}\in\mathbb{N}$, and $\delta>0$. Writing $\mathcal{K}^{0}=\mathcal{K}^{0}(R,M,\delta)$ and $\mathcal{K}=\mathcal{K}(R,M,k_{1},k_{2},\delta)$, there exists a constant $c_{1}>0$ such that if $(Y^{0},Z^{0})\in\mathcal{K}^{0}$, $(Y,Z)\in\mathcal{K}$, and $|\operatorname{\mathbb{E}{}}Y-1|\leqslant c_{1}$, then for all $N\geqslant 1$ we have

[TABLE]

where, defining $\Psi$ and $\Phi$ as in (2.43)–(2.44) and $x^{*}$ as in Lemma 2.15, we have

[TABLE]

and

[TABLE]

Moreover, the implicit constants in (2.5)–(2.8) depend only on $R,M,k_{1},k_{2}$ and $\delta$.

The remainder of this section is devoted to the proof of Theorem 2.2. To this end we fix $R>1$, $M<\infty$, $k_{1},k_{2}\in\mathbb{N}$, and $\delta>0$, and write $\mathcal{K}^{0}=\mathcal{K}^{0}(R,M,\delta)$ and $\mathcal{K}=\mathcal{K}(R,M,k_{1},k_{2},\delta)$ to avoid clutter. Let $|\mathfrak{X}^{L}|$ and $|\mathfrak{X}^{S}|$ denote the total numbers of type-$L$ and type-$S$ particles in $\mathfrak{X}$, so $|\mathfrak{X}|=|\mathfrak{X}^{L}|+|\mathfrak{X}^{S}|$, and set

[TABLE]

Of course, $p_{n,m}$ depends on the distributions of $(Y,Z)$ and $(Y^{0},Z^{0})$. In Section 2.1 we establish a simple integral formula for $p_{n,m}$. Then, in Section 2.2 we use a version of the saddle point method to estimate this integral asymptotically. Finally, in Section 2.3 we prove (2.5) by summing all $p_{n,m}$ with $n+m=N$.

2.1 An integral formula for $p_{n,m}$

In this section we derive an explicit integral formula for $p_{n,m}$, see (2.14). We start with a simple conditional version of the classical Otter–Dwass formula (see e.g. Dwass [8]), which hinges on the random walk representation of a branching process and a well-known random-walk hitting time result.

Lemma 2.3**.**

For all integers $n\geqslant 1$ and $m,n_{0},m_{0}\geqslant 0$,

[TABLE]

Proof.

Let $(Y_{j},Z_{j})_{j\geqslant 1}$ be independent with each pair having the same distribution as $(Y,Z)$. Since particles of type $S$ do not have any children, by exploring the branching process $\mathfrak{X}$ in the usual way (i.e., revealing the offspring of the particles of type $L$ one-by-one until none are left to explore), we have

[TABLE]

That the right-hand side of the above expression equals (2.10) is surely folklore (by conditioning on $\sum_{1\leqslant j\leqslant n}Z_{j}=m-m_{0}$ this also follows directly from [17, Theorem 7]); we include a short argument. Namely, by a version of the well-known Cyclic Lemma (sometimes also called Spitzer’s combinatorial lemma), see, e.g., [13, Lemma 15.3] or [18, Lemma 6.1], for any sequence $(y_{1},\ldots,y_{n})$ with $y_{i}\in\{-1,0,1,2,\ldots\}$ and $n_{0}+\sum_{1\leqslant i\leqslant n}y_{i}=0$, there are exactly $n_{0}$ cyclic shifts of $(y_{1},\ldots,y_{n})$ for which all corresponding partial sums $s_{i}=y_{1}+\cdots+y_{i}$ of length $i\leqslant n-1$ satisfy $n_{0}+s_{i}>0$. Hence, taking a uniformly random cyclic shift of the $n$ independent variables $(Y_{j}-1,Z_{j})$, the formula (2.10) follows. ∎

Remark 2.4**.**

This two-type version of the Otter–Dwass formula is a simple variation of the usual one-type case; this is because one type is barren and can essentially be ignored. For a much more complicated formula in the general multi-type case, see Chaumont and Liu [5].

The probability on the right-hand side of (2.10) can be expressed using generating functions as

[TABLE]

For $n\geqslant 1$ and $m\geqslant 0$, recalling the notation (2.9) and summing (2.10) over all $n_{0},m_{0}$, we thus obtain

[TABLE]

where

[TABLE]

For later use, we also define

[TABLE]

Remark 2.5**.**

Let G(y,z):=\operatorname{\mathbb{E}{}}\bigl{(}y^{|\mathfrak{X}^{L}|}z^{|\mathfrak{X}^{S}|}\bigr{)} be the bivariate generating function for the size of the branching process $\mathfrak{X}$, and let G_{1}(y,z):=\operatorname{\mathbb{E}{}}\bigl{(}y^{|\mathfrak{X}^{1,L}|}z^{|\mathfrak{X}^{1,S}|}\bigr{)} be the corresponding generating function when starting with a single particle of type $L$. Then $G(y,z)=g_{0}(G_{1}(y,z),z)$ and $G_{1}(y,z)=yg(G_{1}(y,z),z)$, and the formula (2.12) can alternatively be obtained by the Lagrange inversion formula in the Bürmann form, see e.g. [9, A.(14)], regarding the generating functions as (formal) power series in $y$ with coefficients that are power series in $z$. We omit the details.

The extraction of coefficients in (2.12) can be performed by complex integration in the usual way (e.g., using Cauchy’s integral formula to evaluate \frac{\partial^{n+m}}{\partial y^{n}\partial z^{m}}\bigl{(}\tilde{g}_{0}(y,z)g(y,z)^{n}\bigr{)}\big{|}_{y=z=0}=n!\,m!\,np_{n,m} as in the textbook proof of Cauchy’s estimates), yielding the formula

[TABLE]

where we integrate (for example) over two circles with centre [math] and radii such that $\tilde{g}_{0}(y,z)$ and $g(y,z)$ are defined. In particular, if $(Y,Z)$ and $(Y^{0},Z^{0})$ are both in $\mathcal{K}^{0}$, then for any $\alpha,\beta<\log R$ we can integrate over $|y|=e^{\alpha}$ and $|z|=e^{\beta}$, and the standard change of variables $y=e^{\alpha+\mathrm{i}u}$, $z=e^{\beta+\mathrm{i}v}$ then yields

[TABLE]

Remark 2.6**.**

Alternatively, (2.14) can be obtained from (2.10) by first considering suitably tilted versions of the random variables (cf. Cramér [6]), and then passing to characteristic functions and making a Fourier inversion.

Remark 2.7**.**

It is not hard to write an integral formula for the final probability $p_{N}=\sum_{m+n=N}p_{n,m}$ that we are aiming to estimate. For example, multiplying (2.12) by $x^{n}/n$ and summing we see that $p_{n,m}=[x^{n}y^{n}z^{m}]H(x,y,z)$, where $H(x,y,z)=-\tilde{g}_{0}(y,z)\log(1-xg(y,z))$. Thus one can find $p_{N}$ by extracting the coefficient of $w^{0}t^{N}$ in $H(w,t/w,t)$. However, the corresponding integral does not obviously lend itself to asymptotic evaluation by methods such as those used here. Still, a direct estimate of $p_{N}$ may perhaps be possible by appropriate singularity analysis.

2.2 An asymptotic estimate of $p_{n,m}$

In this section we estimate the integral (2.14) asymptotically (see Theorem 2.11 below), using parameters defined in terms of the moment generating function f(y,z)=f_{Y,Z}(y,z)=\operatorname{\mathbb{E}{}}\bigl{(}e^{yY+zZ}\bigr{)}. Whenever $f$ is defined and non-zero, let

[TABLE]

taking the principal value of the logarithm; we shall only consider $\varphi$ on domains on which $|f-1|\leqslant 1/2$. The next lemma simply states that in suitable domains, $f$, $\varphi$ and their (partial) derivatives are all bounded.

Lemma 2.8**.**

There exist constants $0<c_{2}\leqslant(\log R)/2$ and $C_{1}^{(m)}$, $m\in\mathbb{N}$, such that if $(Y,Z)\in\mathcal{K}^{0}$ and $m\in\mathbb{N}$, then the following hold.

1. (i)

If $\alpha,\beta,u,v\in\mathbb{R}$ with $|\alpha|,|\beta|\leqslant c_{2}$, then $\|D^{m}f(\alpha+\mathrm{i}u,\beta+\mathrm{i}v)\|\leqslant C_{1}^{(m)}$. 2. (ii)

If, in addition, $|u|,|v|\leqslant c_{2}$, then $\varphi(\alpha+\mathrm{i}u,\beta+\mathrm{i}v)$ is defined, and $\|D^{m}\varphi(\alpha+\mathrm{i}u,\beta+\mathrm{i}v)\|\leqslant C_{1}^{(m)}$. 3. (iii)

If $|\alpha|,|\beta|\leqslant c_{2}$, then $\frac{\partial}{\partial y}f(\alpha,\beta)\geqslant\delta/2$.

Proof.

(i): When $|y|,|z|\leqslant R$, then $|g(y,z)|=|\operatorname{\mathbb{E}{}}(y^{Y}z^{Z})|\leqslant\operatorname{\mathbb{E}{}}(|y|^{Y}|z|^{Z})\leqslant\operatorname{\mathbb{E}{}}R^{Y+Z}$, which is at most $M$ by assumption. Thus $|f(y,z)|\leqslant M$ when $\operatorname{Re}(y)\leqslant\log R$ and $\operatorname{Re}(z)\leqslant\log R$. Recall that $R>1$ by assumption, so $\log R>0$. For any $c_{2}\leqslant(\log R)/2$, say, for suitable $C_{1}^{(m)}>0$ statement (i) follows by standard Cauchy estimates

(ii): Let $C=C_{1}^{(1)}$ denote the constant from the above proof of (i). Set $c_{2}:=\min\{(\log R)/2,1/(8C)\}$. Since $f(0,0)=g(1,1)=1$, it follows from (i) that if $|\alpha|,|\beta|,|u|,|v|\leqslant c_{2}$, then

[TABLE]

so $\varphi(\alpha+\mathrm{i}u,\beta+\mathrm{i}v)$ is defined and bounded. Furthermore, after decreasing $c_{2}$ and increasing $C_{1}^{(m)}$, if necessary, the bounds for the derivatives now again follow by Cauchy’s estimates.

(iii): Let $\tilde{f}=\frac{\partial}{\partial y}f$. By our assumption (2.2), $\tilde{f}(0,0)=\operatorname{\mathbb{E}{}}Y\geqslant\delta$. Furthermore, $D\tilde{f}(\alpha,\beta)=DD_{1}f(\alpha,\beta)=O(1)$ for $|\alpha|,|\beta|\leqslant c_{2}$ by part (i). Consequently, after reducing $c_{2}$ if necessary, we have $\tilde{f}(\alpha,\beta)\geqslant\frac{1}{2}\delta$ for $|\alpha|$, $|\beta|\leqslant c_{2}$. ∎

The next lemma expresses, in a quantitative form, the unsurprising fact that if we evaluate the probability generating function $g(y,z)=g_{Y,Z}(y,z)=\operatorname{\mathbb{E}{}}(y^{Y}z^{Z})$ at $y$, $z$ which are not positive real numbers, then there is significant cancellation, i.e., $|g(y,z)|$ is significantly smaller than $g(|y|,|z|)$. It will be more convenient to write this in terms of the moment generating function $f=f_{Y,Z}$ rather than $g$.

Lemma 2.9**.**

There exists a constant $c_{3}>0$ such that if $(Y,Z)\in\mathcal{K}$ and $\alpha,\beta,u,v\in\mathbb{R}$ with $|\alpha|,|\beta|\leqslant c_{2}$ and $|u|,|v|\leqslant\pi$, then

[TABLE]

Proof.

Let $\pi_{k,l}:=\operatorname{\mathbb{P}{}}(Y=k,Z=l)$. Then

[TABLE]

and thus $f(\alpha,\beta)>0$. Then

[TABLE]

Each term on the right-hand side is non-negative, and considering just the cases $(k,l,m,n)=(k_{1},k_{2},k_{1}+1,k_{2})$ and $(k_{1},k_{2},k_{1},k_{2}+1)$, recalling (2.3) we obtain

[TABLE]

since $1-\cos x=\Omega(x^{2})$ for $|x|\leqslant\pi$. Moreover, by Lemma 2.8(i), $f(\alpha,\beta)=O(1)$. Consequently

[TABLE]

for some constant $c_{3}>0$, and thus

[TABLE]

establishing (2.17) since $f(\alpha,\beta)>0$. ∎

We next establish that the symmetric bilinear form $D^{2}\varphi(\alpha,\beta)$ is positive-definite; a variant of the lower bound (2.18) could also be proved by first considering $D^{2}\varphi(0,0)$ and then using continuity. For the interpretation of $D^{2}\varphi{(\alpha,\beta)}[u,v]$, see (1.3).

Lemma 2.10**.**

If $(Y,Z)\in\mathcal{K}$ and $\alpha,\beta\in\mathbb{R}$ with $|\alpha|,|\beta|\leqslant c_{2}$, then $D^{2}\varphi(\alpha,\beta)\geqslant c_{3}I$, i.e.,

[TABLE]

In particular, $\operatorname{Det}(D^{2}\varphi(\alpha,\beta))\geqslant c_{3}^{2}$.

Proof.

We first consider only $|u|,|v|\leqslant c_{2}$, so Lemma 2.8(ii) applies. Then the estimate (2.17) can be written

[TABLE]

A Taylor expansion yields

[TABLE]

Since $\varphi(\alpha,\beta)$ is real for real $\alpha$ and $\beta$, all derivatives $D^{m}\varphi(\alpha,\beta)$ are real. Hence, when taking the real part, the linear term vanishes, and (2.19) implies

[TABLE]

Exploiting bilinearity, by replacing $(u,v)$ with $(tu,tv)$ and letting $t\to 0$, we now obtain (2.18) for all $u,v\in\mathbb{R}$, with room to spare.

Finally, by (1.4), note that (2.18) can be written $D^{2}\varphi(\alpha,\beta)\geqslant c_{3}I$. This says that both eigenvalues are $\geqslant c_{3}$, and thus the determinant is $\geqslant c_{3}^{2}$. ∎

For $|\alpha|,|\beta|\leqslant c_{2}$, define

[TABLE]

We are now ready to estimate the integral (2.14) for $p_{n,m}$ using a (two-dimensional) version of the saddle point method (see, e.g., [9, Chapter VIII]). We defer the problem of finding suitable $(\alpha,\beta)$ satisfying equation (2.21) to Section 2.3. Recall that \tilde{f}_{0}(y,z)=\frac{\partial}{\partial y}f_{Y^{0},Z^{0}}(y,z)=\operatorname{\mathbb{E}{}}\bigl{(}Y^{0}e^{yY^{0}+zZ^{0}}\bigr{)}, see (2.13).

Theorem 2.11**.**

Suppose that $(Y^{0},Z^{0})\in\mathcal{K}^{0}$ and $(Y,Z)\in\mathcal{K}$. Suppose further that $n\geqslant 1$, $m\geqslant 0$ are integers and that $\alpha$, $\beta$ are real numbers with $|\alpha|,|\beta|\leqslant c_{2}$ such that

[TABLE]

Then

[TABLE]

where the implicit constant depends only on the parameters $R,M,k_{1},k_{2},\delta$ of $\mathcal{K}^{0}$ and $\mathcal{K}$.

Proof.

We write (2.14) as

[TABLE]

where

[TABLE]

Using assumption (2.21) we have $\psi(\alpha,\beta)=\varphi(\alpha,\beta)-\alpha-\beta m/n$, so

[TABLE]

We shall estimate (2.24) using Laplace’s method (in two dimensions), cf. e.g. [9, Appendix B.6]. Roughly speaking, the idea is as follows. We view the integrand as a product of a term independent of $n$ with a term that is exponential in $n$. As we shall see, the condition (2.21) ensures that the exponent has a stationary point, in fact a maximum, at $u=v=0$. It turns out that the main contribution is near to this point, and here the exponent may be approximated by a quadratic, leading to a (two-dimensional) Gaussian integral.

Applying Lemma 2.8(i) to $(Y^{0},Z^{0})$ shows that $\tilde{f}_{0}(\alpha,\beta)=O(1)$. Since \operatorname{Det}\bigl{(}D^{2}\varphi(\alpha,\beta)\bigr{)}=\Omega(1) by Lemma 2.10, and $\psi(\alpha,\beta)=O(1)$ by (2.20) and Lemma 2.8(ii), the conclusion (2.22) holds for any fixed $n$ simply by taking the implicit constant large enough. Thus we may assume that $n$ is at least any given constant $n_{0}$, and in particular that $n^{-0.4}\leqslant c_{2}$.

Applying Lemma 2.8(i) to $(Y^{0},Z^{0})$ also shows that $\tilde{f}_{0}(\alpha+\mathrm{i}u,\beta+\mathrm{i}v)=O(1)$. Hence, if $|u|\geqslant n^{-0.4}$ or $|v|\geqslant n^{-0.4}$, then by Lemma 2.9 the integrand in (2.24) is O\bigl{(}e^{-c_{3}n\cdot n^{-0.8}}\bigr{)}=O\bigl{(}e^{-c_{3}n^{0.2}}\bigr{)}=O\bigl{(}n^{-99}\bigr{)}. On the other hand, if $|u|$, $|v|\leqslant n^{-0.4}$ then, since $n^{-0.4}\leqslant c_{2}$, Lemma 2.8(ii) shows that $\varphi(\alpha+\mathrm{i}u,\beta+\mathrm{i}v)$ is defined and we obtain

[TABLE]

with

[TABLE]

Considering a Taylor expansion of $\varphi$ around $(\alpha,\beta)$, and noting that the linear terms cancel by our assumption (2.21), we have

[TABLE]

where we used Lemma 2.8(ii) to bound the error term. For $|u|$, $|v|\leqslant n^{-0.4}$, note that Lemma 2.8(ii) implies $nD^{3}\varphi(\alpha,\beta)[u,v]=O(n(|u|+|v|)^{3})=O(n^{-0.2})=O(1)$, and $O(n(|u|+|v|)^{4})=O(n^{-0.6})=O(1)$. Hence, writing for brevity

[TABLE]

the exponential factor in (2.27) is

[TABLE]

Recalling $\tilde{f}_{0}=\frac{\partial}{\partial y}f_{Y^{0},Z^{0}}$, using Lemma 2.8(i) we also have the Taylor expansion

[TABLE]

Multiplying together (2.29) and (2.30), the integrand in (2.27) is thus

[TABLE]

When we integrate, the terms with $D\tilde{f}_{0}$ and $D^{3}\varphi$ are odd functions of $(u,v)$ so their integrals vanish. Hence,

[TABLE]

Recalling that $Q=D^{2}\varphi(\alpha,\beta)$, by Lemma 2.10 we have $Q[u,v]=\Omega(u^{2}+v^{2})$. Since for $k\in\{1,2,3\}$ we have $\iint_{\mathbb{R}^{2}}e^{-a(u^{2}+v^{2})}(u^{2}+v^{2})^{k}\,\mathrm{d}u\,\mathrm{d}v=O(a^{-(k+1)})$, it follows that

[TABLE]

Since $Q=D^{2}\varphi(\alpha,\beta)$ is symmetric and positive-definite by Lemma 2.10, we have the following standard Gaussian integral over $\mathbb{R}^{2}$:

[TABLE]

Since $Q[u,v]=\Omega(u^{2}+v^{2})$, the contribution of the range $\max\{|u|,|v|\}\geqslant n^{-0.4}$ to the above integral (2.32) is again exponentially small. Hence

[TABLE]

The result follows by combining (2.23), (2.25), (2.26) and (2.33). ∎

We next estimate the exponent in (2.22), without assuming that equation (2.21) holds.

Lemma 2.12**.**

There exists a constant $0<c_{4}\leqslant c_{2}$ such that if $(Y,Z)\in\mathcal{K}$ and $\alpha,\beta\in\mathbb{R}$ with $|\alpha|,|\beta|\leqslant c_{4}$, then

[TABLE]

Moreover, $\psi(0,0)=0$, $D\psi(0,0)=0$ and $D^{2}\psi(0,0)\leqslant-c_{3}I$.

Proof.

We have $\psi(0,0)=\varphi(0,0)=0$. Furthermore, differentiating (2.20) yields

[TABLE]

and thus $D\psi(0,0)=(0,0)$. Differentiating again shows that $D_{ij}\psi(0,0)=-D_{ij}\varphi(0,0)$ for all $i,j\in\{1,2\}$. Hence, using Lemma 2.10,

[TABLE]

Moreover, it follows from Lemma 2.8(ii) that $\|D^{3}\psi(\alpha,\beta)\|=O(1)$ for $|\alpha|,|\beta|\leqslant c_{2}$. Consequently, a Taylor expansion yields (2.34) for $c_{4}$ sufficiently small. ∎

2.3 Summing $p_{n,m}$: proof of Theorem 2.2

In this section we prove Theorem 2.2 by summing several different estimates of the point probabilities in

[TABLE]

Throughout we consider, as in (2.22), only real inputs $\alpha$, $\beta$ for the various functions $f$, $\varphi$ etc. Thus, all relevant functions are treated as mapping from (subdomains in) $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$ for suitable $n$, $m$.

An individual of type $L$ has on average $\operatorname{\mathbb{E}{}}Y$ children of type $L$ and $\operatorname{\mathbb{E}{}}Z$ children of type $S$. So, in the near-critical case $\operatorname{\mathbb{E}{}}Y\approx 1$, we expect that the overall fraction of type $L$ individuals in $\mathfrak{X}$ should be close to

[TABLE]

This suggests that the contribution from terms in (2.35) with $n/N$ far from $x_{0}$ will be negligible, and we shall later confirm this by standard Chernoff-like estimates. Below our main focus is thus on the terms where $n/N$ is close to $x_{0}$. Here the plan is to rewrite the asymptotic estimate (2.22) for $p_{n,N-n}$ using the following version of the inverse function theorem, where we explicitly state uniformity for a set of functions. We define

[TABLE]

Lemma 2.13** (Inverse function theorem).**

Let $d\geqslant 1$ be an integer and $r>0$ a real number. For every $0<A<\infty$, there exist $\sigma>0$ and $0<r_{1}<r$, both depending only on $A,r$, such that if $F:B^{d}_{r}\to\mathbb{R}^{d}$ is twice continuously differentiable and satisfies

1. (i)

$F(0)=0$, 2. (ii)

$DF(0)$* is invertible and $\|DF(0)^{-1}\|\leqslant A$, and* 3. (iii)

$\|D^{2}F(x)\|\leqslant A$* for all $x\in B^{d}_{r}$,*

then there exists a twice continuously differentiable function $G:B^{d}_{\sigma}\to B^{d}_{r}$ with $G(0)=0$ and $F(G(y))=y$ for $y\in B^{d}_{\sigma}$. Furthermore, for each $y\in B^{d}_{\sigma}$, $x=G(y)$ is the unique $x\in\mathbb{R}^{d}$ with $|x|\leqslant r_{1}$ such that $F(x)=y$. Moreover, $\|DG(y)\|=O(1)$ and $\|D^{2}G(y)\|=O(1)$, uniformly for $y\in B^{d}_{\sigma}$ and all such $F$, and if $F$ is infinitely differentiable or (real) analytic, then so is $G$.

Proof.

This follows by a standard proof of the inverse function theorem; we give some details for completeness.

First, let $r_{1}:=\frac{1}{2}\min\{r,A^{-2}\}$. If $|x|\leqslant r_{1}$, then by the mean-value theorem $\|DF(x)-DF(0)\|\leqslant A|x|\leqslant Ar_{1}$. Hence, $\|DF(0)^{-1}DF(x)-I\|\leqslant A^{2}r_{1}\leqslant\frac{1}{2}$, and thus $DF(0)^{-1}DF(x)$ is invertible and its inverse has norm at most $2$ (e.g., by the von Neumann series representation of the inverse). Consequently, $DF(x)$ is invertible and

[TABLE]

Next, let $\sigma:=r_{1}/(2A)$. If $|y|<\sigma$, define inductively $x_{0}:=0$ and $x_{n+1}:=\Gamma(x_{n})$, where

[TABLE]

Using $\|DF(0)^{-1}\|\leqslant A$ and that $\|D\Gamma(x)\|=\|I-DF(0)^{-1}DF(x)\|\leqslant\frac{1}{2}$ if $|x|\leqslant r_{1}$, it is easy to show by induction that $|x_{n}|\leqslant(1-2^{-n})r_{1}$ and $|x_{n+1}-x_{n}|\leqslant 2^{-n}A\sigma\leqslant 2^{-n-1}r_{1}$. Hence $x_{n}$ is defined for all $n\geqslant 0$, and converges to some $x$ with $|x|\leqslant r_{1}<r$. Furthermore, $y-F(x_{n})=DF(0)(x_{n+1}-x_{n})\to 0$ as ${n\to\infty}$, and thus by continuity $F(x)=y$. Define $G(y):=x$.

This shows that the inverse function $G$ exists in $B^{d}_{\sigma}$. The uniqueness statement is immediate, since any $x\in\mathbb{R}^{d}$ satisfying $F(x)=y$ is a fixed point of $\Gamma(x)$, which is a contraction for $|x|\leqslant r_{1}$. Differentiability (and analyticity when $F$ is analytic) follows in the usual way (or by appealing to a standard version of the inverse function theorem, locally at $G(y)$). Finally, $DG(y)=DF(x)^{-1}$, and thus $\|DG(y)\|\leqslant 2A$ by (2.36). Another differentiation (using the chain rule) then yields $\|D^{2}G(y)\|=O(1)$. ∎

Our next aim is to construct an (implicit) solution $(\alpha,\beta)=h(n/N)$ to equation (2.21) when $N=n+m$ and $n/N$ is close to $x_{0}=1/(1+\operatorname{\mathbb{E}{}}Z)$. We start by applying Lemma 2.13 to the function $F:B_{c_{4}}\to\mathbb{R}^{2}$ defined by

[TABLE]

Note that $D_{2}\varphi(\alpha,\beta)=D_{2}f(\alpha,\beta)/f(\alpha,\beta)\geqslant 0$, and thus $F(\alpha,\beta)$ is well-defined. Furthermore, $D\varphi(0,0)=(\operatorname{\mathbb{E}{}}Y,\operatorname{\mathbb{E}{}}Z)$, and thus $F(0,0)=(0,0)$. Moreover, using matrix form (where the first column is $\frac{\partial}{\partial\alpha}$ of the vector valued function $F$ and the second is $\frac{\partial}{\partial\beta}$), we have

[TABLE]

It follows from Lemma 2.10 that \bigl{\|}(D^{2}\varphi(\alpha,\beta))^{-1}\bigr{\|}=O(1), and then (2.38) together with Lemma 2.8 yields

[TABLE]

Lemma 2.8 also implies $\|D^{2}F(\alpha,\beta)\|=O(1)$. Consequently, Lemma 2.13 applies (with $d=2$) and yields a constant $\sigma=c_{5}>0$ and a function $G:B_{c_{5}}\to B_{c_{4}}$ such that

[TABLE]

Recall that $x_{0}=1/(1+\operatorname{\mathbb{E}{}}Z)$. Since $\operatorname{\mathbb{E}{}}Z=D_{2}f(0,0)=O(1)$ by Lemma 2.8 and $\operatorname{\mathbb{E}{}}Z\geqslant\delta>0$ by (2.4), there exists a constant $c>0$ such that $c\leqslant x_{0}\leqslant 1-c$. Let

[TABLE]

Suppose that $|\operatorname{\mathbb{E}{}}Y-1|<c_{6}$. If also $|x-x_{0}|<c_{6}$, then \bigl{(}\operatorname{\mathbb{E}{}}Y-1,x-x_{0}\bigr{)}\in B_{c_{5}}; we then define

[TABLE]

Furthermore, $|x-x_{0}|<c_{6}\leqslant\frac{1}{2}c\leqslant\frac{1}{2}x_{0}$ implies $x\geqslant\frac{1}{2}x_{0}\geqslant c_{6}$ and $1-x\geqslant c_{6}$. Now suppose that $0<n\leqslant N$ and that $|n/N-x_{0}|<c_{6}$, and let $m:=N-n$ and (\alpha,\beta):=h\bigl{(}n/N\bigr{)}. Then, by (2.41) and (2.40),

[TABLE]

Definition (2.37) shows that (2.21) holds. Hence, by Theorem 2.11, (2.22) holds. For $|x-x_{0}|<c_{6}$ define

[TABLE]

Recall that $h(x)\in B_{c_{4}}\subseteq B_{c_{2}}$, and note that Lemma 2.10 implies \operatorname{Det}\bigl{(}D^{2}\varphi(h(x))\bigr{)}\geqslant c_{3}^{2}; thus $\Psi(x)$ and $\Phi(x)$ are well-defined. Then, still assuming $|\operatorname{\mathbb{E}{}}Y-1|<c_{6}$, $|n/N-x_{0}|<c_{6}$ and (\alpha,\beta):=h\bigl{(}n/N\bigr{)}, we see that (2.22) can be written

[TABLE]

(Here, we use $|n/N-x_{0}|<c_{6}\leqslant x_{0}/2$ to bound $n\geqslant c_{6}N$, so an $O(n^{-1})$ error term is $O(N^{-1})$.)

We next show that, in the relevant domains, the functions $\Phi$, $\Psi$ and their (partial) derivatives are all bounded.

Lemma 2.14**.**

For each $m\geqslant 0$, there exists a constant $C_{2}^{(m)}$ such that if $|\operatorname{\mathbb{E}{}}Y-1|<c_{6}$ and $|x-x_{0}|<c_{6}$, then $\|D^{m}\Phi(x)\|\leqslant C_{2}^{(m)}$ and $\|D^{m}\Psi(x)\|\leqslant C_{2}^{(m)}$.

Proof.

We saw in the proof of Lemma 2.13 that DG(y)=\bigl{(}DF(G(y)\bigr{)}^{-1}, which is bounded for $y\in B_{c_{5}}$ by (2.36). By further differentiations, using the chain rule, Lemma 2.8(ii) and induction, it follows that for each $m\geqslant 0$,

[TABLE]

when $|\operatorname{\mathbb{E}{}}Y-1|<c_{6}$ and $|x-x_{0}|<c_{6}$. Hence the definition (2.41) yields $|D^{m}h(x)|=O(1)$, and the result follows by (2.43)–(2.44) together with the chain rule and Lemmas 2.8 and 2.10. ∎

Note for later than since $G(0)=0$ and $\|DG(y)\|=O(1)$ in $B_{c_{6}}$, we have

[TABLE]

if $(w,x-x_{0})\in B_{c_{6}}$.

We now analyze the exponential term $e^{N\Psi(n/N)}$ of the formula (2.45) for $p_{n,N-n}$, which is valid for $|n/N-x_{0}|<c_{6}$. The next result in particular implies that $\Psi(x)\leqslant 0$ is a concave function with a unique maximizer $x^{*}$ close to $x_{0}$. As we shall see, this essentially means that the dominant contribution to the sum of the $p_{n,N-n}$ comes from the terms with $n/N$ close to $x^{*}$, which is in turn close to $x_{0}$.

Lemma 2.15**.**

There exist constants $c_{7},c_{8}>0$ with $c_{8}\leqslant c_{7}<\frac{1}{3}c_{6}$ such that if $|\operatorname{\mathbb{E}{}}Y-1|\leqslant c_{8}$, then the following hold.

1. (i)

If $x\in\mathbb{R}$ with $|x-x_{0}|\leqslant 3c_{7}$, then

[TABLE] 2. (ii)

There exists $x^{*}\in\mathbb{R}$ with $|x^{*}-x_{0}|=O(|\operatorname{\mathbb{E}{}}Y-1|)$ and $|x^{*}-x_{0}|<c_{7}$ such that $\Psi^{\prime}(x^{*})=0$. 3. (iii)

$\Psi^{\prime\prime}(x)=-\Omega(1)$* for every $x$ with $|x-x_{0}|\leqslant 3c_{7}$.* 4. (iv)

$\Phi(x^{*})=\Omega(1)$.

As a consequence, $x^{*}$ is the unique maximum point of $x\mapsto\Psi(x)$ in $[x_{0}-3c_{7},x_{0}+3c_{7}]$.

Proof.

For $|w|,|x-x_{0}|\leqslant c_{6}$, let

[TABLE]

so that $\Psi(x)=\widehat{\Psi}(\operatorname{\mathbb{E}{}}Y-1,x)$. In the proofs below we assume that $c_{7}$ and $c_{8}$ are positive constants, chosen later, with $c_{8}\leqslant c_{7}<\frac{1}{3}c_{6}$, and that $|w|\leqslant c_{8}$ and $|x-x_{0}|\leqslant 3c_{7}$.

(i): Since $|w|+|x-x_{0}|\leqslant 4c_{7}<2c_{6}\leqslant c_{5}$, and $G$ maps $B_{c_{5}}$ into $B_{c_{4}}$, we have

[TABLE]

Since $F(0)=0$ and $\|DF(y)\|=O(1)$ in $B_{c_{4}}$, using $(w,x-x_{0})=F(G(w,x-x_{0}))$ we also have $|(w,x-x_{0})|=O(|G(w,x-x_{0})|)$. This and Lemma 2.12 imply

[TABLE]

Furthermore, as remarked above, $|x-x_{0}|\leqslant 3c_{7}<c_{6}$ implies $x\geqslant c_{6}$. Hence, recalling (2.49),

[TABLE]

which yields (2.48) since $\Psi(x)=\widehat{\Psi}(\operatorname{\mathbb{E}{}}Y-1,x)$.

(iii): Using $G(0)=0$, which is shorthand for $G(0,0)=(0,0)$, we have

[TABLE]

Together with (2.51), it follows that, for some constant $c>0$,

[TABLE]

The same proof as for Lemma 2.14 shows that

[TABLE]

for every fixed $m\geqslant 0$. Using (2.55) with $m=3$ and (2.54), we see that if $c_{7}$ and hence $c_{8}\leqslant c_{7}$ is small enough, then

[TABLE]

when $|w|\leqslant c_{8}$ and $|x-x_{0}|\leqslant 3c_{7}$. In particular, recalling $\Psi(x)=\widehat{\Psi}(\operatorname{\mathbb{E}{}}Y-1,x)$, by taking $w=\operatorname{\mathbb{E}{}}Y-1$ we have

[TABLE]

(ii): Similarly, (2.53) and (2.55) with $m=2$ imply that $D\widehat{\Psi}(w,x)=O(|w|+|x-x_{0}|)$. In particular, $\Psi^{\prime}(x_{0})=D_{2}\widehat{\Psi}(\operatorname{\mathbb{E}{}}Y-1,x_{0})=O(|\operatorname{\mathbb{E}{}}Y-1|)$. Hence we may choose $c_{8}$ sufficiently small such that $|\operatorname{\mathbb{E}{}}Y-1|\leqslant c_{8}$ implies $|\Psi^{\prime}(x_{0})|\leqslant cc_{7}/3$. Then the mean value theorem and (2.56) imply $\Psi^{\prime}(x_{0}-c_{7})>0$ and $\Psi^{\prime}(x_{0}+c_{7})<0$, so $\Psi^{\prime}(x^{*})=0$ for some $x^{*}\in(x_{0}-c_{7},x_{0}+c_{7})$. Moreover, by the mean value theorem and (2.56) we also have $|x^{*}-x_{0}|\leqslant\frac{2}{c}|\Psi^{\prime}(x_{0})|$, so (ii) holds.

(iv): Since $|x^{*}-x_{0}|\leqslant c_{7}<c_{6}$, by (2.50) and the definition (2.41) of $h$ we have $|h(x^{*})|\leqslant c_{4}\leqslant c_{2}$, so Lemma 2.8(iii) applied to $(Y^{0},Z^{0})$ gives $\tilde{f}_{0}(h(x^{*}))\geqslant\frac{1}{2}\delta$. The other factors in (2.44) are bounded below, using $x^{*}\leqslant x_{0}+c_{7}\leqslant 1+c_{7}$ and Hadamard’s inequality together with Lemma 2.8(ii), and thus (iv) follows. ∎

The following technical lemma will be useful for expanding the sum of the $p_{n,N-n}$ estimates (2.45) around $n/N\approx x^{*}$ (it is easy to give a much more precise formula for $T_{2j}$, but we do not need this).

Lemma 2.16**.**

For $a>0$, $y\in\mathbb{R}$ and an integer $j\geqslant 0$, let

[TABLE]

Then, uniformly for all $0<a\leqslant 1$ and $y\in\mathbb{R}$,

[TABLE]

and for every fixed integer $i\geqslant 0$,

[TABLE]

Proof.

We first consider $T_{0}=\sum_{n\in\mathbb{Z}}e^{-a(n-y)^{2}}$. Applying the well-known Poisson summation formula [26, (II.13.4) or (II.13.14)] and then using the Gaussian integral $\int_{-\infty}^{\infty}e^{-(ax^{2}+bx+c)}\,\mathrm{d}x=\sqrt{\frac{\pi}{a}}e^{b^{2}/(4a)-c}$, a short standard calculation yields the identity

[TABLE]

which for $a\leqslant 1$, say, implies (2.58). (In fact, (2.61) is equivalent to a well-known identity for the theta function $\theta_{3}$, see [16, (20.7.32)].)

Moreover, taking the partial derivative of (2.57) with respect to $y$ we obtain

[TABLE]

In particular, $2aT_{1}=\frac{\partial}{\partial y}T_{0}$, and termwise differentiation of the right-hand side in (2.61) (noting that the main term, $n=0$, is constant) yields

[TABLE]

Repeated differentiation of (2.62) and induction now yield (2.59) and (2.60). ∎

We also have to estimate the sum of the $p_{n,N-n}$ in (2.35) where $n/N$ is far from $x_{0}$. Based on simple Chernoff-type arguments, the next result shows that their contribution is negligible.

Lemma 2.17**.**

If $|n/N-x_{0}|\geqslant c_{7}$, then $p_{n,N-n}\leqslant e^{-\Omega(N)}$.

Proof.

For any $u,v>0$, from (2.12) we have

[TABLE]

Take $u=1$ and $v=e^{t}$, with $|t|\leqslant\log R$, and define

[TABLE]

For any $0\leqslant n\leqslant N$, (2.63) yields

[TABLE]

Note that $\gamma(0)=1$ and $\gamma^{\prime}(0)=0$. Since $g(1,e^{t})=f(0,t)$ and $\operatorname{\mathbb{E}{}}Z=D_{2}f(0,0)$, by Lemma 2.8(i) there is a constant $C_{3}>0$ such that $\gamma^{\prime\prime}(t)\leqslant C_{3}$ whenever $|t|\leqslant c_{2}$, and so

[TABLE]

By assumption, $|n-Nx_{0}|\geqslant c_{7}N$. Recalling that $x_{0}=1/(1+\operatorname{\mathbb{E}{}}Z)$ and $\operatorname{\mathbb{E}{}}Z\geqslant 0$, it follows that

[TABLE]

We now choose $t=\pm c$ where $c:=\min\{\frac{1}{2}c_{7}/C_{3},c_{2}\}$, and the sign is such that $t(n-N+n\operatorname{\mathbb{E}{}}Z)<0$. Using (2.64)–(2.66) and $n\leqslant N$, we infer

[TABLE]

completing the proof for $n\geqslant 1$.

Finally, in the remaining case $n=0$ we have $|\mathfrak{X}^{S}|=Z^{0}$, since $|\mathfrak{X}^{L}|=0$ if and only if $Y^{0}=0$. Hence

[TABLE]

completing the proof (since $R>1$). ∎

We are now ready to prove Theorem 2.2.

Proof of Theorem 2.2.

We suppose throughout that $c_{1}\leqslant c_{8}$ and that $|\operatorname{\mathbb{E}{}}Y-1|\leqslant c_{1}$.

We start by considering the quantities $\xi$ and $\theta$ defined in (2.6) and (2.7). By Lemma 2.15, $\Psi(x)$ has a local maximum point $x^{*}\in(x_{0}-c_{7},x_{0}+c_{7})$. As in (2.6) and (2.7), let

[TABLE]

By Lemmas 2.14 and 2.15(iii), $\Psi^{\prime\prime}(x^{*})=-\Theta(1)$. By (2.48) we have $\xi=-\Psi(x^{*})=\Omega(|\operatorname{\mathbb{E}{}}Y-1|^{2})$. Recalling that $\Psi(x^{*})=\widehat{\Psi}(\operatorname{\mathbb{E}{}}Y-1,x^{*})$, see (2.49), by combining (2.52), (2.53) and (2.55) (with $m=2$) together with Lemma 2.15(ii), it follows that

[TABLE]

Hence \xi=\Theta\bigl{(}|\operatorname{\mathbb{E}{}}Y-1|^{2}\bigr{)}, as claimed. That $\theta=\Theta(1)$ follows from the bound $|\Psi^{\prime\prime}(x^{*})|=\Theta(1)$ above and Lemmas 2.14 and 2.15(iv), which give $\Phi(x^{*})=\Theta(1)$.

Since $\xi$ and $\theta$, which do not depend on $N$, are both $O(1)$, for any fixed $N$, (2.5) holds trivially simply by taking the implicit constant large enough. Thus we may assume throughout that $N^{-0.4}\leqslant c_{7}$.

We have $\operatorname{\mathbb{P}{}}(|\mathfrak{X}|=N)=\sum_{n=0}^{N}p_{n,N-n}$. We estimate this sum by Laplace’s method, similarly to the argument in the proof of Theorem 2.11, but now for a sum instead of a two-dimensional integral.

We consider first $n$ such that $|n/N-x^{*}|<N^{-0.4}$, which includes the main terms in the sum. Suppose that $|x-x^{*}|<N^{-0.4}$. Using Lemma 2.14, a Taylor expansion then yields, cf. (2.28),

[TABLE]

which by exponentiation and a Taylor expansion of $\Phi(x)$ yields, cf. (2.31),

[TABLE]

Similar, but simpler, reasoning also shows that if $|x-x^{*}|<N^{-0.4}$, then

[TABLE]

Consequently, since $\Psi^{\prime\prime}(x^{*})\leqslant 0$, if we define

[TABLE]

then (2.45) yields

[TABLE]

(The odd sums $S_{1}$ and $S_{3}$ do not vanish as the corresponding integrals in the proof of Theorem 2.11 do, but we shall see that they are exponentially small.) Recall (from the start of the proof) that $\Psi^{\prime\prime}(x^{*})=-\Theta(1)$. It follows that if we extend the summation in the definition (2.68) to all $n\in\mathbb{Z}$, and denote the result by $S_{j}^{\prime}$, then $S_{j}-S^{\prime}_{j}$ is O\bigl{(}e^{-\Omega(N^{0.2})}\bigr{)} for each fixed $j$. Let $a=|\Psi^{\prime\prime}(x^{*})|/2N$. In the notation of Lemma 2.16, $S_{j}^{\prime}=N^{-j}T_{j}(a,Nx^{*})$. The error terms of the form $O(a^{-O(1)}e^{-\pi^{2}/a})$ in the conclusion of Lemma 2.16 are $e^{-\Theta(N)}$ and so negligible. Thus, from Lemma 2.16 and (2.69), recalling the definitions (2.6) and (2.7) of $\xi$ and $\theta$, we find

[TABLE]

Next, consider $n$ such that $N^{-0.4}\leqslant|n/N-x^{*}|\leqslant 2c_{7}$, and recall that $3c_{7}<c_{6}$. If $N^{-0.4}\leqslant|x-x^{*}|\leqslant 2c_{7}$, then Lemma 2.15 implies that $|x-x_{0}|\leqslant 3c_{7}$ and $\Psi(x)\leqslant\Psi(x^{*})-\Omega((x-x^{*})^{2})\leqslant\Psi(x^{*})-\Omega(N^{-0.8})=-\xi-\Omega(N^{-0.8})$. Hence, by (2.45) and Lemma 2.14, if $N^{-0.4}\leqslant|n/N-x^{*}|\leqslant 2c_{7}$, then Lemma 2.15 implies that

[TABLE]

The sum over such $n$ is easily absorbed into the error term we are aiming for: we have, say,

[TABLE]

Finally, since $|x^{*}-x_{0}|\leqslant c_{7}$ by Lemma 2.15(ii) and $0\leqslant n\leqslant N$, using Lemma 2.17 there exists a constant $c>0$ such that, say,

[TABLE]

Recalling that $|\operatorname{\mathbb{E}{}}Y-1|\leqslant c_{1}$, by (2.67) we may choose $c_{1}\leqslant c_{8}$ sufficiently small so that $\xi<c$, and then (2.5) follows from (2.70), (2.71) and (2.72). ∎

3 Application to branching process families

In this section we apply the main result of Section 2 (Theorem 2.2) to a family of branching processes. The goal is to prove Theorem 3.4 below, giving estimates for the point probabilities $\operatorname{\mathbb{P}{}}(|\mathfrak{X}|=N)$ in a form suitable for the application to Achlioptas processes in [22].

3.1 Properties of general parameterized families

By a branching process family $(\mathfrak{X}_{Y_{u},Z_{u},Y^{0}_{u},Z^{0}_{u}})_{u\in I}$ we simply mean a family of branching processes of the type in Definition 1.1, one for each $u$ in some interval $I\subset\mathbb{R}$. Given such a family, we write

[TABLE]

for the corresponding probability generating functions. Note that the branching process family is fully specified by the interval $I$ and the functions $g_{u}$ and $g^{0}_{u}$.

The following auxiliary result shows that the associated parameters $\xi_{u}=\xi_{Y_{u},Z_{u}}$ and $\theta_{u}=\theta_{Y_{u},Z_{u},Y^{0}_{u},Z^{0}_{u}}$ defined as in Theorem 2.2 vary smoothly in $u$. This will later allow us to compare the parameters $\xi_{Y,Z}$ and $\theta_{Y,Z,Y^{0},Z^{0}}$ resulting from different probability distributions $(Y^{0},Z^{0})\in\mathcal{K}^{0}$ and $(Y,Z)\in\mathcal{K}$ (by integrating linear mixtures that interpolate between them); here the extra $|\operatorname{\mathbb{E}{}}Y_{u}-1|=O(1)$ factor in (3.2) is crucial.

Lemma 3.1**.**

Suppose that $R>1$, $M<\infty$, $k_{1},k_{2}\in\mathbb{N}$, and $\delta>0$. Set $\mathcal{K}^{0}=\mathcal{K}^{0}(R,M,\delta)$ and $\mathcal{K}=\mathcal{K}(R,M,k_{1},k_{2},\delta)$. Let $(\mathfrak{X}_{u})_{u\in I}=(\mathfrak{X}_{Y_{u},Z_{u},Y^{0}_{u},Z^{0}_{u}})_{u\in I}$ be a branching process family such that, for every $u\in I$, we have $(Y^{0}_{u},Z^{0}_{u})\in\mathcal{K}^{0}$, $(Y_{u},Z_{u})\in\mathcal{K}$, and $|\operatorname{\mathbb{E}{}}Y_{u}-1|\leqslant c_{1}$, where $c_{1}>0$ is the constant appearing in Theorem 2.2. Suppose that $g_{u}(y,z)$ and $g_{u}^{0}(y,z)$ are analytic as functions of $(u,y,z)$ in the domain

[TABLE]

and that for some $\lambda$,

[TABLE]

for all $(u,y,z)\in\mathcal{D}_{I,R}$. Let

[TABLE]

be defined as in Theorem 2.2. Then $\xi_{u}$ and $\theta_{u}$ are (real) analytic as functions of $u\in I$. Furthermore,

[TABLE]

where the implicit constants in (3.2) and (3.3) depend only on $R,M,k_{1},k_{2},\delta$.

Proof.

By assumption, the conditions of Theorem 2.2 hold for each $u\in I$. For any of the quantities or functions defined in previous sections for a single branching process, we use a subscript $u$ to denote the corresponding quantity or function associated to $\mathfrak{X}_{u}$. As in previous sections, $\alpha$ and $\beta$ always denote real numbers.

The idea of the proof is as follows. For a given $u$, the functions defined in the previous sections are defined, either explicitly or implicitly, in terms of $g_{u}$ and $g_{u}^{0}$ (or their reparameterizations $f_{u}$ and $f_{u}^{0}$). Roughly speaking, since $g_{u}$ and $g_{u}^{0}$ vary analytically in $u$ by assumption (and with $u$-derivative $O(\lambda)$), it follows that the same is true for the derived quantities. There are various steps where we must be slightly careful; for example, when taking logs (there is no problem as we stick to the domain $|z-1|\leqslant\frac{1}{2}$), or dividing by the square root of a certain determinant (there is no problem since this determinant is $\Omega(1)$ by Lemma 2.10). We must also be careful with the implicit definitions of $G_{u}$ and $x^{*}_{u}$; the hardest part of the argument is to establish (3.2) with $O(\lambda|\operatorname{\mathbb{E}{}}Y_{u}-1|)$ instead of $O(\lambda)$.

Turning to the details, from (3.1) and standard Cauchy estimates we see that for each fixed $m$ we have

[TABLE]

whenever $|y|,|z|\leqslant R^{1/2}$, say. (Here and below, $D$ does not include derivatives with respect to $u$.) Since $c_{4}\leqslant c_{2}\leqslant(\log R)/2$, the same estimates hold for the derivatives of $f_{u}(y,z)=g_{u}(e^{y},e^{z})$ and $f_{u}^{0}(y,z)=g^{0}_{u}(e^{y},e^{z})$ in the domain $B_{c_{4}}\subset\mathbb{R}^{2}$; from now on we work over the reals. Recalling the definition (2.37) and $\varphi_{u}=\log f_{u}$, from (3.4) it follows that $\|\frac{\partial}{\partial u}F_{u}(\alpha,\beta)\|=O(\lambda)$ for $(\alpha,\beta)\in B_{c_{4}}$.

From the definition (2.37), the function $F_{u}(\alpha,\beta)$ is a (real) analytic function of $(u,\alpha,\beta)\in I\times B_{c_{4}}$. For each $u\in I$, by (2.40) we have an inverse $G_{u}:B_{c_{5}}\to B_{c_{4}}$ of the 2-variable function $F_{u}$. Applying a standard version of the implicit function theorem locally, we see that $G_{u}(\alpha,\beta)$ is analytic as a function of $(u,\alpha,\beta)\in I\times B_{c_{5}}$.111Fix $u_{0}$ and $y_{0}=(\alpha_{0},\beta_{0})$ in the relevant domain. By (2.36), $DF_{u}$ is invertible at $x_{0}=G_{u_{0}}(y_{0})$, so there is an analytic function $\widehat{G}_{u}(y)$ defined in a neighbourhood of $(u_{0},y_{0})$ such that $F_{u}(\widehat{G}_{u}(y))=y$. By local uniqueness, $G_{u}(y)=\widehat{G}_{u}(y)$ near $(u_{0},y_{0})$, so $G$ is indeed analytic at this point.

Noting \operatorname{\mathbb{E}{}}Y_{u}=\frac{\partial}{\partial y}g_{u}(y,z)\big{|}_{y=z=1}, by definition (2.41) and $|\operatorname{\mathbb{E}{}}Y_{u}-1|\leqslant c_{1}$ it follows that $h_{u}(x)$ is an analytic function of $u,x$ for $u\in I$ and $|x-x_{0,u}|<c_{6}$; we consider in the sequel only such $u$ and $x$. Inspecting the definitions (2.43) and (2.44), using Lemma 2.10 (to ensure that the determinant is not degenerate), we see that $\Psi_{u}(x)$ and $\Phi_{u}(x)$ are well-defined compositions of analytic functions, and thus analytic as functions of $u,x$.

Since $F_{u}(G_{u}(y))=y$ is independent of $u$, writing $x=G_{u}(y)$ and differentiating yields $\frac{\partial}{\partial u}F_{u}(x)+DF_{u}(x)(\frac{\partial}{\partial u}G_{u}(y))=0$ and thus, recalling (2.39), for $y\in B_{c_{5}}$,

[TABLE]

Recalling the definition (2.41), note that (3.5) implies $\|\frac{\partial}{\partial u}h_{u}(x)\|=O(\lambda)$. Since $\psi_{u}$ is defined in terms of $\varphi_{u}=\log f_{u}=\log f_{Y_{u},Z_{u}}$ and its derivatives, see (2.20), using $(Y_{u},Z_{u})\in\mathcal{K}$ it follows that $\|D\psi_{u}(\alpha,\beta)\|=O(1)$. Furthermore, since estimates analogous to (3.4) also hold for $f_{u}=f_{Y_{u},Z_{u}}$, we have $\frac{\partial}{\partial u}\psi_{u}(y)=O(\lambda)$. Hence, recalling (2.43) and writing $y=h_{u}(x)$, we have

[TABLE]

Recalling the definitions (2.43) and (2.44), and the estimates in Section 2.3, we similarly deduce $\frac{\partial}{\partial u}\Phi_{u}(x)=O(\lambda)$, $\frac{\partial}{\partial u}\Psi^{\prime}_{u}(x)=O(\lambda)$ and $\frac{\partial}{\partial u}\Psi_{u}^{\prime\prime}(x)=O(\lambda)$.

Since $x^{*}_{u}$ is defined by $\Psi_{u}^{\prime}(x^{*}_{u})=0$, we have $(\frac{\partial}{\partial u}\Psi_{u}^{\prime})(x^{*}_{u})+\Psi^{\prime\prime}_{u}(x^{*}_{u})\frac{\mathrm{d}}{\mathrm{d}u}x^{*}_{u}=0$. It follows, using the lower bound $\Psi_{u}^{\prime\prime}(x^{*}_{u})=-\Omega(1)$ from Lemma 2.15(iii) and the implicit function theorem, that $x^{*}_{u}$ is an analytic function of $u$, and that

[TABLE]

As (3.5) implies $\|\frac{\partial}{\partial u}h_{u}(x)\|=O(\lambda)$, and $\|h^{\prime}_{u}(x)\|=O(1)$ by (2.46), it follows that

[TABLE]

Similarly, from the definitions (2.6) and (2.7), using (3.7) and the bounds above on $\frac{\partial}{\partial u}\Psi_{u}$, $\frac{\partial}{\partial u}\Psi_{u}^{\prime\prime}$ and $\frac{\partial}{\partial u}\Phi_{u}$ it follows that $\xi_{u}$ and $\theta_{u}$ are analytic functions of $u$, with $\frac{\mathrm{d}}{\mathrm{d}u}\xi_{u}=O(\lambda)$ and $\frac{\mathrm{d}}{\mathrm{d}u}\theta_{u}=O(\lambda)$. It remains only to establish (3.2).

For this final step, recalling (2.20) and $\varphi_{u}(\alpha,\beta)=\log f_{u}(\alpha,\beta)=\log g_{u}(e^{\alpha},e^{\beta})$, note that $\|D\frac{\partial}{\partial u}\psi_{u}(x)\|=O(\lambda)$ follows from (3.4). Since $\frac{\partial}{\partial u}\psi_{u}(0)=0$, we thus have \frac{\partial}{\partial u}\psi_{u}(x)=O\bigl{(}\lambda|x|\bigr{)}. Similarly, as a consequence of Lemma 2.12, \psi_{u}(x)=O\bigl{(}|x|\bigr{)} and \|D\psi_{u}(x)\|=O\bigl{(}|x|\bigr{)}. Writing $y^{*}_{u}$ for $h_{u}(x^{*}_{u})$, it follows from the definition (2.43) and (3.7)–(3.8) that

[TABLE]

Since $y^{*}_{u}=h_{u}(x^{*}_{u})$, from the definition (2.41) of $h_{u}$, the bound (2.47), and, for the final step, Lemma 2.15(ii), it follows that

[TABLE]

completing the proof of the lemma. ∎

3.2 A specific result suitable for application to Achlioptas processes

In this section we use Theorem 2.2 and Lemma 3.1 to prove the case $p_{{\mathcal{R}}}=1$, $K=0$ of Theorem A.10 of [22], used there for the analysis of Achlioptas processes. To formulate this main application, i.e., our point probability result for certain (perturbed) branching process families, we need some some further definitions.

Definition 3.2**.**

Let $t_{0}<t_{\mathrm{c}}<t_{1}$ be real numbers. The branching process family $(\mathfrak{X}_{t})_{t\in(t_{0},t_{1})}=(\mathfrak{X}_{Y_{t},Z_{t},Y^{0}_{t},Z^{0}_{t}})_{t\in(t_{0},t_{1})}$ is $t_{\mathrm{c}}$-critical if the following hold:

1. (i)

There exist $\delta>0$ and $R>1$ with $(t_{\mathrm{c}}-\delta,t_{\mathrm{c}}+\delta)\subseteq(t_{0},t_{1})$ such that the probability generating functions

[TABLE]

are defined and analytic on the domain

[TABLE] 2. (ii)

We have

[TABLE] 3. (iii)

There exists some $k_{0}\in\mathbb{N}$ such that

[TABLE]

Definition 3.3**.**

Let $(\mathfrak{X}_{t})_{t\in(t_{0},t_{1})}$ be a $t_{\mathrm{c}}$-critical branching process family, and let $\delta$, $R$ and $k_{0}$ be as in Definition 3.2. Given $t,\eta\geqslant 0$ with $|t-t_{\mathrm{c}}|<\delta$, we say that the branching process $\mathfrak{X}_{Y,Z,Y^{0},Z^{0}}$ is of type $(t,\eta)$ (with respect to $(\mathfrak{X}_{t})$, $\delta$, $R$, and $k_{0}$) if the following hold:

1. (i)

Writing $\mathcal{N}:=\{(y,z)\in\mathbb{C}^{2}:\,|y|,|z|<R\}$, the expectations

[TABLE]

are defined (i.e., the expectations converge absolutely) for all $(y,z)\in\mathcal{N}$. 2. (ii)

For all $(y,z)\in\mathcal{N}$ we have

[TABLE]

Note that $\mathfrak{X}_{t}=\mathfrak{X}_{Y_{t},Z_{t},Y^{0}_{t},Z^{0}_{t}}$ is itself of type $(t,\eta)$ for any $\eta\geqslant 0$. The following result relates the point probabilities from $\mathfrak{X}_{t}$ with those from branching processes $\mathfrak{X}$ of type $(t,\eta)$. A key feature is the form of the uniform $O(\eta|t-t_{\mathrm{c}}|+\eta^{2})$ error term in (3.15). In (3.14) and (3.15) below, we have $\xi_{Y_{t},Z_{t}}=\psi(t)=\Theta((t-t_{\mathrm{c}})^{2})$ and $\theta_{Y_{t},Z_{t},Y^{0}_{t},Z^{0}_{t}}=\theta(t)=\Theta(1)$ for $\mathfrak{X}=\mathfrak{X}_{t}$ (using $\eta=0$), and $\xi_{Y,Z}\sim\psi(t)$ and $\theta_{Y,Z,Y^{0},Z^{0}}\sim\theta(t)$ for any branching process $\mathfrak{X}$ of type $(t,\eta)$ with $\eta\ll|t-t_{\mathrm{c}}|\leqslant\varepsilon_{0}$. In the near-critical case $t=t_{\mathrm{c}}\pm\varepsilon$, the size–$N$ point probabilities of $\mathfrak{X}_{t}$ and $\mathfrak{X}$ thus both decay exponentially in $\Theta(\varepsilon^{2}N)$.

Theorem 3.4** (Point probabilities of $\mathfrak{X}$ of type $(t,\eta)$).**

Let $(\mathfrak{X}_{t})_{t\in(t_{0},t_{1})}$ be a $t_{\mathrm{c}}$-critical branching process family. Then there exist constants $\varepsilon_{0},\eta_{0}>0$ and analytic functions $\theta$, $\psi$ on the interval $I=[t_{\mathrm{c}}-\varepsilon_{0},t_{\mathrm{c}}+\varepsilon_{0}]$ such that

[TABLE]

uniformly over all $N\geqslant 1$, $t\in I$, $0\leqslant\eta\leqslant\eta_{0}$ and all branching processes $\mathfrak{X}=\mathfrak{X}_{Y,Z,Y^{0},Z^{0}}$ of type $(t,\eta)$ (with respect to $(\mathfrak{X}_{t})$), where the parameters $\xi_{Y,Z}$ and $\theta_{Y,Z,Y^{0},Z^{0}}$, which depend on the distributions of $(Y,Z)$ and of $(Y^{0},Z^{0})$, satisfy

[TABLE]

Moreover, $\theta(t)>0$, $\psi(t)\geqslant 0$, $\psi(t_{\mathrm{c}})=\psi^{\prime}(t_{\mathrm{c}})=0$, and $\psi^{\prime\prime}(t_{\mathrm{c}})>0$.

Proof.

Fix a $t_{\mathrm{c}}$-critical branching process family $(\mathfrak{X}_{t})_{t\in(t_{0},t_{1})}$, and let $\delta>0$ and $R>1$ be as in the definitions above. We pick $0<\varepsilon_{0}<\delta$, and decrease $R$ slightly, keeping $R>1$. Then $g(t,y,z)=g_{t}(y,z)$ and $g^{0}(t,y,z)=g^{0}_{t}(y,z)$ are continuous on the compact domain $|t-t_{\mathrm{c}}|\leqslant\varepsilon_{0}$, $|y|,|z|\leqslant R$ and so bounded, say by $M_{1}$. Let $M=M_{1}+1$. Then, provided $|t-t_{\mathrm{c}}|\leqslant\varepsilon_{0}$, by (3.13) any $\mathfrak{X}$ of type $(t,\eta)$ with $\eta\leqslant 1$ satisfies

[TABLE]

For any integers $k,\ell$, we have

[TABLE]

Since $g_{t}(y,z)$ is analytic in $t,y,z$, this probability varies continuously in $t$. Moreover, since $\operatorname{\mathbb{P}{}}(Y=k,\,Z=\ell)$ can analogously be written as a derivative of $\tilde{g}$ evaluated at $(0,0)$, using standard Cauchy estimates and (3.13) we infer

[TABLE]

A similar argument shows that \operatorname{\mathbb{E}{}}Y_{t}=\frac{\partial}{\partial y}g_{t}(y,z)\big{|}_{y=z=1} is continuous in $t$, and that Cauchy’s estimates imply

[TABLE]

Analogous reasoning applies to $\operatorname{\mathbb{E}{}}Y^{0}_{t}$ and $\operatorname{\mathbb{E}{}}Y^{0}$.

By definition of a $t_{\mathrm{c}}$-critical branching process family, there is some $\delta>0$ such that for $t=t_{\mathrm{c}}$ all of $\operatorname{\mathbb{E}{}}Y_{t}^{0}$, $\operatorname{\mathbb{P}{}}(Y_{t}=k_{0},Z_{t}=k_{0})$, $\operatorname{\mathbb{P}{}}(Y_{t}=k_{0}+1,Z_{t}=k_{0})$ and $\operatorname{\mathbb{P}{}}(Y_{t}=k_{0},Z_{t}=k_{0}+1)$ are at least $2\delta$, say. Furthermore, at $t=t_{\mathrm{c}}$ we have $\operatorname{\mathbb{E}{}}Y_{t}=1$. From the argument above these quantities all vary continuously in $t$, and change by $O(\eta)$ when we move from $\mathfrak{X}_{t}$ to some $\mathfrak{X}$ of type $(t,\eta)$. It follows that there is a constant $\eta_{0}>0$ such that, after reducing $\varepsilon_{0}$ if necessary, whenever $|t-t_{\mathrm{c}}|\leqslant\varepsilon_{0}$ and $\eta\leqslant\eta_{0}$, then any $\mathfrak{X}$ of type $(t,\eta)$ satisfies the conditions of Theorem 2.2, namely that $(Y^{0},Z^{0})\in\mathcal{K}^{0}$, $(Y,Z)\in\mathcal{K}$ and $|\operatorname{\mathbb{E}{}}Y-1|\leqslant c_{1}$.

Now, applying Theorem 2.2 to each branching process in the family $(\mathfrak{X}_{t})_{t\in[t_{\mathrm{c}}-\varepsilon_{0},t_{\mathrm{c}}+\varepsilon_{0}]}$, and Lemma 3.1 to the family itself, establishes the $\eta=0$ case of Theorem 3.4 with $\theta(t)=\theta_{t}$ and $\psi(t)=\xi_{t}$. Indeed, Theorem 2.2 gives that $\theta=\Theta(1)$, so we do have $\theta(t)>0$, while (2.8) gives $\psi(t)=\xi_{t}=\Theta(|\operatorname{\mathbb{E}{}}Y_{t}-1|^{2})$, which is $\Theta(|t-t_{c}|^{2})$ since (3.10) implies, after reducing $\varepsilon_{0}$ if necessary, that

[TABLE]

It follows that $\psi(t_{\mathrm{c}})=\psi^{\prime}(t_{\mathrm{c}})=0$ and $\psi^{\prime\prime}(t_{\mathrm{c}})>0$.

To complete the proof, assume now that $\mathfrak{X}=\mathfrak{X}_{Y,Z,Y^{0},Z^{0}}$ is of type $(t,\eta)$, with $0\leqslant\eta\leqslant\eta_{0}$ and $|t-t_{\mathrm{c}}|\leqslant\varepsilon_{0}$. As noted above, Theorem 2.2 applies to $\mathfrak{X}$, giving (3.14); it remains to establish (3.15). We do this by interpolating between $\mathfrak{X}$ and $\mathfrak{X}_{t}$, and applying Lemma 3.1. Consider the branching process family $(\bar{Y}_{u},\bar{Z}_{u},\bar{Y}^{0}_{u},\bar{Z}^{0}_{u})_{u\in[0,1]}$ defined by the mixtures

[TABLE]

(As noted earlier, the probability generating functions $\bar{g}_{u}$, $\bar{g}^{0}_{u}$ and the interval $I=[0,1]$ fully specify the family.) Since the assumptions of Theorem 2.2 are preserved by taking mixtures, every branching process in this family satisfies these assumptions. (In fact, they are all clearly of type $(t,\eta)$ too.) Moreover, the assumption (3.13) implies that (3.1) holds with $\lambda=\eta$, and since \operatorname{\mathbb{E}{}}\bar{Y}_{u}=\frac{\partial}{\partial y}\bar{g}_{u}(y,z)\big{|}_{y=z=1} we have

[TABLE]

by (3.16) and (3.17). Thus we may apply Lemma 3.1, and, by integrating (3.2) with $\xi_{u}=\xi_{\bar{Y}_{u},\bar{Z}_{u}}$, we infer

[TABLE]

Finally, $\theta-\theta_{t}=O(\eta)$ follows similarly by integrating (3.3). ∎

Theorem 3.4 immediately implies the key case $p_{{\mathcal{R}}}=1$, $K=0$ of Theorem A.10 of [22] with any positive value of the constant $c$. Indeed, after reducing $\varepsilon_{0}$ if necessary, the assumption $\eta\leqslant c|t-t_{\mathrm{c}}|$ in the latter theorem implies the assumption $\eta\leqslant\eta_{0}$ of Theorem 3.4. Moreover, the same assumption $\eta\leqslant c|t-t_{\mathrm{c}}|$ together with (3.15) implies the bound $\xi_{Y,Z}=\psi(t)+O(\eta|t-t_{\mathrm{c}}|)$ in Theorem A.10 of [22].

4 The survival probability

In this section we study the survival probability of the branching process $\mathfrak{X}=\mathfrak{X}_{Y,Z,Y^{0},Z^{0}}$ from Definition 1.1 and the branching process family $(\mathfrak{X}_{u})_{u\in I}$ from Section 3. The goal is to prove Theorem 4.5 below, i.e., to give estimates for $\operatorname{\mathbb{P}{}}(|\mathfrak{X}|=\infty)$ suitable for the application to Achlioptas processes in [22].

Our strategy mimics the general approach used in Sections 2–3 for point probabilities, though the technical details are much simpler. In Section 4.1 we first prove a technical result for the survival probability $\operatorname{\mathbb{P}{}}(|\mathfrak{X}|=\infty)$ of a single branching process (Lemma 4.2). Then we show that in a branching process family $(\mathfrak{X}_{u})_{u\in I}$ certain parameters related to the survival probability vary smoothly in $u$ (Lemma 4.4). Finally, in Section 4.2 we combine these two auxiliary results to prove Theorem 4.5.

4.1 Properties of a single process and general parameterized families

As far as the survival of $\mathfrak{X}_{Y,Z,Y^{0},Z^{0}}$ is concerned, particles of type $S$ are irrelevant and may be ignored, so we may consider a standard single-type Galton–Watson branching process with offspring distribution $Y$ and initial distribution $Y^{0}$, which we henceforth denote by $\mathfrak{X}_{{Y,Y^{0}}}$. Thus

[TABLE]

Writing $\mathbbm{1}$ as shorthand for the distribution with constant value one, it similarly follows that

[TABLE]

Throughout this section, we shall work with the univariate probability generating functions $g_{Y}(y):=\operatorname{\mathbb{E}{}}y^{Y}=g_{Y,Z}(y,1)$ and $g_{Y^{0}}(y):=g_{Y^{0},Z^{0}}(y,1)$. By standard branching process arguments (see, e.g., [10, Theorem 5.4.5]), we have

[TABLE]

where the extinction probability $1-\rho_{Y}$ is the smallest non-negative solution to

[TABLE]

Fix $R>1$, $M$, $k_{1}$, $k_{2}$ and $\delta>0$. We henceforth assume that $(Y,Z)\in\mathcal{K}=\mathcal{K}(R,M,k_{1},k_{2},\delta)$ and $(Y^{0},Z^{0})\in\mathcal{K}^{0}=\mathcal{K}^{0}(R,M,\delta)$. Since $(Y,Z)\in\mathcal{K}$, by (2.1) the function $g_{Y}(y)$ is analytic in $\{y\in\mathbb{C}:|y|<R\}$, with $g_{Y}(1)=1$. A Taylor expansion of $g_{Y}(y)$ at $y=1$ yields, for $|x|<R-1$,

[TABLE]

Define

[TABLE]

removing the removable singularity at $x=0$. Then $h_{Y}$ is analytic in $\{x\in\mathbb{C}:|x|<R-1\}$, and

[TABLE]

Observe that if $\rho_{Y}>0$, then (4.4) is equivalent to $h_{Y}(\rho_{Y})=1$. Furthermore,

[TABLE]

We next derive bounds on the derivatives of $h_{Y}$ valid for small $x$.

Lemma 4.1**.**

Suppose that $R>1$, $M<\infty$, $k_{1},k_{2}\in\mathbb{N}$, and $\delta>0$. There exist constants $0<c_{9}\leqslant\min\{R-1,1\}/3$ and $C_{4}^{(m)}$ such that if $(Y,Z)\in\mathcal{K}=\mathcal{K}(R,M,k_{1},k_{2},\delta)$, then the following hold.

1. (i)

If $m\in\mathbb{N}$ and $|x|\leqslant c_{9}$, then $|D^{m}h_{Y}(x)|\leqslant C_{4}^{(m)}$. 2. (ii)

If $\operatorname{\mathbb{E}{}}Y\geqslant 1-\delta$, then $h_{Y}^{\prime}(0)\leqslant-\delta$ and $\operatorname{\mathbb{P}{}}(Y\geqslant 2)>0$. 3. (iii)

If $\operatorname{\mathbb{E}{}}Y\geqslant 1-\delta$ and $|x|\leqslant c_{9}$, then $h_{Y}^{\prime}(x)\leqslant-\delta/2$.

Proof.

(i): By (4.6) and (2.1), $h(x)=O(1)$ if $|x|=(R-1)/2$, say. Hence the result, with $c_{9}:=\min\{R-1,1\}/3$, say, follows by Cauchy’s estimates.

(ii): If (2.3) holds with $k_{1}\geqslant 1$, then $\operatorname{\mathbb{P}{}}(Y\geqslant 2)\geqslant\operatorname{\mathbb{P}{}}(Y=k_{1}+1)\geqslant\pi_{k_{1}+1,k_{2}}\geqslant\delta$, and thus $\operatorname{\mathbb{E}{}}(Y(Y-1))\geqslant 2\delta$.

If instead (2.3) holds with $k_{1}=0$, then $\operatorname{\mathbb{P}{}}(Y=0)\geqslant\pi_{k_{1},k_{2}}+\pi_{k_{1},k_{2}+1}\geqslant 2\delta$. Since $\operatorname{\mathbb{E}{}}Y\geqslant 1-\delta$, then $Y\in\mathbb{N}$ implies $\operatorname{\mathbb{E}{}}(\mathbbm{1}_{\{{Y\geqslant 2}\}}(Y-1))=\operatorname{\mathbb{E}{}}(Y-1)+\operatorname{\mathbb{P}{}}(Y=0)\geqslant\delta$, and thus $\operatorname{\mathbb{E}{}}(Y(Y-1))\geqslant 2\operatorname{\mathbb{E}{}}(\mathbbm{1}_{\{{Y\geqslant 2}\}}(Y-1))\geqslant 2\delta$.

In both cases, $h_{Y}^{\prime}(0)\leqslant-\delta$ follows by (4.8), and $\operatorname{\mathbb{P}{}}(Y\geqslant 2)>0$ holds, too.

(iii): Follows by (ii) and (i) (with $m=2$), replacing $c_{9}$ by $\min\{c_{9},\delta/2C_{4}^{(2)}\}$. ∎

We next characterize the survival probability $\rho_{Y}$ in terms of the (unique) solution to $h_{Y}(\hat{\rho})=1$.

Lemma 4.2**.**

Suppose that $R>1$, $M<\infty$, $k_{1},k_{2}\in\mathbb{N}$, and $\delta>0$. There exists a constant $0<c_{10}\leqslant\delta$ such that the following holds. If $(Y,Z)\in\mathcal{K}(R,M,k_{1},k_{2},\delta)$ and $|\operatorname{\mathbb{E}{}}Y-1|<c_{10}$, then there is a unique $\hat{\rho}=\hat{\rho}_{Y}\in\{x\in\mathbb{R}:|x|<c_{9}\}$ such that

[TABLE]

Furthermore, $\rho_{Y}=\max\{\hat{\rho},0\}$, $\operatorname{sign}(\hat{\rho})=\operatorname{sign}(\operatorname{\mathbb{E}{}}Y-1)$, and $|\hat{\rho}|=\Theta(|\operatorname{\mathbb{E}{}}Y-1|)$, where the implicit constants depend only on $R,M,k_{1},k_{2}$ and $\delta$.

Proof.

We apply the inverse function theorem, Lemma 2.13, with $d=1$, $r=c_{9}$ and

[TABLE]

using (4.8) and Lemma 4.1 to verify the assumptions; we shall ensure that $c_{10}\leqslant\delta$, so $|\operatorname{\mathbb{E}{}}Y-1|<c_{10}$ implies $\operatorname{\mathbb{E}{}}Y\geqslant 1-\delta$. Writing $B_{r}=B^{1}_{r}=\{x\in\mathbb{R}:|x|<r\}$ to avoid clutter (as before), Lemma 2.13 shows the existence of a constant $c_{10}>0$, which we may assume to be at most $\delta$, and an inverse function $G:B_{c_{10}}\to B_{c_{9}}$ with $F(G(x))=x$ and $G(0)=0$. We define

[TABLE]

so that $h_{Y}(\hat{\rho})=F(\hat{\rho})+\operatorname{\mathbb{E}{}}Y=1$. Since $\|DG(y)\|=O(1)$ in $B_{c_{10}}$ by Lemma 2.13 and $\|DF(x)\|=O(1)$ in $B_{c_{9}}$ by Lemma 4.1(i), using $G(0)=F(0)=0$ we have $|\hat{\rho}|=|G(1-\operatorname{\mathbb{E}{}}Y)|=O(|\operatorname{\mathbb{E}{}}Y-1|)$ and $|\operatorname{\mathbb{E}{}}Y-1|=|F(\hat{\rho})|=O(\hat{\rho})$, establishing $|\hat{\rho}|=\Theta(|\operatorname{\mathbb{E}{}}Y-1|)$.

We relate $\hat{\rho}$ and $\rho_{Y}$ by a variant of the usual fixed point analysis of $g_{Y}(x)=x$ in $[0,R]$. Since $\operatorname{\mathbb{P}{}}(Y\geqslant 2)>0$ by Lemma 4.1(ii), $g_{Y}$ is strictly convex on $[0,R]$, which implies that $g_{Y}(x)=x$ has at most two solutions in this interval, and exactly one solution if $\operatorname{\mathbb{E}{}}Y=1$, since $g_{Y}(1)=1$ and $g^{\prime}_{Y}(1)=\operatorname{\mathbb{E}{}}Y$. Now $x=1$ and $x=1-\rho_{Y}\in[0,1]$ are solutions. Since $h_{Y}(\hat{\rho})=1$, $x=1-\hat{\rho}$ is also a solution (see (4.6)); since $|\hat{\rho}|<c_{9}<\min\{R-1,1\}$, we have $1-\hat{\rho}\in(0,R)$.

If $\hat{\rho}>0$, then $1-\hat{\rho}\in(0,1)$ and $1$ are two distinct solutions; thus $1-\rho_{Y}=1-\hat{\rho}$, and $g^{\prime}_{Y}(1)>1$ by strict convexity. Similarly, if $\hat{\rho}<0$, then $1-\hat{\rho}\in(1,R)$ and thus $1-\rho_{Y}=1$, and $g^{\prime}_{Y}(1)<1$ by strict convexity. Finally, if $\hat{\rho}=0$, then $\operatorname{\mathbb{E}{}}Y=h_{Y}(0)=h_{Y}(\hat{\rho})=1$ by (4.8), so that $1-\rho_{Y}=1$ (since then $x=1$ is the only solution to $g_{Y}(x)=x$ in $[0,R]$). Hence $\rho_{Y}=\max\{\hat{\rho},0\}$ in all cases. It follows also that $\hat{\rho}$ is unique, and that $\hat{\rho}$ has the same sign as $\operatorname{\mathbb{E}{}}Y-1=g^{\prime}_{Y}(1)-1$. ∎

Remark 4.3**.**

Since $F^{\prime}(0)=h_{Y}^{\prime}(0)=-\operatorname{\mathbb{E}{}}(Y(Y-1))/2$, when $\operatorname{\mathbb{E}{}}Y>1$ it follows easily that $\rho_{Y}=\frac{2(\operatorname{\mathbb{E}{}}Y-1)}{\operatorname{\mathbb{E}{}}(Y(Y-1))}+O(|\operatorname{\mathbb{E}{}}Y-1|^{2})$. In particular $\rho_{Y}\sim\frac{2(\operatorname{\mathbb{E}{}}Y-1)}{\operatorname{\mathbb{E}{}}(Y(Y-1))}$ as $\operatorname{\mathbb{E}{}}Y\searrow 1$, assuming, as always here, that $(Y,Z)\in\mathcal{K}$. This holds under much weaker conditions on $Y$, see [12] and [2] for precise conditions; see also [11, Section 3].

We next consider a branching process family $(\mathfrak{X}_{u})_{u\in I}=(\mathfrak{X}_{Y_{u},Z_{u},Y^{0}_{u},Z^{0}_{u}})_{u\in I}$ as in Section 3; as there we indicate the parameter $u$ by subscripts. Thus, for example, $\hat{\rho}_{u}=\hat{\rho}_{Y_{u}}$ is defined as in Lemma 4.2, with $(Y,Z)$ replaced by $(Y_{u},Z_{u})$. Furthermore, in analogy to (4.3), we also define

[TABLE]

Thus, by combining (4.3) with Lemma 4.2, when $\operatorname{\mathbb{E}{}}Y_{u}\geqslant 1$ we have $\hat{\rho}_{u}=\rho_{Y_{u}}$ and $\hat{\rho}_{Y_{u},Y^{0}_{u}}=\rho_{Y_{u},Y^{0}_{u}}$. Mimicking Lemma 3.1, the following auxiliary result shows that $\hat{\rho}_{u}$ and $\hat{\rho}_{Y_{u},Y^{0}_{u}}$ both vary smoothly in $u$.

Lemma 4.4**.**

Suppose that $R>1$, $M<\infty$, $k_{1},k_{2}\in\mathbb{N}$, and $\delta>0$. Set $\mathcal{K}^{0}=\mathcal{K}^{0}(R,M,\delta)$ and $\mathcal{K}=\mathcal{K}(R,M,k_{1},k_{2},\delta)$. Let $(\mathfrak{X}_{u})_{u\in I}=(\mathfrak{X}_{Y_{u},Z_{u},Y^{0}_{u},Z^{0}_{u}})_{u\in I}$ be a branching process family satisfying the assumptions of Lemma 3.1, with $|\operatorname{\mathbb{E}{}}Y_{u}-1|\leqslant c_{1}$ replaced by $|\operatorname{\mathbb{E}{}}Y_{u}-1|\leqslant c_{10}$. Let $\hat{\rho}_{u}$ and $\hat{\rho}_{Y_{u},Y^{0}_{u}}$ be defined as in Lemma 4.2 and (4.9). Then $\hat{\rho}_{u}$ and $\hat{\rho}_{Y_{u},Y^{0}_{u}}$ are analytic functions of $u\in I$. Furthermore,

[TABLE]

where the implicit constants depend only on $R,M,k_{1},k_{2}$ and $\delta$.

Proof.

Let $h_{u}(x)=h_{Y_{u}}(x):=(1-g_{u}(1-x))/x$ be the equivalent of (4.6) for $\mathfrak{X}_{u}$, again removing the removable singularity at $x=0$. Then $h_{u}(x)$ is an analytic function of $(u,x)\in I\times\{x\in\mathbb{C}:|x|<R-1\}$. Note that (3.1) implies $|\frac{\partial}{\partial u}h_{u}(x)|=O(\lambda)$ if $|x|=(R-1)/3$, say. Since $c_{9}\leqslant(R-1)/3$, by the maximum modulus principle (applied with $u$ fixed) it follows that

[TABLE]

for all $u\in I$ and $|x|\leqslant c_{9}$.

By Lemma 4.2, for every $u\in I$ there is a unique $\hat{\rho}_{u}\in\mathbb{R}$ with $|\hat{\rho}_{u}|<c_{9}$ such that

[TABLE]

Since $|h_{u}^{\prime}(\hat{\rho}_{u})|\geqslant\delta/2$ by Lemma 4.1(iii) and $|\operatorname{\mathbb{E}{}}Y_{u}-1|\leqslant c_{10}\leqslant\delta$, the implicit function theorem shows that $\hat{\rho}_{u}$ is an analytic function of $u\in I$. That $\hat{\rho}_{Y_{u},Y^{0}_{u}}$ is analytic then follows from (4.9) and the assumption that $g_{u}^{0}(y,z)$ is analytic. By differentiating (4.12) we obtain $\frac{\partial h_{u}}{\partial u}(\hat{\rho}_{u})+h_{u}^{\prime}(\hat{\rho}_{u})\cdot\frac{\mathrm{d}}{\mathrm{d}u}\hat{\rho}_{u}=0$. So, using $|h_{u}^{\prime}(\hat{\rho}_{u})|\geqslant\delta/2$ and (4.11),

[TABLE]

Finally, $g_{Y_{u}^{0}}^{\prime}(1-\hat{\rho}_{u})=O(1)$ follows from (2.1) and Cauchy’s estimates (recall that $|\hat{\rho}_{u}|<c_{9}\leqslant(R-1)/3$). By differentiating (4.9) and then using (3.1) and (4.13), we obtain

[TABLE]

completing the proof. ∎

4.2 A specific result suitable for application to Achlioptas processes

We are now ready to prove our main result, concerning the $t$-dependence of the survival probability of $\mathfrak{X}_{t}$ when $(\mathfrak{X}_{t})_{t\in I}$ is a $t_{\mathrm{c}}$-critical branching process family, as well as the survival probability of branching processes $\mathfrak{X}=\mathfrak{X}_{Y,Z,Y^{0},Z^{0}}$ of type $(t,\eta)$; see Section 3.2 for the relevant definitions. Two key features are the convergent power series expansion (4.14), and the uniform $O(\eta)$ error term in (4.15). In particular, we have $\tilde{\rho}\sim\rho(t_{\mathrm{c}}+\varepsilon)=\Theta(\varepsilon)$ for any branching process $\mathfrak{X}$ of type $(t_{\mathrm{c}}+\varepsilon,\eta)$ with $\eta\ll\varepsilon\leqslant\varepsilon_{0}$. In the supercritical case $t=t_{\mathrm{c}}+\varepsilon$, the survival probabilities of $\mathfrak{X}_{t}$ and $\mathfrak{X}$ thus both grow linearly in $\varepsilon$.

Theorem 4.5** (Survival probabilities).**

Let $(\mathfrak{X}_{t})_{t\in(t_{0},t_{1})}=(\mathfrak{X}_{Y_{t},Z_{t},Y^{0}_{t},Z^{0}_{t}})_{t\in(t_{0},t_{1})}$ be a $t_{\mathrm{c}}$-critical branching process family. Then there exist constants $\varepsilon_{0},c>0$ with the following properties. Firstly, the survival probability $\rho(t):=\operatorname{\mathbb{P}{}}(|\mathfrak{X}_{t}|=\infty)$ is zero for $t_{\mathrm{c}}-\varepsilon_{0}\leqslant t\leqslant t_{\mathrm{c}}$, and is positive for $t_{\mathrm{c}}<t\leqslant t_{\mathrm{c}}+\varepsilon_{0}$. Secondly, $\rho(t)$ is analytic on $[t_{\mathrm{c}},t_{\mathrm{c}}+\varepsilon_{0}]$. More precisely, there are constants $a_{i}$ with $a_{1}>0$ such that

[TABLE]

for $0\leqslant\varepsilon\leqslant\varepsilon_{0}$. Thirdly, for any $t$, $\eta$ with $|t-t_{\mathrm{c}}|\leqslant\varepsilon_{0}$ and $\eta\leqslant c|t-t_{\mathrm{c}}|$, and any branching process $\mathfrak{X}=\mathfrak{X}_{Y,Z,Y^{0},Z^{0}}$ of type $(t,\eta)$ (with respect to $(\mathfrak{X}_{t})$), the survival probability $\tilde{\rho}:=\operatorname{\mathbb{P}{}}(|\mathfrak{X}|=\infty)$ is zero if $t\leqslant t_{\mathrm{c}}$, and is positive and satisfies

[TABLE]

if $t>t_{\mathrm{c}}$, where the implicit constant depends only on the family $(\mathfrak{X}_{t})$, not on $t$ or $\mathfrak{X}$. Moreover, analogous statements hold for the survival probabilities $\rho_{1}(t):=\operatorname{\mathbb{P}{}}(|\mathfrak{X}^{1}_{Y_{t},Z_{t}}|=\infty)$ and $\tilde{\rho}_{1}:=\operatorname{\mathbb{P}{}}(|\mathfrak{X}^{1}_{Y,Z}|=\infty)$.

Proof.

We argue as in the proof of Theorem 3.4. In particular, we may assume that $(Y,Z)\in\mathcal{K}$ and $(Y^{0},Z^{0})\in\mathcal{K}^{0}$ for some $R,M,k_{1},k_{2},\delta$. We shall also assume that $c\leqslant 1$.

We consider only $t$ with $|t-t_{\mathrm{c}}|\leqslant\varepsilon_{0}$; we may assume that $\varepsilon_{0}$ is small enough that this implies $t\in(t_{0},t_{1})$, and, by (3.10), that $\operatorname{\mathbb{E}{}}Y^{0}_{t}>0$, and that

[TABLE]

By (4.2) and Lemma 4.2 and it follows that $\rho_{1}(t)=\rho_{Y_{t}}$ is zero for $t_{\mathrm{c}}-\varepsilon_{0}\leqslant t\leqslant t_{\mathrm{c}}$, and positive for $t_{\mathrm{c}}<t\leqslant t_{\mathrm{c}}+\varepsilon_{0}$. Since $\operatorname{\mathbb{P}{}}(Y_{t}^{0}\geqslant 1)>0$, now (4.1) and (4.3) imply an analogous statement for $\rho(t)=\rho_{Y_{t},Y^{0}_{t}}$. Lemmas 4.2 and 4.4 also imply that

[TABLE]

are both analytic for $t_{\mathrm{c}}\leqslant t\leqslant t_{\mathrm{c}}+\varepsilon_{0}$. Hence (4.14) holds if $\varepsilon_{0}$ is sufficiently small.

Next, for a branching process of type $(t,\eta)$, by (3.16) we have $|\operatorname{\mathbb{E}{}}Y_{t}-\operatorname{\mathbb{E}{}}Y|=O(\eta)$. Since $\eta\leqslant c|t-t_{\mathrm{c}}|$, it follows from (4.16) that if $c$ is small enough, then $\operatorname{sign}(\operatorname{\mathbb{E}{}}Y-1)=\operatorname{sign}(t-t_{\mathrm{c}})$. Moreover, since $\eta\leqslant c|t-t_{\mathrm{c}}|\leqslant|t-t_{\mathrm{c}}|\leqslant\varepsilon_{0}$, using (4.16) we also have $|\operatorname{\mathbb{E}{}}Y-1|<c_{10}$ if $\varepsilon_{0}$ is small enough. Mimicking the above reasoning for $\rho_{1}(t)$ and $\rho(t)$, using (4.1)–(4.3) and Lemma 4.2 it follows for $\eta\leqslant c|t-t_{\mathrm{c}}|$ that $\tilde{\rho}_{1}=\rho_{Y}$ and $\tilde{\rho}=\rho_{{Y,Y^{0}}}$ satisfy $\tilde{\rho}_{1}=\tilde{\rho}=0$ if $t_{\mathrm{c}}-\varepsilon_{0}\leqslant t\leqslant t_{\mathrm{c}}$, and $\tilde{\rho}_{1},\tilde{\rho}>0$ if $t_{\mathrm{c}}<t\leqslant t_{\mathrm{c}}+\varepsilon_{0}$; furthermore,

[TABLE]

for $t_{\mathrm{c}}\leqslant t\leqslant t_{\mathrm{c}}+\varepsilon_{0}$ and $\eta\leqslant c|t-t_{\mathrm{c}}|$.

Finally, we consider the interpolating branching process family $(\bar{Y}_{u},\bar{Z}_{u},\bar{Y}^{0}_{u},\bar{Z}^{0}_{u})_{u\in[0,1]}$ defined by (3.18), for which, as noted in Section 3.2, (3.1) holds with $\lambda=\eta$ and $I=[0,1]$. Note that (3.19) and $\eta\leqslant|t-t_{\mathrm{c}}|\leqslant\varepsilon_{0}$ imply $|\operatorname{\mathbb{E}{}}\bar{Y}_{u}-1|<c_{10}$ provided $\varepsilon_{0}$ is small enough. Integrating (4.10) of Lemma 4.4 over $u\in[0,1]$ similarly to (3.20) in the proof of Theorem 3.4, using the identities (4.17)–(4.18) we infer $\tilde{\rho}_{1}-\rho_{1}(t)=\hat{\rho}_{Y}-\hat{\rho}_{Y_{t}}=O(\eta)$ and $\tilde{\rho}-\rho(t)=\hat{\rho}_{{Y,Y^{0}}}-\hat{\rho}_{Y_{t},Y^{0}_{t}}=O(\eta)$ for $t_{\mathrm{c}}\leqslant t\leqslant t_{\mathrm{c}}+\varepsilon_{0}$ and $\eta\leqslant c|t-t_{\mathrm{c}}|$, completing the proof. ∎

Theorem 4.5 immediately implies the key case $p_{{\mathcal{R}}}=1$, $K=0$ of Theorem A.11 of [22], used there for the analysis of Achlioptas processes.

Acknowledgement. The last two authors are grateful to Christina Goldschmidt for useful pointers to the local limit theorem literature, which were helpful for the developing parts of the slightly more involved (large deviation based) point probability analysis contained in an earlier version of [22].