On the largest component in the subcritical regime of the Bohman-Frieze process

Kang, Perkins and Spencer showed that the size of the largest component of the Bohman-Frieze process at a fixed time $t$ smaller than $t_c$, the critical time for the process is $L_1(t)=\Omega(\log n/(t_c-t)^2)$ with high probability. They also conjectured that this is the correct order, that is $L_1(t)=O(\log n/(t_c-t)^2)$ with high probability for fixed $t$ smaller than $t_c$. Using a different approach, Bhamidi, Budhiraja and Wang showed that $L_1(t_n)=O((\log n)^4/(t_c-t_n)^2)$ with high probability for $t_n\leq t_c-n^{-\gamma}$ where $\gamma\in(0,1/4)$. In this paper, we improve their result by showing that for any fixed $\lambda>0$, $L_1(t_n)=O(\log n/(t_c-t_n)^2)$ with high probability for $t_n\leq t_c-\lambda n^{-1/3}$. In particular, this settles the conjecture of Kang, Perkins and Spencer. We also prove some generalizations for general bounded size rules.


Introduction
Initiated by a question of Dimitris Achlioptas, the study of modified Erdős-Rényi processes (called Achlioptas processes) has grown into a large area of research in the past decade. At each step of an Achlioptas process, two randomly chosen edges are presented and one of these edges is added to the current random graph according to some selection rule. The behavior of the random graph process (e.g. point of phase transition, appearance of Hamiltonian cycles) depends on the selection rule. The Erdős-Rényi process, for example, is an Achlioptas process where the first edge chosen is always added to the random graph. An account of the literature on properties of several Achlioptas processes can be found in [3,4,5,6,7,8,9,11,12,13,14,15] and the references therein.
One such Achlioptas process, called the Bohman-Frieze process has received a great deal of attention and will be of particular interest to us. We first describe a continuous time version of the Bohman-Frieze process. Consider the complete graph G n = (V n , E n ) on the vertex set V n = {1, . . . , n}. Consider independent Poisson processes P e indexed by e = (e 1 , e 2 ) ∈ E n × E n each having rate 2/n 3 . Let ∪ e∈En×En P e = {u 1 < u 2 < . . .}. Then the dynamics of the continuous time Bohman-Frieze process (BF n (·)) are given as follows.
BF n (u) is the empty graph on V n for 0 ≤ u < u 1 .
If the Poisson process P e has a point at u i where e = (e 1 , e 2 ) and the endpoints of e 1 are isolated vertices in BF n (u i −), set BF n (u) = BF n (u i −) ∪ e 1 for u ∈ [u i , u i+1 ). Otherwise, set BF n (u) = BF n (u i −) ∪ e 2 for u ∈ [u i , u i+1 ). The expected number of edges at u = 1 is thus the time normalization is the one corresponding to the Erdős-Rényi process. The corresponding discrete time version (DBF n (·)) of the Bohman-Frieze process evolves as follows.
DBF n (u) is the empty graph on V n for 0 ≤ u < 2/n. At time 2(k + 1)/n, two edges e 1 and e 2 are selected uniformly (with replacement) from E n . If the endpoints of e 1 are isolated vertices in DBF n (2k/n), set DBF n (u) = DBF n (2k/n) ∪ e 1 for u ∈ [2(k + 1)/n, 2(k + 2)/n). Otherwise, set DBF n (u) = DBF n (2k/n) ∪ e 2 for u ∈ [2(k + 1)/n, 2(k + 2)/n). We shall denote by L BF 1 (t) (resp. L DBF 1 (t)), the size of the largest component of BF n (t) (resp. DBF n (t)). It is known that the phase transition for the discrete time Bohman-Frieze process happens at time t c > 1. (It is easy to see that t c is also the critical time for the continuous time Bohman-Frieze process and hence we shall refer to it as the critical time for the Bohman-Frieze process.) Theorem 4 of [13] shows that for any fixed t ∈ (0, t c ), for some constant K free of t. (Actually, the version of the Bohnman-Frieze process studied in [13] is slightly different from DBF n (·) since we are sampling the edges at time 2(k + 1)/n from E n whereas, in [13], only the edges not present in the graph at time 2k/n are allowed, but this difference is negligible, for details see [12].) Kang, Perkins and Spencer conjecture that this is indeed the correct order, i.e.
for any fixed t ∈ (0, t c ) and some constant K ′ free of t (Conjecture 1 in [13]). Bhamidi, Budhiraja and Wang [4,5] independently show that for γ ∈ (0, 1/4), there exists a constant C = C(γ) such that (1.1) P(L BF 1 (t n ) ≥ C(log n) 4 /(t c − t n ) 2 ) → 0 for t n ≤ t c − n −γ by connecting the dynamics of BF n (·) to an inhomogeneous random graph model. In this work, we take the approach in [4] and go through a more careful analysis to prove Conjecture 1 of [13] (Theorem 1 and Corollary 2). Our result is true for t n ≤ t c − λn −1/3 (for any fixed λ > 0) and thus closes the gap between the critical window and the interval 0 < t ≤ t c − n −γ , γ < 1/4 where the bound in [4] is valid.
Spencer and Wormald [12] introduced a generalization of the Bohman-Frieze process called bounded size rules. For K ≥ 0, we define The symbol ω represents "numbers bigger than K" (see [12]). For a graph G on V n and v ∈ V n , let C(v, G) denote the size of the component of G containing v. Let Consider independent Poisson processes {P e : e ∈ V 4 n } of intensity 1/2n 3 and let ∪ e∈V 4 n P e = {u 1 < u 2 < . . .}. Then the continuous time bounded size rule process (BSR(·)) associated with F can be described in the following way.
Define BSR n (u) to be the empty graph on V n for 0 ≤ u < u 1 .
If the Poisson process P e has a point at ). The discrete time version, DBSR n (·) can be defined in the same way we defined DBF n (·). In [5], it was shown that (1.1) holds if we replace L BF 1 (t n ) by L BSR 1 (t n ), the size of the largest component in BSR(t n ). We improve this result in Theorem 3 (see also Corollary 4).
The precise statements of our results are given in Section 2. The proofs for the Bohman-Frieze process and the bounded size rules are given in Section 3 and Section 4 respectively. Although bounded size rules generalize the Bohman-Frieze rule, we choose to treat the more important Bohman-Frieze process separately in Section 3 to retain clarity in the (notationally simpler) proof. In Section 4, we give an outline of the proof for general bounded size rules highlighting only those parts of the proof which are slightly different.

Main Results
The following theorem is our main result. Theorem 1. Let t c be the critical time for the Bohman-Frieze process. Let t = t(n) satisfy t ≤ t c − λn −1/3 for a fixed λ > 0. Let L BF 1 (t) denote the size of the largest component of BF n (t). Then there exists a positive constant C depending only on λ such that An immediate consequence of Theorem 1 is the analogous result for the discrete time Bohman-Frieze process.
Corollary 2. Let t c be the critical time for the Bohman-Frieze process. Let t = t(n) satisfy t ≤ t c − λn −1/3 for a fixed λ > 0. Let L DBF 1 (t) denote the size of the largest component of DBF n (t). Then there exists a positive constant C depending only on λ such that The arguments used in the proof of Theorem 1 can be generalized to general bounded size rules. Recall the definition of Ω K from Section 1.
K and consider the bounded size rule associated with F . Let t c be the critical time for the process. Let L BSR 1 (u) denote the size of the largest component of BSR n (u) for u ≥ 0. Then, there exists ζ = ζ(F ) > 1/4 having the following property: for every fixed ζ ′ ∈ (0, ζ), there exists a constant C depending only on ζ ′ and F such that An analogous result is true for the discrete time process.
Remark 5. It will follow from our arguments that the conclusions of Theorem 1 and Corollary 2 remain true for any bounded size rule with K = 1.
We shall see in Section 4.3 that if K = 2 and F satisfies some conditions, then the constant ζ(F ) is at least 1/3, hence the upper bounds in Theorem 3 and Corollary 4 hold up to the critical window.
However, if K ≥ 3 then the constant ζ that we get from our proof will be smaller than 1/3. So, the interval where the stated upper bound holds falls shy of the critical window.

Proofs for the Bohman-Frieze process
We shall present the proofs of Theorem 1 and Corollary 2 in this section. We start off with 3.1. Connection between the Bohman-Frieze process and an Inhomogeneous Random Graph Model. The analysis in [4] is carried out by relating the dynamics of the Bohman-Frieze (BF) process to an inhomogeneous random graph model. Since this plays a crucial role in the proof, we briefly describe this connection and introduce the necessary notations. A more detailed discussion of these and related models can be found in [1], [4] and [10].
Let X n (v) be the number of singletons (i.e. isolated vertices) in the BF process at time v and set x n (v) = X n (v)/n. An edge added in the BF process at time v can be of the following types: (I) both its endpoints were isolated vertices in BF n (v−); (II) only one of its endpoints was an isolated vertex in BF n (v−); (III) none of its endpoints were isolated in BF n (v−).
Two singletons are added in the BF process (i.e. an edge of type I is created) if one of the following happens: (i) the first edge selected connects two isolated vertices or (ii) the first edge selected does not connect two isolated vertices but the second edge selected joins two isolated vertices.
Hence two singletons are added in the BF process at a rate 2 n 3 where a n (y) = a 0 (y) + O(1/n) and a 0 : [0, 1] → R + is the function a 0 (y) = y 2 − y 4 /2. One can similarly show that a given non-singleton vertex (i.e. a vertex which is not isolated) in BF n (v) gets connected to some isolated vertex (i.e. an edge of type II is created) at a rate c n (x n (v)) where c n (y) = c 0 (y) + O(1/n) and the function c 0 : [0, 1] → R + is given by c 0 (y) = (1 − y 2 )y.
Finally, an analogous computation will show that two given non-singleton vertices are joined (i.e. an edge of type III is created) at a rate b n (x n (v))/n where b n (y) = b 0 (y) + O(1/n) and the function b 0 : [0, 1] → R + is given by b 0 (y) = 1 − y 2 .
The function x n (v) is highly concentrated around x(v) (see [12,4]), which satisfies the ODE As a result, the rate functions, say for example a n (x n (v)) lies very close to a 0 (x(v)) with high probability. Hence, the BF process can be approximated with high probability by a random graph model with deterministic rate functions which we describe next.
The last three events happen independently. This model (introduced in [4]) is a generalization of the RGIV model of [1] and is a good approximation of the BF process when For fixed v, IA n (a, b, c) v can be obtained by the following two step construction: (Step 1) Run a Poisson process P with rate na(s)ds in the interval [0, v]. At each arrival of this Poisson process a new doubleton is born. A doubleton born at time s ∈ [0, v) grows its own component independent of other components according to the following Markov process: if w(.) denotes the size of the component of the doubleton born at time s as a function of time, then given {w(u)} s≤u≤s 1 , the component size grows by one in (s 1 , s 1 + ds 1 ] with rate w(s 1 )c(s 1 ). Continue this process till time v. In the end we shall have a collection of pairs is the function such that w i (u) denotes the size at time u, of the component of the doubleton born at time s i with the convention w i (u) = 0 for u < s i . ( Step 2) Conditional on step 1, add an edge between two clusters x = (s 1 , w 1 ), y = (s 2 , w 2 ) with probability p n,v (x, y) : The distribution of the component sizes of the graph obtained via the two step process is the same as that of IA n (a, b, c) v . This allows one to view IA n (a, b, c) as an inhomogeneous random graph (IRG) model whose construction we recall next.
A triplet (X, T , µ) where X is Polish, T is the Borel σ-field and µ is a finite measure is called a type space.
A measurable function k : A non-negative measurable function φ on X is called a weight function. Given a type space (X, T , µ), a weight function φ and a sequence of kernels k n , we construct a random graph by letting its vertices be points of a Poisson process P on X with intensity nµ(dx) and then join any two points x, y ∈ P with probability kn(x,y) n ∧ 1. The volume of a connected component C is given by volume(C) := x∈C φ(x).
Then IA n (a, b, c) v corresponds to the IRG model associated with (X, µ, k n,v , φ v ) where X = [0, T ] × W and W = D([0, T ] : Z ≥0 ) is equipped with the Skorohod topology; µ(d(s, w)) = a(s)ds ν s (dw) where ν s is the law of the function denoting the size of a cluster born at time s in IA n (a, b, c); k n,v (x, y) = np n,v (x, y) and φ v (s, w) = w(v).
We also define the kernel k v by Note that for constructing RG n,v (a, b, c), we have taken the constant sequence of kernels where each element is k v . Sometimes we shall write µ(a, b, c) to express the dependence of µ on a, b, c (µ as a measure depends only on the functions a and c, but we prefer to write it this way to put emphasis on the underlying IRG model). We shall write ρ v (a, b, c) to denote the norm of the operator

3.2.
Proof of Theorem 1. We present the proof of Theorem 1 in this section. Throughout the proof C, C ′ etc. will denote positive universal constants whose values may change from line to line. Special constants will be indexed, as for example C 1 , C 2 etc. In the proof we shall assume λ = 1 since it will not make a difference.
The first few steps consist of reducing the problem to getting an upper bound on the total progeny of a continuum type branching process. Assume that t = t(n) satisfies t ≤ t c − n −1/3 . Let C 0 n (t) denote the component of the first doubleton appearing in BF n (t). Fix γ ∈ (1/3, 1/2) and define E n : and similar upper bounds hold for b n (x n (u)) and c n (x n (u)). Set δ = δ n = C 2 n −γ and define a n,δ (u) = (a 0 (x(u)) + δ n ) for t ∈ [0, T ]. Define b n,δ (u) and c n,δ (u) similarly. Hence, a n (x n (u)) ≤ a n,δ (u) on E c n and similar upper bounds hold for b n (x n (u)) and c n (x n (u)). Note that sup Consider the first immigrating doubleton in IA n (a n,δ , b n,δ , c n,δ ) and let C IA n,δ (t) denote the size of the component of IA n (a n,δ , b n,δ , c n,δ ) t which contains the first immigrating doubleton. Let ν s,δ denote the measures on W such that µ(a n,δ , b n,δ , c n,δ )(d(s, w)) = a n,δ (s)ds ν s,δ (dw). Consider the IRG model RG n,t (a n,δ , b n,δ , c n,δ ) conditioned on having a point (0, w) where w is distributed according to ν 0,δ . Let C RG n,δ (t) be the volume of the component containing (0, w). The assertions in the next Lemma follow from Lemma 5.1, Lemma 5.2, Lemma 6.1, Lemma 6.4 and Lemma 6.10 of [4].
(ii) Connection between BF process, IA n (a n,δ , b n,δ , c n,δ ) and RG n,t (a n,δ , b n,δ , c n,δ ): We have (iii) Properties of operator norms: The function f (u) := ρ u (a 0 (x(·)), b 0 (x(·)), c 0 (x(·))) is strictly increasing and satisfies f (t c ) = 1. Further, there exists some positive constant η such that From Lemma 6, we get Here, P w 0 (·) = P(·| (0, w 0 ) ∈ RG n,t (a n,δ , b n,δ , c n,δ )) for fixed w 0 ∈ W . Consider now a branching process on [0, t] × W as follows. Write k t,δ for k t (a n,δ , b n,δ , c n,δ ) and µ δ for µ(a n,δ , b n,δ , c n,δ ). Define x 0 = (0, w 0 ) to be generation zero of the branching process. For k ≥ 0, denote by N k , the total number of points in generation k. N k in generation k gives birth to its own offsprings according to a Poisson processes with intensities k t,δ (x (k) i , y) µ δ (dy) respectively independent of the other points in generation k.
i ) denote the total progeny. By a breadth first search argument (see Lemma 6.12 in [4]), we have From (3.1) and (3.2), we have The following Lemma is an improvement over Lemma 6.9 of [4]. Recall that T = 2t c .
The proof of Lemma 7 is given in the Section 3.
Denote by K t,δ , the integral operator associated with the kernel k t,δ . Let X t := [0, t] × W . Let µ := µ(1, 1, 1), i.e. the measure on [0, T ] × W associated with the functions which are identically equal to one and let ν s be the family of measures on W such that µ(d(s, w)) = ds ν s (dw).
We shall use the following lemmas to get an upper bound on the right side of (3.3).
Hence we have From (3.19) and (3.3), we see that this completes the proof of Theorem 1.
From (3.7) and (3.21), Note that This together with the simple inequality e x − 1 ≤ xe x yields Since the tail of w 1 (T ) decays exponentially, we conclude that for large n (so that δ = δ n is sufficiently small) Note that inf [0,T ] a(s) = m 1 > 0 and 0 ≤ a δ (s) − a(s) ≤ δ. Hence, The integrand corresponding to L 2 can be handled in the same way.
x ′ (t) > 0 (see [12]) for some small ǫ > 0 and hence for a positive constant m 2 . Further, c(t) is increasing in an interval [0, t 0 ] and is bounded away from zero on [t 0 , T ]. Hence Hence, on the set w 1 (T ) ≥ 3, Hence, Here we choose p, q > 1 so that p −1 + q −1 = 1 and 2q = 2 + θ with 0 < θ < 1. Define The last inequality is a consequence of (3.27). Since θ < 1, the last integral is finite. A similar analysis can be carried out for the integrand corresponding to Finally, From (3.31) and (3.32), we get which is the desired bound.

3.4.
Proof of Corollary 2. Let X n (s) denote the number of edges in BF n (s). Define a process BF n (·) by BF n (2X n (s)/n) := BF n (s) for s ≥ 0 and extend the definition to R + by right continuity. Then BF n (·) has the same distribution as DBF n (·). Let L BF 1 (s) denote the size of the largest component of BF n (s). Let us assume that t c /2 ≤ t ≤ t c − λn −1/3 , since for t ≤ t c /2 the desired bound will follow directly. We have Hence P(L BF 1 (t) ≥ m) ≤ P(L BF 1 (t + log n/ √ n) ≥ m) + P(X n (t + log n/ √ n) < nt/2). (3.33) Let Z 1 , . . . , Z n be i.i.d. Poisson random variables with mean µ n := 1 2 (1 − 1/n) 2 (t + log n/ √ n).

Proofs for general bounded size rules
As in the case of the Bohman-Frieze process, the proofs for general bounded size rules also proceed through the analysis of an inhomogeneous random graph model. We start with some definitions.
Fix K ≥ 2 and F ⊂ Ω 4 K . Consider the bounded size rule associated with F . For j = (j 1 , j 2 , j 3 , j 4 ) ∈ F and i ∈ Ω K , let When j ∈ F c , we replace j 1 , j 2 in the above expressions by j 3 , j 4 respectively and define ∆( j, i) in the same way. For the motivation behind this definition, see [12]. Consider now the system of ODEs with the initial condition x i (0) = I{i = 1}. Spencer and Wormald [12] showed that this system has a solution {x i (·) : i ∈ Ω K } defined for v ≥ 0. Further, the proportion of vertices of DBSR n (v) in components of size i converges to x i (v) in probability. For i 1 , i 2 ∈ Ω K , we define the functions We now define some rate functions which are analogous to the functions a 0 (x(·)), b 0 (x(·)) and c 0 (x(·)) appearing in the proof of Theorem 1. For i ∈ {1, . . . , K}, let and let Let BSR * n (v) be the subgraph of BSR n (v) consisting of those components of BSR n (v) which contain at least K + 1 vertices. Then, for 1 ≤ i ≤ K, na i (v) approximates the rate at which a component of size K + i appears in BSR * n (v), i.e. the rate at which two components of sizes smaller than K + 1 merge to form a component of size K + i. Given two vertices in BSR * n (v), b(v)/n approximates the rate at which these two vertices are joined by an edge. Given a vertex in BSR * n (v) and 1 ≤ i ≤ K, c i (v) approximates the rate at which a component of size i gets linked to this vertex by an edge. For the actual computations leading to these formulas for the rate functions, see [5].
In general, for any nonnegative continuous functions {a i } 1≤i≤K , b and {c i } 1≤i≤K and v ∈ [0, T ], we denote by k v ({a i } i≤K , b, {c i } i≤K ) (resp. µ({a i } i≤K , b, {c i } i≤K )), the kernel (resp. the measure) on X constructed in a manner similar to the construction of k v,δ (resp. µ δ ) using the We shall simply write Since a i is real analytic around zero and x j (t) > 0 for t > 0 and j ∈ Ω K (Theorem 2.1 in [12]), m a i is necessarily finite. Define m c i for 1 ≤ i ≤ K similarly. Let The following Lemma is the analogue of Lemma 7.
Lemma 10. Let α be as above. Then for any α ′ ∈ (0, α) we have The constant C depends only on α ′ .
Let C RG 1 (t) be the largest component of RG n,t (= RG n,t ({a i,δ } i≤K , b δ , {c i,δ } i≤K )) and let L RG 1 (t) := x∈C RG 1 (t) φ t (x) be the volume of C RG 1 (t). From Lemma 4.4 of [5], it follows that (4.6) P(L BSR 1 (t) ≥ m) ≤ P(L RG 1 (t) ≥ m) + C exp(−C ′ n 1−2γ ) for m ≥ K + 1. For x = (s, w) ∈ X, define I(x) = s. Let P n be a Poisson process in X with rate nµ δ and let N n,t := {x ∈ P n : I(x) ≤ t}. Then for any A > 0 (4.7) P(L RG 1 (t) ≥ m) ≤ P(L RG 1 (t) ≥ m, N n,t ≤ nA) + P(N n,t > nA). Let I 1 ≤ . . . ≤ I Nn,t be an ordering of elements of {I(x) : x ∈ P n , I(x) ≤ t}. Then Define the event F j as follows, where C RG (I j , t) is the component in RG n,t containing the point x j such that I(x j ) = I j . From (4.8), it follows that (4.9) For any w 0 ∈ W with w 0 (0) = 2K, define a branching process on [0, t] × W starting from x 0 = (0, w 0 ) exactly as in the proof of Theorem 1 using the kernel k t,δ and the measure µ δ . Let G(x 0 ) be as in the proof of Theorem 1. Consider the event E w 0 := {x 0 ∈ P n }. On the event E w 0 , let C RG (x 0 , t) denote the component of x 0 in RG n,t . Then, as in the proof of Theorem 1, It is easy to see that the analogues of Lemma 8 and Lemma 9 remain true in the present setup with ∆ = 1 − ρ t,δ . Hence, we can proceed as before to conclude that for some positive constant C 5 independent of w 0 , E exp(η∆ 2 G(x 0 )) ≤ exp(C 5 ηw 0 (T )) whenever η ≤ η 0 , an absolute constant free of w 0 . Hence for η small enough. Since ∆ ≥ C(t c − t), we conclude from (4.6), (4.7), (4.9), (4.10) and (4.11) that  As in the proof of Lemma 7 (see (3.32)), we have It is easy to see that if K ≥ 2, then m a K ≥ 2 and m c K ≥ 1 (see the discussion in Section 4.3).