On the critical probability in percolation

For percolation on finite transitive graphs, Nachmias and Peres suggested a characterization of the critical probability based on the logarithmic derivative of the susceptibility. As a first test-case, we study their suggestion for the Erd\H{o}s-R\'enyi random graph G_{n,p}, and confirm that the logarithmic derivative has the desired properties: (i) its maximizer lies inside the critical window p=1/n+\Theta(n^{-4/3}), and (ii) the inverse of its maximum value coincides with the \Theta(n^{-4/3})-width of the critical window. We also prove that the maximizer is not located at p=1/n or p=1/(n-1), refuting a speculation of Peres.

In the language of mathematical physics, G n,p interpreted as percolation on the complete n-vertex graph is a mean-field model. Hence, we expect that the percolation phase transition of many 'high dimensional' finite graphs is similar, with the hypercube and various tori being examples of great interest (see, e.g., [2,5,18,7,8,9,16]). To fix notation, we assume that G is a given transitive n-vertex graph, and we write G p ⊆ G for the binomial random subgraph where each edge is included independently with probability p. As pointed out by Nachmias and Peres [29], in this general percolation setting it is a challenging problem to find a good definition of the critical probability p c , such that for a suitable critical window around p c , for example, the size of the largest component is not concentrated.
The folklore average degree heuristic p c = 1/(deg G (v) − 1) is a natural first guess (the graph G is assumed to be transitive and thus regular, so the choice of the vertex v does not matter). For the hypercube with vertex set {0, 1} m , and thus degree m, Ajtai, Komlós and Szemerédi [2] showed that there is a critical threshold (1 + o(1))/m; this was sharpened by Bollobás, Kohayakawa, and Luczak [5], who raised the question whether the critical probability might be exactly 1/(m − 1). However, Borgs, Chayes, van der Hofstad, Slade and Spencer [7,8] and van der Hofstad and Nachmias [16,17] have shown that there is critical window of width Θ(n −1/3 p c ) = Θ(2 −m/3 /m) about a critical probability p c , which by van der Hofstad and Slade [18] satisfies p c = 1/(m − 1) + 3.5m −3 + O(m −4 ); since the width of the window is o(m −3 ), the value 1/(m − 1) is outside the critical window.
A more sophisticated suggestion for the critical probability was pioneered by Borgs, Chayes, van der Hofstad, Slade and Spencer [7,8] (and used for the hypercube result just described). They essentially proposed to define p c = p c (G) as the unique solution to the polynomial equation where the susceptibility χ G (p) denotes the expected size of the component C(v) containing a fixed vertex v in G p . (This is a widely studied key parameter in percolation theory and random graph theory, see, e.g., [1,13,22,25,32,33]. Since G is assumed to be transitive, the choice of v does not matter.) The aforementioned technical definition is guided by Erdős-Rényi mean-field type behaviour. Indeed, in the subcritical phase we expect that C(v) closely mimics a subcritical branching process, which suggests that typically |C 1 | ≈ ( χ G (p)) 2 up to logarithmic corrections (see, e.g., Section 1.2 in [15] or Proposition 5.1 in [1]). Furthermore, in the supercritical phase we expect that the largest component dominates all other components, which by transitivity suggests that χ G (p) ≈ E p |C 1 | 2 /n. Assuming that inside the critical window we can observe subcritical and supercritical features, it thus seems plausible that the critical probability should roughly satisfy χ G (p) ≈ E p |C 1 | 2 /n ≈ χ G (p) 4 /n, motivating the choice of equation (1.1). Borgs et al. [7,8] showed that (a minor variant of) the discussed definition is very useful in combination with the so-called finite triangle condition: they recovered many Erdős-Rényi features under such generic mean-field assumptions (see [16,17] for some more recent developments). As pointed out by Peres [31], the suggestion of Borgs et al. [7,8] builds the mean-field scaling Θ(n 1/3 ) into the definition of the critical probability. It would be desirable to have a useful general definition that recovers this scaling for n-vertex mean-field graphs G = G n , rather than having separate definitions for each different scaling behaviour (or, in mathematical physics jargon, for each 'universality class'). With this aim in mind, Nachmias and Peres [29] suggested to define p c = p c (G) as the value of p which maximizes the logarithmic derivative d dp To motivate this definition, note that by the Margulis-Russo formula [28,34] the derivative d dp E p |C(v)| intuitively counts the expected (weighted) number of edges of G p which can affect the size of |C(v)|, see also Section 2. In other words, p c equals the probability where the addition of a random edge has maximum relative impact on the component size |C(v)|. Denoting the maximum value of (1.2) by M = M (G), Warnke [36] conjectured that for mean-field graphs G the width of the critical window is of order Θ(1/M ). This is motivated by the fact that log( χ G (p 2 )/ χ G (p 1 )) = p2 p1 d dp log χ G (p)dp (p 2 − p 1 )M entails that the susceptibility satisfies χ G (p 2 ) = Θ( χ G (p 1 )) for p 2 − p 1 = O(1/M ).

Main results
In this paper we investigate, as a first test-case, the suggested definition of Nachmias and Peres [29] for the Erdős-Rényi random graph G n,p i.e., the case G = K n (as proposed by Peres [31]). Here our first main result confirms that their definition of the critical probability p c has the desired properties, i.e., that for G n,p = (K n ) p the logarithmic derivative d dp log χ Kn (p) satisfies the following: (i) its maximizer lies inside the critical window p = 1/n + O(n −4/3 ), and (ii) the inverse of its maximum value coincides with the Θ(n −4/3 )-width of the critical window. Having established the qualitative behaviour of the logarithmic derivative for G n,p , it is intriguing to investigate the finer scaling behaviour inside critical window. By symmetry considerations it might be tempting to believe that p = 1/n or p = 1/(n − 1) could be the maximizer of d dp log χ Kn (p), as speculated by Peres [31]. Our second main result refutes this tantalizing belief, instead strengthening the general feeling that λ = 0 is no special point inside the critical window of form p = 1/n + λn −4/3 . Theorem 1.3 (Scaling inside the critical window of G n,p ). Given λ ∈ R, for p = 1/n + λ + o(1) n −4/3 we have, as n → ∞, The definition of the function f appearing in Theorem 1.3 is quite involved, since it intuitively needs to capture the contribution of components with arbitrary numbers of cycles. It is easy to find asymptotics as λ → ±∞; we have f (λ) ∼ |λ| −1 as λ → −∞ and f (λ) ∼ 4λ 2 as λ → +∞; hence log f (λ) = − log |λ| + o(1) as λ → −∞ and log f (λ) = 2 log λ + O(1) as λ → +∞; furthermore, d dλ log f (λ) = O(1/|λ|) for all λ ∈ R, see Appendix A for proofs. Theorem 1.3 also extends to convergence of higher derivatives, see Appendix C.
It would be interesting to know whether the logarithmic derivative d dλ log f (λ) has a unique maximizer λ * , and whether it is unimodal. Figure 1 below (which is obtained by numerical integrations) suggests that this is the case, with λ * ≈ 1 (we conjecture λ * > 1 based on our limited precision numerical data). , where f is as in Theorem 1.3. It provides some evidence for our belief that d dλ log f (λ) has a unique maximizer λ * ≈ 1.
The high-level structure of our proofs is as follows. For Theorem 1.1 our starting point is the Margulis-Russo Formula, which allows us to write d dp χ Kn (p) in terms of sums involving the squared component sizes of G n,p . Using ideas from random graph theory we then estimate these sums, combining correlation inequalities and the 'symmetry rule' (also called 'discrete duality principle') with results for the largest component and the susceptibility of G n,p , which eventually implies (1.3)-(1.4); see Section 2. For Theorem 1.3 with p = 1/n + λ + o(1) n −4/3 , our starting point is the well-known fact that X n,2 = j 1 |C j | 2 /n 4/3 d → W λ,2 for some random variable W λ,2 . Using technical arguments we then justify taking expectations and derivatives, which in view of χ Kn (p)/n 1/3 = E p X n,2 eventually establishes (1.5)-(1.6) with f (λ) = EW λ,2 ; see Section 3.1. For inequality (1.7) we show that f and its derivatives can be computed at λ = 0 by series expansions (exploiting recursive formulas for the area under a normalized Brownian excursion). Since these series converge exponentially, we can then numerically verify (1.7) by finite truncation; see Section 3.2.

Remarks on some other graphs
In the present paper we discuss only the Erdős-Rényi random graph G n,p = (K n ) p , i.e., percolation on the complete n-vertex graph. In particular, Theorem 1.1 shows that the definition of the critical probability p c suggested by Nachmias and Peres [29,31] 'works' in this case. It is an interesting open problem to establish analogous results for other finite graphs.
For example, consider again the hypercube with vertex set {0, 1} m discussed above, see [7,8,16,17]. In the subcritical phase p = (1 − ε)p c with ε 3 n → ∞, [7, Proposition A.1] combined with [8, Theorem 1.3 and Theorem 1.5] show that d dp χ G (p) ∼ m( χ G (p)) 2 and χ G (p) ∼ ε −1 , and thus d dp In the supercritical phase p = (1 + ε)p c with ε 3 n → ∞ and ε = ε(n) → 0, we have χ G (p) ∼ 4ε 2 n according to [16,Theorem 1.1]; hence it is natural to conjecture that the logarithmic derivative satisfies d dp in the supercritical phase too, and, moreover, that the logarithmic derivative has a maximum of order Θ(n 1/3 m) which is attained inside the critical window. Proving this, however, remains a challenging problem. Another important example would be random d-regular graphs with d = d(n) → ∞. Moreover, it would be conceptually very interesting to start with the maximizer of (1.2) and then derive properties of the phase transition of G p (rather than, as in this paper, using known results for G p to verify properties of the maximizer).
It also seems highly desirable to better understand the critical probability p c for finite transitive graphs G which do not exhibit the mean-field behavior of the complete graph K n or the hypercube {0, 1} m . Here the perhaps simplest example is percolation on the n-vertex cycle, n 3, for which it is not difficult to check that there are three different phases: (i) for 1 − p = o(n −1 ) we typically have |C 1 | = n, (ii) for 1 − p = ω(n −1 ) we typically have |C 1 | = o(n), and (iii) for 1 − p = Θ(n −1 ) the rescaled sizes |C 1 |/n, · · · , |C r |/n of the r = Θ(1) largest components are not concentrated. Hence the critical window is parametrized by p = 1 − λ n n −1 with λ n = Θ(1). For p ∈ (0, 1) it is routine to see that A short calculation shows that E p |C(v)| = Θ(n 1/3 ) implies p = 1 − Θ(n −1/3 ). Furthermore, d dp log E p |C(v)| = Θ(n) for 1 − p = Θ(n −1 ), and d dp log For the critical probability p c of n-vertex cycles, it follows that the mean-field definition of Borgs et al. fails (as expected, since cycles are not 'high dimensional'). By contrast, the definition based on the maximizer of the logarithmic derivative of the susceptibility does correctly predict p c = 1 − Θ(n −1 ) and the Θ(n −1 )-width of the critical window, supporting the hope that this definition might work beyond the mean-field case.

Some notation
For emphasis, we will often use P n,p and E n,p for probability and expectation with respect to G n,p . We let C i denote the components of G n,p in order of decreasing sizes, |C 1 | |C 2 | · · · (resolving ties by taking the component with the smallest vertex label first, for definiteness). Finally, convergence in distribution is denoted d →, and unspecified limits are as n → ∞.

Maximizer of the logarithmic derivative
In this section we prove Theorem 1.1. Our arguments combine the Margulis-Russo formula with results and ideas from random graph theory. For mathematical convenience we shall work with the 'rescaled' susceptibility parameters where the component C(v) is with respect to G n,p , as usual, and Recall that χ Kn (p) := E n,p |C(v)|, which is the same for every v ∈ [n] by symmetry, and thus by (2.1)-(2.2) which implies d dp log χ Kn (p) = d dp log S n (p). There is a constant C > 0 such that, for all n 1 and p ∈ (0, 1), d dp log S n (p) C · min |p − 1/n| −1 , n 4/3 .

(2.5)
Furthermore, for every λ ∈ R there is a constant D λ > 0 such that, for all n 2 and p = 1/n+λn −4/3 ∈ (0, 1), The remainder of this section is devoted to the proof of Theorem 2.1, and we start by studying a combinatorial form of d dp S n (p). Writing v ↔ w for the event that v and w are connected (which trivially holds if v = w), note that S(G) = v,w∈V (G) 1 {v↔w} and thus, by taking the expectation, see (2.2), We now record the following simple monotonicity property, which is obvious from (2.7).
Lemma 2.2. If p p and n n , then S n (p) S n (p ).
We say that an edge e ∈ E(K n ) is pivotal for v ↔ w, if v ↔ w in G n,p + e and v ↔ w in G n,p − e (i.e., in the possibly modified graphs where e is added and removed, respectively). Recalling the form of (2.7), for p ∈ (0, 1) the Margulis-Russo Formula [28,34] gives Let P e,v,w denote the event that (i) e ∈ G n,p and (ii) e is pivotal for v ↔ w. Since being pivotal does not depend on the status of e, it follows that d dp An edge not present in G n,p is pivotal for v ↔ w if and only if one of its endpoints is in C(v) and the other and thus, by (2.9), d dp which eventually allows us to bring random graph theory into play.

Upper bounds
In this subsection we prove the upper bound (2.5) from Theorem 2.1.
Lemma 2.5. For all n 1 and p ∈ [0, 1], Proof. We start with (2.16) and fix any vertex v ∈ [n]. Conditioning on the vertex set of C(v) in G n,p , the remaining graph with vertex set [n] \ C(v) has the same distribution as G n−|C(v)|,p (up to relabeling of the vertices). Since S n−|C(v)| (p) S n (p) by Lemma 2.2, using (2.1) it follows that Taking the expectation and summing over all vertices v ∈ [n], we obtain, recalling (2.10) and (2.1)-(2.2),  [13, (6.85)-(6.96)] for a modern exposition). As noted in [1, p. 123], their proofs apply directly to percolation on any finite transitive graph. For any integer k 1 and vertex v ∈ [n], these inequalities state (in our notation) that which together with (2.16) establishes (2.17).
This is a more difficult range. We shall be guided by the so-called 'symmetry rule', which intuitively states the following: after removing the largest component from the supercritical random graph G n,p with np = 1 + ε, the remaining graph resembles a subcritical random graph G n ,p with suitable n and n p = 1 − ε , see [21,Section 5.6]. Let so that (n − α(ε)n) · p 1 − cε by (2.15). Using the subcritical estimate (2.12) of Theorem 2.3, it follows that for Note that |C 1 | α(ε)n is a decreasing event, and that S(G n,p ) = i |C i | 2 and thus S(G n,p ) 2 = i,j |C i | 2 |C j | 2 are increasing functions of the edge indicators. By Harris's inequality (a special case of the FKG-inequality), it follows that Combining (2.25) with (2.23), the tail estimate (2.14) and the inequality (2.17), using the upper bound (2.13) for S n (p), it follows that where we used e −x (x + x 3 ) 2 for the last inequality (and that a, δ, D are constants).
Conditioning on (the vertex set of) the largest component C 1 of G n,p , the remaining graph with vertex set [n] \ C 1 has the same distribution as G n−|C1|,p conditioned on the event D C1 that all components have size at most |C 1 | and that there is no component of size exactly |C 1 | with a smaller vertex label than C 1 . Similarly to (2.18), it follows that (2.27) For any given C 1 , D C1 is a decreasing event for the random graph G n−|C1|,p , while S(G n−|C1|,p ) is an increasing function. Hence, as in (2.25), by Harris's inequality, it follows that and thus, by taking the expectation and using (2.2), where we used ε 3 n Λ 3 1 and S n−α(ε)n (p) S n (p) (see Lemma 2.2) for the final inequality. In view of (2.11), using p 1/2 (for n 2(1 + A), say) our estimates (2.26), (2.30) and (2.31) imply d dp S n (p) = O(ε −1 n) · S n (p), which due to ε −1 n = |p − 1/n| −1 and |p − 1/n| −1 n 4/3 /Λ yields (2.5) in this case too. Case 4: A np n/2. In this range many technicalities from the previous case simplify. By distinguishing the events |C 1 | n/2 and n/2 < |C 1 | n (in which case |C 2 | n − |C 1 | < n/2), using i |C i | = n we infer i =j (2.32) As enpe −np/2 1/2 by the choice of A, standard component counting arguments from random graph theory and Stirling's formula (k! √ 2πk(k/e) k ) yield (2.33) Since np π 0 , by Corollary 2.4 we see that for large n we also have np < n. This is a less interesting range since with very high probability, G n,p is connected and thus i |C i | 2 = |C 1 | 2 = n 2 . To obtain rigorous estimates, let E denote the monotone increasing event that G n,p is 2-edge connected (after deleting any edge the resulting graph remains connected). It is well-known that P n,2(log n)/n (¬E) = o(1) holds (see, e.g., [12]), so a multi-round exposure argument yields P n,p (¬E) P n,2 log n/n (¬E) np/2 log n n −ω (1) . Observe that if E holds, then no edge can be pivotal for the event v ↔ w. Using (2.8) we infer d dp S n (p) n 4 · P n,p (¬E) n −ω(1) , which together with S n (p) 1 and |p − 1/n| 1/n completes the proof of (2.5).

Lower bound
In this subsection we focus on the lower bound (2.6) in Theorem 2.1. Our proof strategy is to consider the event that G n,p contains two distinct components of size Θ(n 2/3 ).
Lemma 2.6. Let L be the event that |C 2 | n 2/3 , i.e., that G n,p contains two distinct components with at least n 2/3 vertices each. For every λ ∈ R there exist constants δ λ , n 0 > 0 such that, for all n n 0 , if p = 1/n + λn −4/3 , then P n,p (L) δ λ . Proof of (2.6) of Theorem 2.1. As for the upper bound, we may assume that n is large enough, since (2.6) trivially holds (if D λ is chosen small enough) for every fixed n 2 because S n (p) and d dp S n (p) are positive functions on (0, 1).

Scaling inside the critical window
In this section we prove Theorem 1.3. Our arguments exploit that inside the critical window, the rescaled sizes of the largest components converge to some random variables (as mentioned in the introduction).
Recall now (2.2), and note that (2.11) can be written as d dp To treat such sums, we first note the following fact, which is stated in [22, Theorem B1 and Remark B2] as an immediate consequence of results of Aldous [3] and Janson and Spencer [23]. 23,22]). Let λ ∈ R and k ∈ N with k 2. Then there exists a random variable W λ,k with

Convergence
In this subsection we prove the convergence results (1.5)-(1.6) of Theorem 1.3 using the distributional convergence (3.8) from Lemma 3.1 and the following auxiliary result.
Theorem 3.2. Let D, λ ∈ R and k, q ∈ N with k 2 and q 1.

10)
where the random variable W λ,k is defined as in Lemma 3.1. Moreover, the limit in (3.10) is a continuous function of λ, and if p = 1/n + λn −4/3 , then the convergence in (3.10) is uniform for λ in any compact interval [λ 1 , λ 2 ] ⊂ R.
Proof. We start with the uniform moment bound (3.9). Since i |C i | k does not decrease if any edge is added, the expectation is a monotone function of p; thus it suffices to consider p = 1/n + Dn −4/3 . As a warm-up, we first consider the special case q = 2. Similarly to (2.10) we have Mimicking the conditioning and monotonicity arguments leading to ( Since j i k 2, (3.12) applies to each factor in each product in (3.11), so (3.9) follows for suitable C = C(D, k, q).
The final claims now follow by the following elementary calculus lemma. Proof. First, suppose that h is discontinuous at some λ. Then there exist ε > 0 and a sequence λ k → λ such that |h(λ k ) − h(λ)| > ε for all k. Since h n (λ k ) → h(λ k ), we may find an increasing sequence n k such that |h n k (λ k ) − h(λ k )| < ε/2. Then |h n k (λ k ) − h(λ)| > ε/2. On the other hand, the assumption implies h n k (λ k ) → h(λ), a contradiction. Similarly, assume that h n (λ) does not converge uniformly to h(λ) on the compact set K. Then there exist ε > 0 and sequences n k → ∞ and λ k ∈ K such that |h n k (λ k ) − h(λ k )| > ε. Since K is compact, we may select a subsequence such that, along this subsequence, λ k → λ for some λ. Then the assumption and the continuity of h just shown imply that, along the subsequence, h n k (λ k ) → h(λ) and h(λ k ) → h(λ), and thus h n k (λ k ) − h(λ k ) → 0, a contradiction. Remark 3.4. Lemma 3.3 is valid for functions on any metric space. (Also the ranges of the functions may be in an arbitrary metric space.) Furthermore, the converse of the lemma also holds (and is easy): if h n (λ) → h(λ) uniformly on compact sets and h is continuous, then h n (λ n ) → h(λ) whenever λ n → λ.

Explicit bounds for λ = 0
In this subsection we complete the proof of Theorem 1.3, and by the arguments of Section 3.1 it remains to prove the following technical lemma.
Remark 3.6. The proof of Lemma 3.5 shows that d 2 dλ 2 log f 2 (0) ≈ 0.296833365232. The idea is to give (in the special case λ = 0) rigorous numerical estimates for the right hand side of (3.18) The derivatives can be computed by (3.5). In the special case λ = 0 we obtain Furthermore, by [23,Remark 6] we also have (in our notation) the identity f 3 (λ) = 2 + 2λf 2 (λ), so that f 3 (0) = 2. Hence (3.18) yields and due to 0 < f 2 (λ) < ∞ our task is reduced to showing that To evaluate the terms in (3.22), note that by Tonelli's theorem, the function f k defined in (3.4) can be written as The plan is to truncate the infinite sum in (3.25) with the help of the following uniform estimates.
Lemma 3.7. For all k 2 and 1 we have I k, , w 0 and The upper bound (3.27) is off from the asymptotic value in [19, (52)] only by a factor 8/3.
Hence, a (5/6) − 0 a 0 and the result follows by summing a geometric series.

Lemma 3.8 thus gives
and the result follows.
The constants w are easily computed by recursion, see, e.g., [19, (4)-(5) or (6)- (7)], so the finite sum 0 0 (2π) −1/2 w I k, can be computed numerically (with arbitrary precision) for any 0 that is not too large. Together with the estimate in Corollary 3.9 of the remainder, which can be made arbitrarily small by choosing a suitable 0 , this enables us to compute f k (0) with arbitrary precision for any k 2.
Proof of Lemma 3.5. Choosing 0 = 75, the right hand side of (3.30) is less than 10 −17 for all 2 k 6, with room to spare. Proceeding as discussed above, we then obtain (using Maple) where . = means equality for all but the last digit (which might be off by one). Hence which shows (3.22) and thus completes the proof of Lemma 3.5. Remark 3.6 follows by inserting (3.34) and (3.31) into (3.21).
A Asymptotics of f (λ) as λ → ±∞ In this appendix we prove the asymptotics of the function f (λ) = f 2 (λ) stated after Theorem 1.3, and extend the results to f k (λ) for arbitrary k 2.
For λ > 0, we use results from [23]. The idea is that as λ → +∞, we approach the supercritical regime, where there is a single giant component C 1 that dominates the sum i |C i | k , and that |C 1 | ≈ 2λ.
It is shown in [23,Lemma 9.5] that as λ → +∞, the intensity Λ (λ) (x) is well approximated by the density function of the normal distribution N (2λ, 2λ −1 ) (except for small x), and (A.2) follows easily by (3.4) and estimates as in the proof of [23, Lemma 9.5]; we omit the details.
For λ → +∞, the estimates for Λ (λ) (x) used in [23] are not precise enough to yield as precise results for derivatives (note that in (3.5) we expect the leading terms of f k+2 (λ)/2 ∼ (2λ) k+2 /2 and λf k+1 ∼ λ(2λ) k+1 to cancel); we conjecture that here too we can take derivatives formally in (A. Part (i) is easy and well-known, and included for completeness. For part (ii), we do not know any reference with a short proof; the bound is proved in [7] as a special case of a more general and involved result. We give here a more direct argument which adapts recent ideas from percolation theory [24,16] to the simpler G n,p case.
We start by recalling some well-known branching processes results (we include proofs for completeness). Let X n,p denote a Galton-Watson branching process with Bin(n, p) offspring distribution, starting with a single individual, and let |X n,p | be its total size. We define X λ and |X λ | analogously, with Bin(n, p) replaced by Po(λ).

C Higher derivatives of the susceptibility
In this appendix we extend the method of proof from Section 3.1 to higher derivatives, using arguments from [22]. The key fact is that, extending (3.15), any mixed moment of X n,k defined in (3.13), k 2, has a derivative that can be expressed as a linear combination of such moments, and thus by induction the same holds for higher derivatives as well. We illustrate the general method by some examples, leaving the details in the general case to the reader. For notational convenience, we write D n,p := n −4/3 (1 − p) d dp .
Note that D n,p = n −4/3 d dt for the parametrization p = 1 − e −t used in [22]. Note also that the factor 1 − p, which is needed in the exact formulas below, disappear asymptotically, since p = o(1), and that apart from this factor, D n,p = d dλ for our usual parametrization p = n −1 + λn −4/3 . First, consider D n,p E n,p (X n,k ) for an arbitrary k 2. As noted in [22, (3.1)], if v ↔ w, then adding the edge vw to the graph increases i |C i | k by (C.1) (And, trivially, the change ∆ vw ( i |C i | k ) = 0 if v ↔ w.) Recalling X n,k = n −2k/3 i |C i | k , see (3.13), it follows by a modification of the argument leading to (2.11) that (similar to [13, Theorem 2.32]), with a factor 1 2 because each edge is counted twice, D n,p E n,p (X n,k ) = 1 2 n −2(k+2)/3