Critical window for the configuration model: finite third moment degrees Souvik Dhara, Remco van der Hofstad, Johan S.H. van Leeuwaarden and Sanchayan Sen
arXiv:1605.02868v1 [math.PR] 10 May 2016
Department of Mathematics and Computer Science, Eindhoven University of Technology May 11, 2016
Abstract We investigate the component sizes of the critical configuration model, as well as the related problem of critical percolation on a supercritical configuration model. We show that, at criticality, the finite third moment assumption on the asymptotic degree distribution is enough to guarantee that the component sizes are O(n2/3 ) and the re-scaled component sizes converge to the excursions of an inhomogeneous Brownian Motion with a parabolic drift. We use percolation to study the evolution of these component sizes while passing through the critical window and show that the vector of percolation cluster-sizes, considered as a process in the critical window, converge to the multiplicative coalescent process in finite dimensions. This behavior was first observed for Erd˝os-Rényi random graphs by Aldous (1997) and our results provide support for the empirical evidences that the nature of the phase transition for a wide array of random-graph models are universal in nature. Further, we show that the re-scaled component sizes and surplus edges converge jointly under a strong topology, at each fixed location of the scaling window.
1
Introduction
Random graphs are the main vehicles to study complex networks that go through a radical change in their connectivity, often called the phase-transition. A large body of literature aims at understanding the properties of random graphs that experience this phase-transition in the sizes of the large connected components for various models. The behavior is well understood for the Erd˝os-Rényi random graphs, thanks to a plethora of results [2, 15]. However, these graphs are often inadequate for modeling real-world networks since the real-world network data often show a power-law behavior of the asymptotic degrees whereas the degree distribution of the Erd˝osRényi random graphs has exponentially decaying tails. See [10, 22] for interesting discussions about internet topology. Therefore, many alternative models have been proposed to capture this power-law tail behavior. An interesting fact, however, is that the behavior, in most of these models, is quite universal in the sense that there is a critical value where the graphs experience a phasetransition and the nature of this phase-transition is insensitive to the microscopic descriptions of the model. Correspondence to: S. Dhara. Email adresses:
[email protected],
[email protected],
[email protected],
[email protected]. 2010 Mathematics Subject Classification. Primary: 60C05, 05C80. Keywords and phrase. Critical configuration model, finite third moment degree, Brownian excursions with parabolic drift, scaling window, multiplicative coalescent, universality.
1
In this work, we focus on the configuration model, the canonical model for generating a random multi-graph with a prescribed degree sequence. This model was introduced by Bollobás [8] to choose a uniform simple d-regular graph on n vertices, when dn is even. The idea was later generalized for general degree sequences d by Molloy and Reed [19] and others. We denote by CMn (d) the multi-graph generated by the configuration model on the vertex set [n] = {1, 2 . . . , n} with the degree sequence d. The configuration model, conditioned on simplicity, yields a uniform simple graph with the same degree sequence. Various features related to the emergence of the giant component phenomenon for this model have been studied recently [11, 12, 14, 16, 19, 21]. We shall give a brief overview of the relevant literature in Section 4.1. Our aim is to obtain precise asymptotics for the component sizes of CMn (d) in the critical window of the phase transition under the optimal assumptions on the degree sequence involving a finite third-moment condition. The scaled limiting vector of component sizes is shown to be distributed as the excursions of certain reflected inhomogeneous Brownian motions with a parabolic drift. This shows that CMn (d), for a large collection of possible d, has similar component sizes as the Erd˝os-Rényi random graphs [2] and the inhomogeneous random graphs [6]. We use percolation on a super-critical configuration model to show the joint convergence of the scaled vectors of component sizes at multiple locations of the percolation scaling window. We also obtain the asymptotic distribution of the number of surplus edges in each component and show that the sequence of vectors consisting of the re-scaled component sizes and surplus converges to a suitable limit under a strong topology as discussed in [5]. These results give very strong evidence in favor of the structural similarity of the component sizes of CMn (d) and Erd˝os-Rényi random graphs at criticality. Our contribution The main innovation of this paper is that we derive the strongest results in the literature under only finite third-moment assumption on the degrees. This finite third- moment assumption is also believed to be necessary for Erd˝os-Rényi type scaling limits, amongst others since the third moment appears in the limiting random variable. In work in progress [9], we consider the infinite third-moment case where the degrees follow a power-law and show that the scaling limit of the cluster sizes is quite different. The joint convergence of the component sizes and the surplus edges has not been proved under such a strong topology for the configuration model, which makes our results the most general in the existing literature. We also study percolation on the configuration model to gain insight about the evolution of the configuration model over the critical scaling window. This is achieved by studying a dynamic process that generates the percolated graphs with different values of the percolation parameter, a problem that is interesting in its own right. Before stating our main results, we need to introduce some notation and concepts.
2
Definitions and notation P
L
→, − → to denote convergence in probability and in distribution We shall use the standard notation − or law, respectively. The topology needed for the distributional convergence will always be specified unless it is clear from the context. A sequence of events (En )n≥1 issaid to occur with high probability (whp) with respect to probability measures (Pn )n≥1 if Pn En → 1. Denote fn = OP (gn ) if P (|fn |/|gn |)n≥1 is tight; fn = oP (gn ) if fn /gn − → 0; fn = ΘP (gn ) if fn = OP (gn ) and gn = OP (fn ). We f write fn = OE (an) (respectively fn = oE (an )) to denote that supn≥1 E a−1 n n < ∞ (respectively −1 limn→∞ E an fn = 0). Denote by ∞ X x2i < ∞ , ℓ2↓ := x = (x1 , x2 , x3 , ...) : x1 ≥ x2 ≥ x3 ≥ ... and i=1
2
(2.1)
the subspace non-increasing sequences of real numbers with square norm metric P of non-negative, 2 )1/2 and let (ℓ2 )k denote the k-fold product space of ℓ2 . With ℓ2 × N∞ , we d(x, y) = ( ∞ (x − y ) i ↓ ↓ ↓ i=1 i denote the product topology of ℓ2↓ and N∞ , where N∞ denotes the collection of the sequences on N, endowed with the product topology. Define also ∞ X 2 ∞ N xi yi < ∞ and yi = 0 whenever xi = 0 ∀i U↓ := ((xi , yi ))∞ ∈ ℓ × : i=1 ↓
(2.2)
i=1
with the metric dU ((x1 , y1 ), (x2 , y2 )) :=
X ∞ i=1
2
(x1i − x2i )
1/2
∞ X x1i y1i − x2i y2i . +
(2.3)
i=1
Further, we introduce U0↓ ⊂ U↓ as U0↓ := ((xi , yi ))∞ i=1 ∈ U↓ : if xk = xm , k ≤ m, then yk ≥ ym .
(2.4) We usually use the boldface notation X for a time-dependent stochastic process X(s) s≥0 , unless stated otherwise, C[0, t] denotes the set of all continuous functions from [0, t] to R equipped with the topology induced by sup-norm || · ||t . Similarly, D[0, t] (resp. D[0, ∞)) denotes the set of all càdlàg functions from [0, t] (resp. [0, ∞)) to R equipped with the Skorohod J1 topology. The inhomogeneous Brownian motion with a parabolic drift is given by √ η ηs2 λ B(s) + λs − 3 (2.5) Bµ,η (s) = µ 2µ where B = B(s) s≥0 is a standard Brownian motion, and µ > 0, η > 0 and λ ∈ R are constants. Define the reflected version of Bλµ,η as λ λ W λ (s) = Bµ,η (s) − min Bµ,η (t). 0≤t≤s
(2.6)
For a function f ∈ C[0, ∞), an interval γ = (l, r) is called an excursion above past minima or simply an excursion of f if f (l) = f (r) = minu≤r f (u) and f (x) > f (r) for all l < x < r. |γ| = r(γ) − l(γ) will denote the length of the excursion γ. Also, define the counting process of marks Nλ = (N λ (s))s≥0 to be a process that has intensity βW λ (s) at time s conditional on (W λ (u))u≤s so that λ
N (s) −
Zs
βW λ (u)du
(2.7)
0
is a martingale (see [2]). For an excursion γ, let N (γ) denote the number of marks in the interval [l(γ), r(γ)]. Remark 1. By [2, Lemma 25] and the Cameron-Martin theorem, almost surely, the excursions of the paths of Bλµ,η can be rearranged in decreasing order of length and the ordered excursion lengths can be considered as a vector in ℓ2↓ . Let γ λ = (|γjλ |)j≥1 be the ordered excursion lengths of Bλµ,η . Then, ((|γjλ |, N (γjλ )))j≥1 can be ordered as an element of U0↓ almost surely by [5, Theorem 4.1] . Denote this element of U0↓ by Z(λ) = (Yjλ , Njλ ) j≥1 obtained from γjλ , N (γjλ ) j≥1 .
Finally, we define a Markov process X := (X(s))−∞ 4. Also, if one has a random degreesequence that satisfies Assumption 1 with high probability, then Theorems 1 and 2 hold conditionally on the degrees. In particular, when the degree sequence consists of an i.i.d. sample from a distribution with E[D 3 ] < ∞ [16], then Assumption 1 is satisfied almost surely. We will later see that degree sequences in the percolation scaling window also satisfy Assumption 1.
3.2 Percolation results Bond percolation on a graph G refers to deleting edges of G independently with equal probability p. Consider bond percolation on CMn (d) with probability pn , yielding CMn (d, pn ). We assume the following: Assumption 2. (i) Assumption 1 (i) and (ii) hold for the degree sequence and the CMn (d) is super-critical, i.e. P E [D(D − 1)] i∈[n] di (di − 1) P νn = > 1. (3.7) →ν= E [D] i∈[n] di (ii) (critical window for percolation) For some λ ∈ R,
1 λ pn = pn (λ) := 1 + 1/3 . νn n
(3.8)
2 (ii), is non-negative for n sufficiently large. Now, Note that pn (λ), as defined in Assumption P √ suppose d˜i ∼ Bin(di , pn ), n+ := i∈[n] (di − d˜i ) and n ˜ = n + n+ . Consider the degree sequence d˜ consisting of d˜i for i ∈ [n] and n+ additional vertices of degree 1, i.e. d˜i = 1 for i ∈ [˜ n] \ [n]. ˜ We shall show later that the degree Dn of a random vertex from this degree sequence satisfies ˜ with E[D ˜ 3 ] < ∞. Now, using Assumption 1 (i), (ii) almost surely for some random variable D λ the notation in Section 2, define γ˜j to be the ordered excursions of the inhomogeneous Brownian motion Bλµ,η , and Nλ with the parameters ˜ µ = E[D],
˜ 2 ], ˜ 3 ]E[D] ˜ − E2 [D η = E[D
˜ β = 1/E[D].
(3.9)
p Let p = 1/ν. Denote the j th largest cluster of CMn (d, pn (λ)) by C(j) (λ). Also, let Zpn (λ) denote the ˜ vector in U0↓ obtained by rearranging critical percolation clusters and their surplus edges and Z(λ) √ λ λ 0 γj |, N (˜ γj )))j≥1 . denote the vector in U↓ obtained by rearranging (( p|˜
5
Theorem 3. Under Assumption 2, L
with respect to the U0↓ topology.
˜ Zpn (λ) − → Z(λ)
(3.10)
Next we consider the percolation cluster for multiple values of λ. There is a very natural way to couple (CMn (d, pn (λ))λ∈R described as follows: for λ < λ′ , perform bond-percolation on CMn (d, pn (λ′ )) with probability pn (λ)/pn (λ′ ). The resulting graph is distributed as CMn (d, pn (λ)). k−1 This can be used to couple (CMn (d, pn (λi ))i=0 for any fixed k ≥ 1. The next theorem shows that the convergence of the component sizes holds jointly in finitely many locations within the critical window, under the above described coupling: p (λ)|)j≥1 , for any fixed k ∈ Theorem 4. Under Assumption 2 and with Cn (λ) = (n−2/3 |C(j) −∞ < λ0 < λ1 < · · · < λk−1 < ∞,
L √ ˜ λ1 , . . . , γ ˜ λk−1 ) → p(˜ Cn (λ0 ), Cn (λ1 ), . . . , Cn (λk−1 ) − γ λ0 , γ
N and (3.11)
with respect to the (ℓ2↓ )k topology where p = 1/ν.
Remark 3. The coupling for the limiting process in Theorem 4 is given by the multiplicative coalescent process described in Section 2. This will become more clear when we describe the ideas of the proof. An intuitive picture is that as we change the value of the percolation parameter from pn (λ) to pn (λ + dλ), exactly one edge is added to the graph and the two endpoints i, j are chosen approximately proportional to the number of half-edges of i and j that were not retained CMn (d, pn (λ)). Define the degree deficiency Di of a component Ci to be the total number of halfedges in a component that were not retained in percolation. Think of Di as the mass of Ci . By the above heuristics, Ci and Cj merge at rate proportional to Di Dj and creates a cluster of mass Di + Dj − 2. Later, we shall show that the degree deficiency of a component is approximately proportional to the component size. Therefore, the component sizes merge approximately like the multiplicative coalescent over the critical scaling window. Remark 4. Janson [12] studied the phase transition of the maximum component size for percolation on a super-critical configuration model. The critical value turns out to be p = 1/ν. This is precisely the reason behind taking pn of the form given by Assumption 2 (ii). The reason behind the width of the critical window being of the order n1/3 does not come as a surprise in light of the existing literature [2, 6, 20, 21]. Remark 5. Theorem 1 and Theorem 2 also hold for configuration models conditioned on simplicity. We do not give a proof here. The arguments in [16, Section 7] can be followed verbatim to obtain a proof of this fact. As a result, Theorem 3 and Theorem 4 also hold, conditioned on simplicity. The rest of the paper is organized as follows: In Section 4.1, we give a brief overview of the relevant literature. This will enable the reader to understand better the relation of this work to the large body of literature already present. Also, it will become clear why the choices of the parameters in Assumption 1 (iii) and Assumption 2 (ii) should correspond to the critical scaling window. We prove Theorems 1 and 2 in Section 5. In Section 6 we find the asymptotic degree distribution in each component. This is used along with Theorem 2 to establish Theorem 3 in Section 7. In Section 8, we analyze the evolution of the component sizes over the percolation critical window and prove Theorem 4.
6
4
Discussion
4.1 Literature overview Erdos-Rényi ˝ type behavior. We first explain what ‘Erd˝os-Rényi type behavior’ means. The study of critical window for random graphs started with the seminal paper [2] on the Erd˝os-Rényi random graphs with p = n−1 (1 + λn−1/3 ). Aldous showed in this regime that all the components are of asymptotic size n2/3 and the ordered component sizes have the same distribution as the excursions of a Brownian motion with a parabolic drift. Aldous also described the component sizes as a dynamic process as λ varies and showed that the dynamic process can be described by a process called the standard multiplicative coalescent. In Theorem 4, we show that similar results hold for the configuration model under a very general set of assumptions. Of course, for general configuration models, there is no obvious way to couple the graphs such that the location parameter in the scaling window varies and percolation seems to be the most natural way to achieve this. By [11, 12], percolation on a configuration model can be viewed as a configuration model with a random degree sequence and this is precisely the reason for studying percolation in this paper. Universality and optimal assumptions. In [6] it was shown that, inside the critical scaling window, the scaled component sizes of an inhomogeneous random graph with −(1 + λn−1/3 )wi wj P (4.1) pij = 1 − exp k∈[n] wk
converge to excursions of an inhomogeneous Brownian motion with a parabolic drift under only finite third-moment assumption on the weight distribution. We establish a counterpart of this for the configuration model in Theorem 1. Later Nachmias and Peres [20] studied the case of percolation scaling window on the random regular graph; for percolation on the configuration model similar results were obtained by Riordan [21] for bounded maximum degrees. Joseph [16] obtained the same scaling limits when the degrees are i.i.d. samples from a distribution having finite third moment. Theorem 2 and Theorem 3 prove stronger versions of all these existing results for the configuration model under less stringent and possibly optimal assumptions. Further, in Theorem 4, we give a dynamic picture for percolation cluster sizes in the critical window and show that this dynamics can be approximated by the multiplicative coalescent. Comparison to branching processes. In [14, 19] the phase transition for the component sizes configuration model was identified in terms of the parameter ν = E[D(D − 1)]/E[D]. Janson and Luczak [14], showed that the configuration model can be locally approximated by a branching process X which has ν as its expected progeny and thus, when ν > 1, CMn (d) has a component Cmax of approximate size ρn, where ρ is the survival probability of X . Further, the progeny distribution of X has finite variance when E[D 3 ] < ∞. Now, for a branching process with mean ≈ 1 + ε and finite variance σ 2 , the survival probability is approximately 2σ −2 ε for small ε > 0. This seems to suggest that the maximum component size under Assumption 1 should be of the order n2/3 since ε = Θ(n−1/3 ). Theorem 1 formalizes this intuition and shows that in fact all the maximal component sizes are of the order n2/3 .
4.2 Proof ideas The proof of Theorem 1 uses standard functional central limit theorem argument. Indeed we associate a suitable semi-martingale with the graph obtained from an exploration process used to explore the connected components of CMn (d). The martingale part is then shown to converge to an inhomogeneous Brownian motion, and the drift part is shown to converge to a parabola. The fact that the component sizes can be expressed in terms of the hitting times of the semi-martingale implies the finite-dimensional convergence of the component sizes. The convergence with respect 7
to ℓ2↓ is then concluded using size-biased point process arguments by Aldous [2]. Theorem 2 requires a careful estimate of the tail probability of the distribution of surplus edges when the component size is small and we obtain this using martingale estimates. Theorem 3 is proved by showing that the percolated degree sequence satisfies Assumption 1 almost surely. Finally, we prove Theorem 4 using a coupling argument. Key challenges here are that, for each fixed n, the components do not merge according to their component size, and that the components do not merge exactly like a multiplicative coalescent over the scaling window. We shall deal with these in Section 8.
4.3 Open problems (a) Theorem 4 proves the joint convergence at finitely many locations of the scaling window. However, the tightness of (Cn (λ))λ∈R in D((−∞, ∞), ℓ2↓ ) should also hold so that we have convergence of the whole process. (b) It is also believed that the connected components, considered as metric spaces with the graph distance, converge to a suitable limiting metric structure under the finite third-moment condition only. This result was proved under exponential moment conditions in [4, Theorem 4.7]. (c) A reason for studying percolation in this paper is to understand the minimal spanning tree of the giant component. For a super-critical configuration model with i.i.d. edge weights, it should be the case that the minimal spanning tree can be described by the critically percolated graph at a very high location of the scaling window. Such results were obtained in [1] for the Erd˝os-Rényi random graph.
5
Proofs of Theorems 1 and 2
5.1 The exploration process Let us explore the graph sequentially using a natural approach outlined in [21]. At step k, divide the set of half-edges into three groups; sleeping half-edges Sk , active half-edges Ak , and dead half-edges Dk . The depth-first exploration process can be summarized in the following algorithm: Algorithm 1 (DFS exploration). At k = 0, Sk contains all the half-edges and Ak , Dk are empty. While (Sk 6= ∅ or Ak 6= ∅) we do the following at stage k + 1: S1 If Ak 6= ∅, then take the smallest half-edge a from Ak . S2 Take the half-edge b from Sk that is paired to a. Suppose b is attached to a vertex w (which is necessarily not discovered yet). Declare w to be discovered, let r = dw −1 and bw1 , bw2 , . . . bwr be the half-edges of w other than b. Declare bw1 , bw2 ,..., bwr , b to be smaller than all other halfedges in Ak . Also order the half-edges of w among themselves as bw1 > bw2 > · · · > bwr > b. Now identify Bk ⊂ Ak ∪ {bw1 , bw2 , . . . , bwr } as the collection of all half-edges in Ak paired to one of the bwi ’s and the corresponding bwi ’s. Similarly identify Ck ⊂ {bw1 , bw2 , . . . , bwr } which is the collection of loops incident to w. Finally, declare Ak+1 = Ak ∪{bw1 , bw2 , . . . , bwr }\ Bk ∪ Ck , Dk+1 = Dk ∪ {a, b} ∪ Bk ∪ Ck and Sk+1 = Sk \ {b} ∪ {bw1 , bw2 , ..., bwr } . Go to stage k + 2. S3 If Ak = ∅ for some k, then take out one half-edge a from Sk uniformly at random and identify the vertex v incident to it. Declare v to be discovered. Let r = dv − 1 and assume that av1 , av2 ,..., avr are the half-edges of v other than a and identify the collection of halfedges involved in a loop/multiple edge/cycle Ck as in Step 2. Order the half-edges of v as av1 > av2 > · · · > avr > a. Set Ak+1 = {a, av1 , av2 ,..., avr } \ Ck , Dk+1 = Dk ∪ Ck , and Sk+1 = Sk \ {a, av1 , av2 , ..., avr }. Go to stage k + 2. 8
In words, we explore a new vertex at each stage and throw away all the half-edges involved in a loop/multiple edge/cycle with the vertex set already discovered before proceeding to the next stage. The ordering of the half-edges is such that the connected components of CMn (d) are explored in the depth-first way. We call the half-edges of Bk ∪ Ck cycle half-edges because they create loops, cycles or multiple edges in the graph. Let Ak := |Ak |,
c(k+1) := (|Bk | + |Ck |)/2,
Uk := |Sk |.
(5.1)
Let d(j) be the degree of the j th explored vertex and define the following process: Sn (0) = 0,
i X (d(j) − 2 − 2c(j) ). Sn (i) =
(5.2)
j=1
The process Sn = (Sn (i))i∈[n] “encodes the component sizes as lengths of path segments above past minima” as discussed in [2]. Suppose Ci is the ith connected component explored by the above exploration process. Define τk = inf i : Sn (i) = −2k . (5.3) Then Ck is discovered between the times τk−1 + 1 and τk and Ck has size τk − τk−1 .
5.2 Size-biased exploration The vertices are explored in a size-biased manner with sizes proportional to their degrees, i.e., if we denote by v(i) the ith explored vertex in Algorithm 1 and by d(i) the degree of v(i) then
P v(i) = j|v(1) , v(2) , ..., v(i−1) = P
dj k ∈V / i−1
dk
=P
dj k∈[n] dk
−
Pi−1
k=1 d(k)
, ∀j ∈ Vi−1 ,
(5.4)
where Vi denotes the first i vertices to be discovered in the above exploration process. The following lemma will be used in the proof of Theorem 1: Lemma 5. Suppose that Assumption 1 holds and denote σr = E[D r ] and µ = as n → ∞, ⌊n2/3 u⌋ σ2 u P −2/3 X d(i) − sup n → 0, − µ u≤t
E[D]. Then for all t > 0, (5.5)
i=1
and
⌊n2/3 u⌋ σ3 u P −2/3 X 2 → 0. sup n d(i) − − µ u≤t
(5.6)
i=1
The proof of this lemma follows from a more general result stated in Proposition 29 and the following observation: Lemma 6. Assumption 1 implies 1 X k→∞ n→∞ n lim lim
1{dj >k} drj = 0, r = 1, 2, 3.
j∈[n]
For r = 3, in particular, this implies d3max = o(n).
9
(5.7)
5.3 Estimate of cycle half-edges The following lemma gives an estimate of the number of cycle half-edges created upto time t. This result is proved in [21] for bounded degrees. In our case, it follows from Lemma 5 as we show below: Lemma 7. For Algorithm 1, if Ak = Ak , Bk := Bk , and Ck := Ck , then
E Bk |Fk = (1 + oP (1))
and
2Ak + OP (n−2/3 ) Uk
E Ck |Fk = OP (n−1 )
(5.8)
(5.9)
uniformly for k ≤ tn2/3 and any t > 0, where Fk is the sigma-field generated by the information revealed up to stage k. Further, all the OP and oP terms in (5.8) and (5.9) can be replaced by OE and oE . Proof. Suppose Uk := Sk . First note that by (5.5) k Uk 1X 1 X dj − d(j) = E[D] + oP (1) = n n n
(5.10)
j=1
j∈[n]
uniformly over k ≤ tn2/3 . Now let a be the half-edge that is being explored at stage k + 1. Now each of the (Ak − 1) half-edges of Ak \ {a} is equally likely to be paired with a half-edge of v(k+1) , thus creating two elements of Bk . Also, given Fk and v(k+1) , the probability that a half-edge of Ak \ {a} is paired to one of the half-edges of v(k+1) is (d(k+1) − 1)/(Uk − 1). Therefore,
E Bk |Fk , v(k+1) = 2(Ak − 1)
Hence,
E Bk |Fk = 2E
Now, using (5.5) and (5.6),
E d(k+1) − 1|Fk =
P
Ak d(k+1) − 1 d(k+1) − 1 = 2 d(k+1) − 1 −2 . Uk − 1 Uk − 1 Uk − 1
Ak E d(k+1) − 1|Fk −2 . d(k+1) − 1|Fk Uk − 1 Uk − 1
j ∈V / k
P
dj (dj − 1) j ∈V / k
dj
P
2 j∈[n] dj
=P
j∈[n] dj
− 1 + oP (1) = 1 + oP (1).
(5.11)
(5.12)
(5.13)
uniformly over k ≤ tn2/3 , where the last step follows from Assumption 1. Further, since p0 < 1, fact that all the OP , oP can be Uk ≥ c0 n uniformly over k ≤ o(n). P(5.8). rThe P P Thus, (5.12) gives r d ≤ replaced by OE , oE follows from j∈[n] drj − kd2max ≤ j ∈V j∈[n] dj for r = 1, 2, together / k j with dmax = o(n1/3 ). To prove (5.9), note that
E Ck |Fk , v(k+1) = 2(d(k+1) − 2)
d(k+1) − 1 . Uk − 1
(5.14)
By Assumption 1 and (5.5) P
j ∈V / k
d3j
j ∈V / k
dj
E[d(k+1) |Fk ] = P 2
uniformly for k ≤ tn2/3 . Therefore,
≤P
P
3 j∈[n] dj
j∈[n] dj
+ oP (n2/3 )
= OP (1),
E Ck |Fk = OP (n−1 )
uniformly over k ≤ tn2/3 . Again, OP term can be replaced by OE , as argued before. 10
(5.15)
(5.16)
5.4 Key ingredients ¯ n := X ¯ n (u) := n−1/3 Xn ⌊n2/3 u⌋ and X ¯ n (u) For any D[0, ∞)-valued process Xn define X . u≥0 λ The following result is the main ingredient for proving Theorem 1. Recall the definition of Bµ,η from (2.5) with parameters given in (3.4). Theorem 8 (Convergence of the exploration process). Under Assumption 1, as n → ∞, L ¯n − S → Bλµ,η
(5.17)
with respect to the Skorohod J1 topology. As in [16], we shall prove this by approximating Sn by a simpler process defined as sn (0) = 0,
sn (i) =
i X (d(j) − 2).
(5.18)
j=1
Note that the difference between the processes Sn and sn is due to the cycles, loops, and multipleedges encountered during the exploration. Following the approach of [16], it will be enough to prove the following: Proposition 9. Under Assumption 1, as n → ∞, L
¯sn − → Bλµ,η
(5.19)
with respect to the Skorohod J1 topology. Remark 6. It will be shown that Proposition 9 implies Theorem 8 by showing that the distributions ¯ n and ¯sn are very close as n → ∞. This is achieved by proving that we shall not see too many of S cycle half-edges upto the time ⌊n2/3 u⌋ for any fixed u > 0. ¯ n and ¯sn by From here onwards we shall look at the continuous versions of the processes S linearly interpolating between the values at the jump points and write it using the same notation. It is easy to see that these continuous versions differ from their càdlàg versions by at most n−1/3 dmax = o(1) uniformly on [0, T ], for any T > 0. Therefore, the convergence in law of the continuous versions implies the convergence in law of the càdlàg versions and vice versa. Before proceeding to show that Theorem 8 is a consequences of Proposition 9, we shall need to bound the difference of these two processes in a suitable way. We need the following lemma. Recall the definition of c(k+1) := (Bk + Ck )/2 from (5.1). Lemma 10. Fix t > 0 and M > 0 (large). Define En (t, M ) := maxs≤t {¯ sn (s) − minu≤s s¯n (u)} < M . Then X (5.20) lim sup E c(k) 1En (t,M ) < ∞. n→∞
k≤tn2/3
Proof. Lemma 10 is similar to [16, Lemma 6.1]. We add a brief proof here. Note that, for all large n, Ak ≤ M n1/3 on En (t, M ), because Ak = Sn (k) − min Sn (j) + 2 = sn (k) − 2 j≤k
k X j=1
c(j) − min Sn (j) + 2 ≤ sn (k) − min sn (j) + O(1), (5.21) j≤k
j≤k
where the last step follows by noting that minj≤k sn (j) ≤ minj≤k Sn (j) + 2
E c(k) 1En (t,M ) ≤
Pk
j=1 c(j) .
M n1/3 M −2/3 + o(n−2/3 ) = n + o(n−2/3 ) µn µ
By Lemma 7 (5.22)
uniformly for k ≤ tn2/3 . Summing over 1 ≤ k ≤ tn2/3 and taking the lim sup completes the proof. 11
Proof of Theorem 8. The argument is a standard and we include the proof for the sake of completeness (see [16, Section 6.2]). For t > 0 and M > 0 define the event En (t, M ) as in Lemma 10. Also, denote by CL [0, t] the set of all bounded Lipschitz functions from [0, t] to R. Since CL [0, t] separates the points in [0, t], and [0, t] is compact, CL [0, t] is dense in C[0, t]. Therefore, to prove Theorem 8, it suffices to prove as n → ∞ (5.23) E f (S¯ n ) − E f (¯sn ) → 0, for all f ∈ CL [0, t]. Choose and fix b > 0 such that z, z1 , z2 ∈ C[0, t] =⇒ |f (z)| ≤ b
f (z1 ) − f (z2 ) ≤ b||z1 − z2 ||∞ .
and
(5.24)
Now, E f (S ¯ n ) − E f (¯sn ) ¯ n )1E (t,M ) − E f (¯sn )1E (t,M ) + E f (S ¯ n )1E (t,M )c − E f (¯sn )1E (t,M )c ≤ E f (S n n n n ≤ bE ¯ Sn − ¯sn 1E (t,M ) + 2bP max s¯n (s) − min s¯n (u) ≥ M t
≤ 2bn−1/3
X
k≤tn2/3
n
u≤s
s≤t
E c(k) 1En (t,M ) + 2bP
max s¯n (s) − min s¯n (u) ≥ M .
u≤s
s≤t
(5.25) The first term in the above sum tends to zero as n → ∞, by Lemma 10. The fact that the reflection of a process is a continuous map from D([0, ∞), R) to D([0, ∞), R) (see [24, Theorem 13.5.1]), Proposition 9 implies L → Wλ , (5.26) s¯n (s) − min s¯n (u) s≥0 − u≤s
where Wλ is as defined in (2.6). By the Portmanteau lemma lim sup P max s¯n (s) − min s¯n (u) ≥ M ≤ P max W λ (s) ≥ M , n→∞
u≤s
s≤t
s≤t
(5.27)
for any t > 0. The proof follows by taking the limit M → ∞. From here onward the main focus of this section will be to prove Proposition 9. We use the martingale functional central limit theorem in a similar manner as Aldous. Proof of Proposition 9. Let {Fi }i∈[n] be the natural filtration defined in Lemma 7. Recall the definition of sn (i) from (5.18). By the Doob-Meyer decomposition [17, Theorem 4.10] we can write s2n (i) = Hn (i) + Bn (i),
sn (i) = Mn (i) + An (i), where Mn (i) =
i X j=1
An (i) =
d(j) − E d(j) |Fj−1 ,
i X j=1
Bn (i) =
i X j=1
(5.29a)
E d(j) − 2|Fj−1 ,
(5.29b)
E d2(j) |Fj−1 − E2 d(j) |Fj−1 .
12
(5.28)
(5.29c)
¯ n (t) = n−1/3 Xn (⌊tn2/3 ⌋). Our Recall that for a discrete time process (Xn (i))i≥1 , we write X result follows from the martingale functional central limit theorem [25, Theorem 2.1] if we can prove the following four conditions: For any u > 0, ηs2 P sup A¯n (s) − st + 3 − → 0, 2µ s≤u
(5.30a)
P η u, 2
¯n (u) − → n−1/3 B
E sup M n(s) − M n (s−) 2 → 0,
(5.30c)
¯n (s) − B ¯n (s−)| → 0. n−1/3 E sup |B
(5.30d)
and
(5.30b)
µ
s≤u
s≤u
Indeed (5.30a) gives rise to the quadratic drift term of the limiting distribution. Conditions (5.30b), (5.30c), (5.30d) are the same as [25, Theorem 2.1, Condition (ii)]. The facts that the jumps of both the martingale and the quadratic-variation process go to zero and that the quadratic variation process is converging to the quadratic variation of an inhomogeneous Brownian Motion, together imply the convergence of the martingale term. The validation of these conditions are given separately in the subsequent part of this section. Lemma 11. The conditions (5.30b), (5.30c), (5.30d) hold. P P Proof. Denote by σr (n) = n1 i∈[n] dri , r = 2, 3 and µ(n) = n1 i∈[n] di . To prove (5.30b), it is enough to prove that P σ3 µ − σ22 u. n−2/3 Bn (⌊un2/3 ⌋) − (5.31) → µ2 P P 3 2/3 , Recall that E[d2(i) |Fi−1 ] = j ∈V / i−1 dj / j ∈V / i−1 dj . Furthermore, uniformly over i ≤ un X
dj =
j ∈V / i−1
X
dj + OP (dmax i) = ℓn + oP (n).
(5.32)
j∈[n]
Assume that, without loss of generality, j 7→ dj is non-increasing. We have, uniformly over i ≤ un2/3 , 2/3 un X X 3 d3j . (5.33) dj − nσ3 (n) ≤ j=1
j ∈V / i−1
For each fixed k, 2/3
2/3
un un 1 X 3 1 X dj ≤ n n j=1
1{dj ≤k} d3j +
j=1
1 X n
j∈[n]
1{dj >k} d3j ≤ k3 un−1/3 +
1 X n
1{dj >k} d3j = o(1),
(5.34)
j∈[n]
where we first let n → ∞ and then k → ∞ and use Lemma 6. Therefore, the right-hand side of (5.33) is o(n) and we conclude that, uniformly over i ≤ un2/3 ,
E d2(i) |Fi−1 =
σ3 + oP (1). µ
(5.35)
E d(i) |Fi−1 =
σ2 + oP (1), µ
(5.36)
A similar argument gives
13
and (5.30b) follows by noting that the error term is oP (1), since we are summing n2/3 terms, scaling by n−2/3 and using the uniformity of errors over i ≤ un2/3 . The proofs of (5.30c) and (5.30d) are rather short and we present them now. For (5.30c), we bound i h h i E sup |M n (s) − M n (s−)|2 = n−2/3 E sup |Mn (k) − Mn (k − 1)|2 s≤u
k≤un2/3
h
= n−2/3 E ≤n
−2/3
E
h
2 i sup d(k) − E[d(k) |Fk−1 ]
k≤un2/3
2
i
sup d(k) + n
k≤un2/3
−2/3
E
h
sup
k≤un2/3
(5.37)
E
2
i d(k) |Fk−1
≤ 2n−2/3 d2max . Similarly, (5.30d) gives ¯n (s) − B ¯n (s−)|2 = n−2/3 E sup |Bn (k) − Bn (k − 1)| n−1/3 E sup |B s≤u
k≤un2/3
= n−2/3 E
sup var d(k) |Fk−1
k≤un2/3
(5.38)
≤ 2n−2/3 d2max , and Conditions (5.30c) and (5.30d) follow from Lemma 6 using dmax = o(n1/3 ). Next, we prove Condition (5.30a) which requires some more work. Note that P / i−1 dj (dj − 2) P E d(i) − 2|Fi−1 = j ∈V j ∈V / i−1 dj P P P P j∈[n] dj (dj − 2) j∈Vi−1 dj (dj − 2) j ∈V / i−1 dj (dj − 2) j∈Vi−1 dj P P P P = − + j∈[n] dj j∈[n] dj j ∈V / i−1 dj j∈[n] dj P P P d2j d2j j∈Vi−1 dj λ j∈V j ∈V / P + P i−1 + o(n−1/3 ), = 1/3 − P i−1 d d d n j∈[n] j j ∈V / i−1 j j∈[n] j (5.39) where the last step follows from Assumption 1 (iii). Therefore, An (k) =
k X i=1
E d(i) − 2|Fi−1
P P P k k 2 2 X X dj kλ j∈V j ∈V / i−1 dj j∈Vi−1 dj P P P i−1 + + o(kn−1/3 ). = 1/3 − d d d n j∈[n] j j ∈V / i−1 j j∈[n] j i=1 i=1
(5.40)
The following lemma estimates the sums on the right-hand side of (5.40): Lemma 12. For all u > 0, as n → ∞, 2/3 i−1 2 X ⌋X −1/3 ⌊sn d(j) σ3 s2 P − sup n →0 − ℓn 2µ2 s≤u
i=1
and
2/3 i−1 X ⌋X −1/3 ⌊sn d(j) σ2 s2 P sup n − − → 0. ℓn 2µ2 s≤u
i=1
(5.41)
j=1
j=1
14
(5.42)
Consequently, P 2/3 P 2 X ⌋ j ∈V d −1/3 ⌊sn σ22 s2 P j j∈Vi−1 dj / i−1 P P sup n − → 0. − 2µ3 s≤u j ∈V / i−1 dj j∈[n] dj
(5.43)
i=1
Proof. Notice that
⌊sn2/3 ⌋ i−1 2 k i−1 2 σ3 k2 −1/3 X X d(j) σ3 s2 −1/3 X X d(j) sup n − − = sup n 2 n4/3 ℓn 2µ2 ℓn 2µ 2/3 s≤u k≤un i=1 j=1 i=1 j=1
≤
1 ℓn
k X i−1 X σ3 (i − 1) d2(j) − sup n−1/3 µ k≤un2/3 i=1 j=1 kσ k2 σ k2 σ3 3 3 + sup + sup − 1/3 1/3 2µ2 n4/3 k≤un2/3 2µℓn n k≤un2/3 2µℓn n
(5.44)
i X σ3 i 1 2 4/3 σ3 n−1/3 1 1 −1/3 2/3 d2(j) − un sup ≤ n + o(1) + − u n ℓn µ 2µ ℓn nµ i≤un2/3 j=1
⌊sn2/3 ⌋ σ3 s u −2/3 X 2 d(j) − ≤ sup n + o(1). µ + o(1) s≤u µ j=1
and the (5.41) follows from (5.6) in Lemma 5. The proof of (5.42) is similar and it follows from 1 P (5.5). We now show (5.43). Recall that σ2 (n) = n i∈[n] d2i and observe 1 X 2 1 X 2 dj = σ2 (n) − dj = σ2 (n) + oP (1) n n
(5.45)
j∈Vi−1
j ∈V / i−1
2/3 where we use Lemma 5 to conclude the uniformity. Similarly, (5.32) uniformly over P i ≤ un 2/3 . Therefore, implies that j ∈V / i−1 dj = ℓn + oP (n) uniformly over i ≤ un
n
−1/3
k X i=1
P
j ∈V / i−1
P
d2j
j ∈V / i−1
P
dj
j∈Vi−1
P
dj
j∈[n] dj
k
nσ2 (n) + oP (n) −1/3 X n = ℓn + oP (n) i=1
P
j∈Vi−1
ℓn
dj
(5.46)
and Assumption 1, combined with (5.42), complete the proof. Lemma 13. Condition (5.30a) holds. Proof. The proof follows by using Lemma 12 in (5.40).
5.5 Finite dimensional convergence of the ordered component sizes Note that the convergence of the exploration process in Theporem 8 implies that, for any large T > 0, the k-largest components explored upto time T n2/3 converge to the k-largest excursions above past minima of Bλµ,η upto time T . Therefore, we can conclude the finite dimensional convergence of the ordered components sizes in the whole graph if we can show that the large components are explored early by the exploration process. The following lemma formalizes the above statement: ≥T Lemma 14. Let Cmax denote the largest component which is started exploring after time T n2/3 . Then, for any δ > 0, ≥T | > δn2/3 = 0. lim lim sup P |Cmax
T →∞ n→∞
15
(5.47)
Let us first state the two main ingredients to complete the proof of Lemma 14: Lemma 15 ([13, Lemma 5.2]). Consider CMn (d) with νn < 1 and let C (Vn ) denote the component containing the vertex Vn where Vn is a random vertex chosen independently of the graph. Then,
E [|C (Vn )|] ≤ 1 + Lemma 16. Define, νn,i = for any T > 0,
P
j ∈V / i−1
dj (dj − 1)/
P
j ∈V / i−1
E [Dn ] 1 − νn
.
(5.48)
dj . There exists some constant C0 > 0 such that
νn,T n2/3 = νn − C0 T n−1/3 + oP (n−1/3 ).
Proof. Using a similar split up as in (5.39), we have P P P j ∈V / i−1 dj (dj − 1) j∈Vi−1 dj j∈Vi−1 dj (dj − 1) P − . νn,i = νn + ℓn ℓn j ∈V / i−1 dj
Now, (5.5) and (5.6) give that, uniformly over i ≤ T n2/3 , P P 2/3 ) j ∈V / i−1 dj (dj − 1) j∈[n] dj (dj − 1) + oP (n P P = = 1 + oP (n−1/3 ), 2/3 ) d d + o (n P j ∈V / i−1 j j∈[n] j X σ3 − 2 (i − 1) + oP (n2/3 ). dj (dj − 2) = µ
(5.49)
(5.50)
(5.51a) (5.51b)
j∈Vi−1
Further, note that σ3 − 2µ = E[D(D − 1)(D − 2)] + E[D(D − 2)] > 0, by Assumption 1(iii) and(iv). Therefore, (5.50) gives (5.49) for some constant C0 > 0. Proof of Lemma 14. Let iT := inf{i ≥ T n2/3 : Sn (i) = inf j≤i Sn (j)}. Thus, iT denotes the first time we finish exploring a component after time T n2/3 . Note that, conditional on the P explored ¯ vertices upto time i , the remaining graph G is still a configuration model. Let ν ¯ = T n i∈G¯ di (di − P ¯ 1)/ i∈G¯ di be the criticality parameter of G. Then, using (5.49), we can conclude that ν¯n ≤ νn − C0 T n−1/3 + oP (n−1/3 ).
(5.52)
Take T > 0 such that λ − C0 T < 0. Thus, with high probability, ν¯n < 1. Denote the component corresponding to a randomly chosen vertex from G¯ by C ≥T (Vn ), and the ith largest component of ≥T ¯ ¯ denote the probability measure conditioned on Fi , and ν¯n < 1 and let E G¯ by C(i) . Also, let P T denote the corresponding expectation. Now, for any δ > 0, X 1 X¯ 1 ¯ 1 ≥T 2 ¯ , P E |C(i)≥T |2 = 2 1/3 E |C(i) | > δ2 n4/3 ≤ 2 4/3 |C ≥T (Vn )| ≤ 2 δ (−λ + C0 T + oP (1)) δ n δ n i≥1 i≥1 (5.53) where the second step follows from the Markov inequality and the last step follows by combining Lemma 15 and (5.52). Noting that ν¯n < 1 with high probability, we get C ≥T lim sup P |Cmax | > δn2/3 ≤ 2 , δ T n→∞ for some constant C > 0 and the proof follows. Theorem 17. The convergence in Theorem 1 holds with respect to the product topology. Proof. The proof follows from Theorem 8 and Lemma 14. 16
(5.54)
5.6 Proof of Theorem 1 The proof of Theorem 1 follows using similar argument as [2, Section 3.3]. However, the proof is a bit tricky since the components are explored in a size-biased manner with sizes being the total degree in the component component sizes as in [2]). For a sequence of random variables P(not the 2 Y = (Yi )i≥1 satisfying i≥1 Yi < ∞ almost surely, define ξ := (ξi )i≥1 such that ξi |Y ∼PExp(Yi ) and the coordinates of ξ are independent conditional on Y. For a ≥ 0, let S (a) := ξi ≤a Yi . Then the size biased point process is defined to be the random collection of points Ξ := {(S (ξi ), Yi )} (see [2, Section 3.3]). We shall use Lemma 8, Lemma 14 and Proposition 15 from [2]. Let C := {C : C is a component of CMn (d)}. Consider the collection ξ := (ξ(C ))C ∈C such that conditional P P on the values ( k∈C dk , |C |)C ∈C , ξ(C ) has an exponential distribution with rate n−2/3 k∈C dk independently over C . Then the order in which Algorithm 1 explores the components can be th obtained by ordering the components according to Ptheir ξ-value. Recall that Ci denotes the i explored component by Algorithm 1 and let Di := k∈Ci dk . Define the size biased point process i X Di , n−2/3 Di Ξn := n−2/3
i≥1
j=1
.
(5.55)
Also define the point processes i X ′ Cj , n−2/3 Ci Ξn := n−2/3
i≥1
j=1
,
Ξ∞ :=
l(γ), |γ| : γ an excursion of Bλµ,η ,
(5.56)
where we recall that l(γ) are the left endpoints of the excursions of Bλµ,η and |γ| is the length of the excursion γ (see (2.6)). Note that Ξ′n is not a size biased point process. However, applying [2, ′
L
Lemma 8] and Theorem 8, we get Ξn − → Ξ∞ . We claim that L
Ξn − → 2Ξ∞ .
(5.57)
To verify the claim, note that (5.5) and Assumption 1 (iii) together imply 2/3
2/3
X⌋ X⌋ −2/3 ⌊un P −2/3 ⌊un σ2 → 0, d(i) − u = sup n sup n d(i) − 2u − µ u≤t u≤t i=1
for any t > 0 since σ2 /µ = E D 2 /E [D] = 2. Therefore, X X D(C ) − 2 |C | = oP (n2/3 ) ξ(C )≤s
(5.58)
i=1
(5.59)
ξ(C )≤s
Thus, (5.57) follows using (5.58) and (5.59). Now, the point process 2Ξ∞ satisfies all the conditions of [2, Proposition 15] as shown by Aldous. Thus, [2, Lemma 14] gives D(i) i≥1 is tight in ℓ2↓ . (5.60)
P This implies that n−2/3 C(i) i≥1 is tight in ℓ2↓ by simply observing that |Ci | ≤ k∈Ci dk + 1. Therefore, the proof of Theorem 1 is complete using Theorem 17.
5.7 Proof of Theorem 2 The proof of Theorem 2 is completed in two separate lemmas. In Lemma 18 we first show that the convergence in Theorem 2 holds with respect to the ℓ2↓ × N∞ topology. The tightness of (Zn )n≥1 with respect to the U0↓ topology is ensured in Lemma 19 and Theorem 2 follows. 17
¯nλ (u) = Nnλ (⌊un2/3 ⌋). Lemma 18. Let Nnλ (k) be the number of surplus edges discovered upto time k and N Then, as n → ∞, L ¯ λn − → Nλ , (5.61) N where Nλ is defined in (2.7).
Proof. Recall of a, b, Ak , Bk , Ck , S k from 5.1. Recall also that Ak := Ak , the definitions Section Bk := Bk , Ck := Ck , Uk := Sk , c(k+1) := ( Bk + Ck )/2 as in Section 5.1. We have Ak = Sn (k) − minj≤k Sn (j) + 2. From Lemma 7 we can conclude that
E c(k+1) |Fk =
Ak + OP (n−1 ). µn
(5.62)
The counting process Nλn has conditional intensity (conditioned on Fk−1 ) given by (5.62). Writing ¯ n , we get that the conditional intensity of the rethe conditional intensity in (5.62) in terms of S λ ¯ scaled process Nn is given by 1 ¯ [Sn (u) − min S¯n (˜ u)] + oP (1). u ˜≤u µ
(5.63)
¯ n . By Theorem 1, Denote by W n (u) := S¯n (u) − minu˜≤u S¯n (˜ u) which is the reflected version S L
Wn − → Wλ ,
(5.64)
where Wλ is as defined in (2.6). Therefore, we can assume that there exists a probability space such that Wn → Wλ almost surely. Using [18, Theorem 1; Chapter 5.3], the continuity of the sample paths of Wλ , we conclude that L ¯λ − N → Nλ , (5.65) n
where
Nλ
is defined in (2.7).
Lemma 19. The vector (Zn )n≥1 is tight with respect to the U0↓ topology. The proof of Lemma 19 makes use of the following crucial estimate of the probability that a component with small size has very large number of surplus edges. Lemma 20. Assume that λ < 0. Let Vn denote a vertex chosen uniformly at random, independent of the graph CMn (d) and let C (Vn ) denote the component containing Vn . Let δk = δk −0.12 . Then, for δ > 0 (small), √ C δ 2/3 2/3 P SP(C (Vn )) ≥ K, |C (Vn )| ∈ (δK n , 2δK n ) ≤ 1/3 1.1 . (5.66) n K where C is a fixed constant independent of n, δ, K. Proof of Lemma 19. To simplify the notation, we write Yin = n−2/3 |C(i) | and Nin =# {surplus edges in C(i) }. Let Yi , Ni denote the distributional limits of Yin and Nin respectively. Recall from Remark 1 that Z(λ) is almost surely U0↓ -valued. Using the definition of dU from (2.3) and Lemma 18, the proof of Lemma 19 is complete if we can show that, for any η > 0 X Yin Nin > η = 0. (5.67) lim lim sup P ε→0 n→∞
Yin ≤ε
18
First, consider the case λ < 0. For every η, ε > 0 sufficiently small X
P
Yin Nin
Yin ≤ε
>η
X X ∞ ∞ n−2/3 1 n n n E ≤ E Yi Ni 1{Yin ≤ε} = |C(i) |Ni 1{|C(i) |≤εn2/3} η η i=1
i=1
i n1/3 h E SP(C (Vn ))1{|C (Vn )|≤εn2/3 } η ∞ 1/3 X (5.68) n X −(i+1) −0.12 2/3 −i −0.12 2/3 P SP(C (Vn )) ≥ k, |C (Vn )| ∈ (2 k n ,2 k n ] = η 0.12 =
k=1 i≥log2 (1/(k
C ≤ η
∞ X k=1
1 k1.1
ε))
X
−(1/2)i
2
i≥log2 (1/(k 0.12 ε))
∞ √ √ CX ε ε). = O( ≤ η k1.04 k=1
where the last but one step follows from Lemma 20. Therefore, (5.67) holds when λ < 0. Now consider the case λ > 0. For T > 0 (large), let Kn := {i : Yin ≤ ε, C(i) is explored before T n2/3 }.
(5.69)
Then, by applying the Cauchy-Schwarz inequality, X
i∈Kn
1/2
(Yin )2
i∈Kn
1/2
×
X
(Yin )2
1/2
× (# surplus edges explored before T n2/3 )
Yin Nin ≤
X
≤
i∈Kn
X
(Nin )2
i∈Kn
(5.70)
Using similar ideas as the proof of Lemma 14, we can run the exploration process till T n2/3 and the unexplored graph becomes a configuration model with negative criticality parameter for large T > 0, by (5.49). Thus, the proof can be completed using (5.70), the ℓ2↓ convergence of the component sizes given by Theorem 1, and Lemma 18. Proof of Lemma 20. To complete the proof of Lemma 20, we shall use martingale techniques coupled with Lemma 15. Fix δ > 0 (small). First we describe another way of exploring C (Vn ) which turns out to be convenient to work with. Algorithm 2 (Exploring C (Vn )). Consider the following exploration of C (Vn ): (S0) Initialize all half-edges to be alive. Choose a vertex from [n] uniformly at random and declare all its half-edges active. (S1) In the next step, take any active half-edge and pair it uniformly with another alive half-edge. Kill these paired half-edges. Declare all the half-edges corresponding to the new vertex (if any) active. Keep repeating (S1) until the set of active half-edges is empty. Unlike Algorithm 1, we need not see a new vertex at each stage and we explore only two half-edges at each stage. Recall that we denote by Dn the degree of a vertex chosen uniformly at random independently of the graph. Define the exploration process s′n by, s′n (0) = Dn , s′n (l) =
X
i∈[n]
di Iin (l) − 2l,
(5.71)
where Iin (l) = 1{i∈Vl } and Vl is the vertex set discovered upto time l. Therefore, s′n (l) counts the number of active half-edges at time l, until C (Vn ) is explored. Note that C (Vn ) is explored when s′n hits zero. We shall use C to denote a positive constant that can be different in different equations. 19
In this proof, Fl shall be used to denote the sigma-field containing information revealed upto stage l by Algorithm 2. For H > 0, let γ := inf{l ≥ 1 : s′n (l) ≥ H or s′n (l) = 0} ∧ 2δn2/3 .
(5.72)
Note that X E s′n(l + 1) − s′n (l)| (Iin (l))ni=1 = / Vl , i ∈ Vl+1 | (Iin (l))ni=1 ) − 2 di P (i ∈ i∈[n]
= =
P
i∈V / l
d2i
ℓn − 2l − 1 λ
n1/3
+ o(n
−2≤
−1/3
P
2 i∈[n] di
ℓn − 2l − 1
−2 P
2l + 1 )+ × ℓn − 2l − 1
(5.73) 2 i∈[n] di
ℓn
≤0
uniformly over l ≤ 2δn2/3 for all small δ > 0 and large n, where the last step follows from the 2δn2/3 is a super-martingale. The optional stopping theorem now fact that λ < 0. Therefore, {s′n (l)}l=1 implies E [Dn ] ≥ E s′n (γ) ≥ H P s′n(γ) ≥ H . (5.74) Thus,
P s′n (γ) ≥ H ≤
E [Dn ] H
.
(5.75)
√ We shall put H = n1/3 K 1.1 / δ. To simplify the writing, we write s′n [0, t] ∈ A to denote that s′n (l) ∈ A, for all l ∈ [0, t]. Notice that P SP(C (Vn )) ≥ K, |C (Vn )| ∈ (δK n2/3 , 2δK n2/3 ) (5.76) ≤ P s′n (γ) ≥ H + P SP(C (Vn )) ≥ K, s′n [0, 2δK n2/3 ] < H, s′n [0, δK n2/3 ] > 0 . Now, P SP(C (Vn )) ≥ K, s′n [0, 2δK n2/3 ] < H, s′n [0, δK n2/3 ] > 0 X P surpluses occur at times l1 , . . . , lK , s′n [0, 2δK n2/3 ] < H, s′n [0, δK n2/3 ] > 0 ≤ 1≤l1 0, the random variable X = f (Z1 , Z2 , . . . , ZN ) satisfies 2
t P X − E[X] > t ≤ 2 exp − PN
2
2 i=1 ci
.
(7.8)
Now let Iij denote the indicator of the j th half-edge corresponding to vertex i to be kept after the √ explosion. Then Iij ∼ Ber( pn ) independently for j ∈ [di ], i ∈ [n]. Let I := (Iij )j∈[di ],i∈[n] and f1 (I) :=
X
i∈[n]
d˜i (d˜i − 1).
(7.9)
P Note that f1 (I) = i∈[˜n] d˜i (d˜i − 1) since the degree one vertices do not contribute to the sum. One can check that, by changing the status of one half-edge corresponding to vertex k, we can change f1 (·) by at most 2(dk + 1). Therefore (7.8) yields X X t2 ˜ ˜ . (7.10) di (di − 1) > t ≤ 2 exp − P di (di − 1) − pn Pp 8 i∈[n] di (di + 1)2 i∈[n]
i∈[n]
23
By setting t = n1/2+ε for some suitably small ε > 0, using the finite third moment conditions and the Borel-Cantelli lemma we conclude that Pp almost surely, X X di (di − 1) + O(n1/2+ε ), (7.11) d˜i (d˜i − 1) = pn i∈[n]
i∈[n]
in particular, X
i∈[˜ n]
d˜i (d˜i − 1) =
X
i∈[n]
d˜i (d˜i − 1) = pn
X
i∈[n]
di (di − 1) + o(n2/3 ).
(7.12)
P ˜ ˜ ˜ Similarly, take f2 (I) = i∈[n] di (di − 1)(di − 2) and note that changing the status of one bond changes f2 (·) by at most [2(dk + 1)]2 . Thus, (7.8) gives X t2 3/2 P di (di − 1)(di − 2) > t ≤ 2 exp − Pp f2 (I) − pn 32 i∈[n] di (di + 1)4 i∈[n] t2 P ≤ exp − , 32dmax (dmax + 1) i∈[n] (di + 1)3 (7.13) which implies that, Pp almost surely, X X X di (di − 1)(di − 2) + o(n), d˜i (d˜i − 1)(d˜i − 2) = p3/2 d˜i (d˜i − 1)(d˜i − 2) = n i∈[˜ n]
(7.14)
i∈[n]
i∈[n]
P since d2max i∈[n] (di + 1)3 = o(n5/3 ). Now, to prove Lemma 23 (1), note that the case r = 1 follows P P by simply observing that i∈˜n d˜i = i∈[n] di . The cases r = 2, 3 follow from (7.12) and (7.14). Finally, to see Lemma 23 (2), note that P P ˜ ˜ pn i∈[n] di di − 1 + o n2/3 i∈[˜ n] di (di − 1) P = ν˜n = P ˜ i∈[n] di i∈[˜ n] di (7.15) P pn i∈[n] di (di − 1) λ −1/3 −1/3 P + o(n ) = 1 + 1/3 + o(n = ), n i∈[n] di by (7.12) and this completes the proof of Lemma 23.
To conclude Theorem 3 we also need to estimate the number of deleted vertices from each ˜ by deleting component. Recall from Remark 8 that CMn (d, pn (λ)) can be obtained from CMn˜ (d) relevant number of degree one vertices uniformly at random. Let v1d (C˜(j) ) be the number of degree ˜ Since the vertices one vertices of C˜(j) that are deleted while creating CMn (d, pn (λ)) from CMn˜ (d). are to be chosen uniformly from all degree one vertices, the number of vertices to be deleted from C˜(j) is asymptotically the total number of degree one vertices in C˜(j) times the proportion of degree one vertices to be deleted. Therefore, n ˜ n+ n+ C˜(j) + oP (n2/3 ) P∞ 1 v1 (C˜(j) ) + oP (n2/3 ) = v1d (C˜(j) ) = n ˜1 n ˜ 1 k=0 k˜ nk √ E[D] 1 − pn ˜ n+ ˜ C(j) + oP (n2/3 ) = C(j) + oP (n2/3 ) = ℓn E[D] √ = 1 − pn C˜(j) + oP (n2/3 ),
(7.16)
where the third equality follows from (6.4). The proof of Theorem 3 is now complete by using the ℓ2↓ convergence in Lemma 22, (7.16) and Remark 10. 24
8
Convergence at multiple locations
We shall prove Theorem 4 in this section. For each percolation cluster let the degree deficiency of the cluster be the number of half-edges of the component that were deleted by percolation. Firstly, we show that the degree deficiency of each component is approximately proportional to the component sizes. We couple the graphs CMn (d, pn (λ)) for λ ≥ λ0 such that the degree deficiencies evolve like an approximate multiplicative coalescent. Then, we describe an exact multiplicative coalescent that is close to the original process. Finally, we conclude the proof of Theorem 4 using known properties of the multiplicative coalescent.
8.1 Estimate of the degree deficiency ˜ is the graph obtained before deleting the red vertices while constructing Recall that CMn˜ (d) ˜ CMn (d, pn (λ)) as described in Algorithm 3. C˜(j) denotes the j th largest component of CMn˜ (d) p p and let C˜(j) denote the residual part of C˜(j) after applying Algorithm 3 (S3). Denote by (dk )k∈[n] p the of CMn (d, pn (λ)) and define the degree-deficiency Di of component C˜(i) to be P degree sequence p k∈C˜p (dk − dk ). Whenever a half-edge is detached from a vertex in Algorithm 3 (S1), we say that (i)
a hole is created on that vertex. Note that X Di = (dk − dpk ) = Hi + Ri ,
(8.1)
p k∈C˜(i)
p where Ri denotes the number of degree one vertices deleted from C˜(i) to obtain C˜(i) and Hi dep notes the number of holes in C˜(i) . We shall now show that the deficiency of a component is approximately proportional to its size. For that we need to estimate Hi and Ri , as in the following lemma:
Lemma 24. There exist constants κ1 , κ2 > 0 such that p + oP (n2/3 ), Hi = κ1 C˜(i)
and
Therefore, for some constant κ > 0,
p + oP (n2/3 ). Ri = κ2 C˜(i) p + oP (n2/3 ). Di = κ C˜(i)
(8.2a) (8.2b) (8.3)
Proof. We use the following notation:
nk,l := #{v ∈ [n] : dv = k, d˜v = l}, nk,l (C ) := #{v ∈ [n] : dv = k, d˜v = l, v ∈ C }, n ˜ l (C ) := #{v ∈ [n] : d˜v = l, v ∈ C }. First, note that nk,l =
Pn k
i=1
√
∼ Bin(k, pn ). Therefore, 1{Xi =l} where Xi i.i.d nk,l P k √ l √ − → pk ( p) (1 − p)r−l = pk,l , l n
(8.4)
where p = 1/ν. Fix any l ≥ 1. Now, conditioned on n ˜ l (C˜(i) ) and (nk,l )k≥1 , the vector (nk,l (C˜(i) ))k≥1 has a multivariate hypergeometric distribution with sample of size n ˜ l (C(i) ), population of size n ˜l = P ˜(i) ) corresponds to an occurrence n that is partitioned into parts of sizes (n ) and n ( C k,l k,l k≥1 k,l k from the (k, l)th element of the partition which has size nk,l . Denote by P2 (·), E2 [·], Var2 (·) the 25
conditional probability, expectation and variance respectively conditioned on (˜ nl (C(i) ))l≥1 and (nk,l )k,l≥1 . Thus, n E2 [nk,l (C˜(i) )] = n˜ l (C˜(i) ) k,l , (8.5a) n ˜l and Var2 (nk,l (C˜(i) )) =
nk,l nk,l n ˜l − n ˜ l (C˜(i) ) ˜ l (C˜(i) ) , 1− n ˜ l (C˜(i) ) = OP n n ˜l n ˜l n ˜l − 1
(8.5b)
where the error term can be chosen to be independent over k, l since nk,l /˜ nl ≤ 1. By Chebyshev’s inequality, q ˜ ˜ (8.6) n ˜ l C˜(i) , nk,l (C(i) ) = n ˜ l C(i) pk,l + OP where the error term is over k, l. Therefore, Hi =
XX l≥1 k≥l
(k − l)nk,l (C˜(i) ) =
dX max l=1
n ˜ l C˜(i)
X (k − l)pk,l + oP (n2/3 ),
(8.7)
k≥l
where the last equality follows by using the Cauchy-Schwartz inequality, dmax = o(n1/3 ) and the tightness of n−2/3 |C˜(i) |. Now, using (6.4), we get l˜ nl n ˜ l C˜(i) = Pdmax k=0
k˜ nk
C˜(i) + OP (n1/3 l−1 ).
(8.8)
P P max −1 Pdmax 1/3 ), where we have used the fact that l Also dl=1 j=0 (k − l)pk,l = O(dmax ) = o(n k≥l (k − l)pk,l = O(l). Thus, (8.9) Hi = κ′1 C˜(i) + oP (n2/3 ),
for some κ′ > 0. Now, the proof of (8.2a) follows from (7.16). To see (8.2b), let n ˜ 1 be the number ˜ n1(i) the number of degree one vertices in C˜(i) , n+ the number of of degree one vertices in CMn˜ (d), ˜ By Remark 8 the red vertices of a component can be generated uniformly red vertices in CMn˜ (d). at random. Thus the central limit theorem yields Ri = n1(i)
n+ + oP (n2/3 ). n ˜1
Using (6.4) to estimate the number of degree one vertices, we conclude Ri = κ′2 C˜(i) + oP (n2/3 ),
(8.10)
(8.11)
for some κ′2 > 0 and the proof follows by appyling (7.16) again.
From here onwards, we shall augment λ with some previously defined notation to emphasize the location parameter in the critical window. Theorem 25. Denote by Dn (λ) := (D(j) (λ))j≥1 the ordered version of (Dj (λ))j≥1 . Then, for every λ ∈ R, √ L γjλ |)j≥1 n−2/3 Dn (λ) − → κ p(|˜
(8.12)
with respect to the ℓ2↓ topology, where (|˜ γjλ |)j≥1 are the lengths of the Brownian excursions defined in (3.10) and p = 1/ν. Proof. Observe that Lemma 24 and Theorem 3 imply the finite-dimensional convergence of Dn (λ). The proof of ℓ2↓ tightness is similar to Section 5.6.
26
8.2 Coupling In this section, we describe the following dynamic process which generates (CMn (d, pn (λ)))λ∈R : (C1) Suppose that, between every two free half-edges e 6= f , there is an Exp(1/(2ℓn − 3)) clock ξ(e, f ) and that all these clocks across pairs are independent. Whenever a clock rings, the corresponding half-edges are paired and all the other exponential clocks corresponding to these two half-edges are discarded. Let Gn (λ) be the graph obtained by upto time − log(1 − pn (λ)). The following lemma ensures that this dynamic process preserves the coupling described before Theorem 4. Lemma 26. The graph Gn (λ) is distributed as CMn (d, pn (λ)) for any λ. Further, for any λ < λ′ , Gn (λ) is distributed as the graph obtained by doing percolation on Gn (λ′ ) with probability pn (λ)/pn (λ′ ). Proof. We shall use the following property of the exponential random variables: Let E1 , E2 , . . . , Er be an i.i.d. sequence of exponential random variables with rate λ. Then, conditional on the event {E1 = min(E1 , . . . , Er )}, E1 is distributed as an exponential random variable with rate rλ. Note that, by construction, Gn (∞) is distributed as CMn (d). Moreover, for any pair of half-edges e, f
P ((e, f ) is an edge in Gn (λ)|(e, f ) is an edge in Gn (∞)) = P ξ(e, f ) < − log(1 − pn (λ))|ξ(e, f ) = ′ min ′ ξ(e′ , f ′ ) e =e, or f =f
(8.13)
= pn (λ),
where we have used the fact stated at the beginning of the proof and the fact that |{(e′ , f ′ ) : e′ = e, or f ′ = f, e′ 6= f ′ }| = 2ℓn − 3. Further, the events {(e, f ) ∈ Gn (λ)} and {(e′ , f ′ ) ∈ Gn (λ)} (e 6= e′ and f 6= f ′ ) are independent conditioned on Gn (∞), because they depend on disjoint sets of ξvalues. To see the second assertion, note that for a random variable X and an event A, satisfying X|A ∼ Exp(1), P (X ≤ x|A, X ≤ y) = (1 − e−x )/(1 − e−y ) for any x ≤ y. Therefore, P (e, f ) is an edge in Gn (λ)|(e, f ) is an edge in Gn (λ′ ) = P ξ(e, f ) < − log(1 − pn (λ))|ξ(e, f ) = ′ min ′ ξ(e′ , f ′ ), ξ(e, f ) < − log(1 − pn (λ′ )) e =e, or f =f (8.14) pn (λ) . = pn (λ′ ) The independence across the edges follow similarly as before and the proof is now complete.
8.3 Proof of Theorem 4 We shall consider the case k = 2 only, since the case for general k can be proved inductively. Fix −∞ < λ0 < λ1 < ∞. Define, ¯ n (λ) := a−1 Dn (λ), D (8.15) n ¯ n (λ) denotes the ordered vector where an = ((νn − 1)(2ℓn − 3)n1/3 )1/2 . Thus n−2/3 an → a > 0. D of the number of free half-edges in each component of Gn (λ0 ) (suitably re-scaled), where by free half-edges we mean the half-edges that were deleted in percolation. Also, the above construction gives a coupling of the graphs (Gn (λ))λ∈R . Moreover, at time λ, the ith and the j th coordinate of ¯ n (λ) = (D¯ λ )i≥1 merge at rate D (i) λ λ D(i) D(j) ×
1 1 1 1 λ λ λ ¯λ ≈ D(i) D(j) × × × = D¯(i) D(j) , 1/3 1/3 2ℓn − 3 (1 − pn (λ))νn n νn − 1 (2ℓn − 3)n 27
(8.16)
λ λ ¯λ ¯λ ¯ and the merged component has size D¯ (i) + D¯ (j) − 2a−1 n ≈ D (i) + D (j) . Thus, (Dn (λ))λ≥λ0 is not an exact multiplicative coalescent but it is close. Firstly notice that
− log(1 − pn (λ)) = − log(1 − 1/νn ) − log 1 −
λ −1/3 + o(n ) . n1/3 (νn − 1)
(8.17)
Using the fact that x − x2 /2 ≤ log(1 − x) ≤ −x for small x > 0, we obtain, for all sufficiently large n, λ + o(n−1/3 ), (8.18) − log(1 − pn (λ)) = − log(1 − 1/νn ) + 1/3 n (νn − 1)
where the error term may depend on λ. But we restrict our attention to λ ∈ [λ0 , λ1 ]. Thus, we can choose εn (only depending upon λ0 , λ1 ) with n1/3 εn → 0 such that − log(1 − pn (λ)) ≤ − log(1 − 1/νn ) +
λ n1/3 (ν
n
− 1)
+ εn .
(8.19)
Define the graph Gˆn (λ) to be the graph obtained by keeping the edge (e, f ) if ξ(e, f ) ≤ − log(1 − ˆ ′n (λ) for Gˆn (λ) is defined as the same quantity as D ¯ n (λ) for Gn (λ). 1/νn )+λn−1/3 (νn −1)−1 +εn . If D We can use (8.18) to conclude that Gn (λ0 − ηn ) ⊂ Gˆn (λ0 ) ⊂ Gn (λ0 + ηn ) for some ηn = o(n−1/3 ) and therefore, ˆ ′ (λ0 ), and D ¯ n (λ0 ) have the same distributional limit. D (8.20) n ˆ ′ (λ) merge exactly according to the product of their size, as λ varies. Note that the co-ordinates of D n ′ ˆ (λ))λ ≤λ≤λ is not a multiplicative coalescent yet due to the depletion of the sizes However, (D n 0 1 ˆ n (λ0 ) = D ˆ ′ (λ0 ) and with each merge. We define an exact multiplicative coalescent below. Let D n consider the following modification of the coupling (C1) defined in Section 8.2 for all the edges that appear after λ0 . After pairing a half-edge e with its neighbors, we do not throw away the exponential clocks between e and other half-edges in the other components. ˆ n (λ))λ ≤λ≤λ the exact multiplicative coalescent version of the above process Let us denote by (D 0 1 ¯ n (λ))λ ≤λ≤λ , (D ˆ ′ (λ))λ ≤λ≤λ and ˆ ′ (λ0 ). The processes (D starting with the initial distribution D n n 0 1 0 1 ˆ (Dn (λ))λ0 ≤λ≤λ1 are all coupled by using the same set of ξ-values. Then, using Theorem 25, (8.20) and the Feller property of the multiplicative coalescent [2, Proposition 5], we conclude √ L ˆ n (λ0 ), D ˆ n (λ1 )) − ˜ λ1 ) γ λ0 , γ (D → κ pa−1 (˜
(8.21)
λ) ˆ n (λ) = (Dˆ(i) ˜ λ = (|˜ with respect to the (ℓ2↓ )2 topology, where γ γjλ |)j≥1 . Write D i≥1 . Now observe that, under the above coupling, for each R ≥ 1, we have X X λ 2 λ 2 (D¯(i) ) ≤ (Dˆ(i) ) (8.22) i≤R
i≤R
for all λ ≥ λ0 , where we have used (8.19). To complete the proof, we need the following two facts, stated separately as lemmas: L
Lemma 27. Suppose Xn , Yn are non-negative random variables such that Xn ≤ Yn a.s. and Xn − → X, L
Yn − → X. Then,
P
→ 0. Yn − Xn −
28
Proof. Note that ((Xn , Yn ))n≥1 is tight in L
R2 . Thus, for any (n′i )i≥1 there exists a subsequence
(ni )i≥1 ⊂ (n′i )i≥1 such that (Xni , Yni ) − → (Z1 , Z2 ). Using the marginal distributional limits we L
L
get Z1 = X, Z2 = X. Also the joint distribution of (Z1 , Z2 ) is concentrated on the line y = x L
in the xy plane. Thus, (Xni , Yni ) − → (X, X). This limiting distribution does not depend on the L
subsequence (ni )i≥1 . Thus the tightness of ((Xn , Yn ))n≥1 implies (Xn , Yn ) − → (X, X). The proof is now complete.
Lemma 28. Suppose Xn := (Xni )i≥1 , and Yn := (Yni )i≥1 are tight random variables. Suppose Yni −
P
L
L
Xni − → 0 for each i. Then, Yn − → X implies Xn − → X.
Now, we have all the ingredients to complete the proof of Theorem 4 and we stitch the proof as follows: Note that (8.21), (8.22), and Lemma 27 implies that the joint vector consisting of only ¯ n (λ0 ), and D ¯ n (λ1 ) has the right limit. Recall the definition of D ¯ n (λ) the first R co-ordinates of D involving Dn (λ) from (8.15). Applying Lemma 28 and Theorem 25, we get √ L ˜ λ1 ), n−2/3 (Dn (λ0 ), Dn (λ1 )) − → κ p(˜ γ λ0 , γ
(8.23)
for any λ0 < λ1 . Now we can use Lemma 24 together with the ℓ2↓ tightness of Cn (λ) from Theorem 2 to conclude that L √ ˜ λ1 ). γ λ0 , γ (8.24) (Cn (λ0 ), Cn (λ1 )) − → p(˜ Thus, we have proved Theorem 4 for k = 2. As remarked at the beginning of this Section, an inductive argument gives the proof for any k > 2.
A
Appendix
In this section we state and prove a general version of Lemma 5 dealing with the size-biased degrees. P Proposition 29. Suppose fn : [n] → R is a function with n−1 i∈[n] fn (i)di → σf (say), where (di )i∈[n] satisfies Assumption 1. Suppose (f˜n (i))i∈[n] is the size-biased ordering of (fn (i))i∈[n] with size (di /ℓn )i∈[n] . P⌊unα ⌋ ˜ Consider Hn (u) := n−α fn (i) for some α ∈ (0, 2 ] and u > 0. Then, i=1
3
σf sup Hn (u) − u = OP (an ∨ bn ∨ n−1/3 ), µ u≤tn
(A.1)
1/2 2 α−1 , as long as t = o(nβ/2 ) for some β satisfywhere an= (tn n−α max n i∈[n] fn (i)) and bn = dmax tn n ing max α/2, 2α − 1 ≤ β < α. P Proof. Denote ℓn = i∈[n] di . Recall the definition of Hn (u). Consider n independent exponential random variables Ti ∼ exp(di /ℓn ). Define X ˜ n (u) = n−α (A.2) fn (i)1{Ti ≤nα u} . H i∈[n]
PN (nα u) α α ˜ n (u) = Therefore, H fn (i) = Hn (N i=1 (n u)) where N (u) := #{j : Tj′ ≤ un }. Consider −β α α Y0 (s) = n (N (sn ) − sn ) for some max α/2, 2α − 1 ≤ β < α. Define, Vs := j : Tj ≤ snα . We have E Y0 (u)|Fs = Y0 (s) + E Y0 (u) − Y0 (s)|Fs 1 = Y0 (s) + β E #{j : Tj ∈ (⌊snα ⌋, ⌊unα ]⌋}|Fs − (u − s)nα n (A.3) 1 X α −1 α = Y0 (s) + β 1 − exp(−dj (t − s)n ℓn − (u − s)n ≤ Y0 (s), n ′ j ∈V / s
29
where the last step follows because 1 − e−x ≤ x. Therefore, {Y0 (s)}s≥0 is a supermartingale. Also noting that E Y0 (0) = 0 and e−x ≤ 1 − x + x2 /2, h X i α −1 E Y0 (u) = −E Y0 (u) = 1 unα − 1 − exp(−un d ℓ ) i n nβ i∈[n]
2 2α i 1 hX −β u n α −1 ≤ n unα di ℓ−1 − 1 − exp(−un d ℓ ) = β i n n n 2 i∈[n] P n u2 2α−1−β n1 i=1 d2i = n 2 , 1 Pn 2 d i i=1 n
and
Var(Y0 (u)) = n−2β var(N (unα )) = n−2β
X
i∈[n]
≤ n−2β
P
2 i∈[n] di ℓ2n
(A.4)
P(Tj ≤ unα )(1 − P(Tj ≤ unα )) (A.5)
X dj unα = unα−2β . ℓn
i∈[n]
Using the maximal inequality in [2, Lemma 12], for any ε > 0 and T > 0, p εP sup |Y0 (s)| > 3ε ≤ 3 E(Y0 (T )) + Var(Y0 (T )) .
(A.6)
s≤T
Therefore,
β sup n−α N (unα ) − u = OP (t−2 n n ).
(A.7)
u≤tn
This, in particular, implies N (2tn nα ) ≥ tn nα whp when tn = o(nβ/2 ). Therefore, (unα ) −α NX σf −α σf u α fn (i) − ≤ sup n n N (n u) sup Hn (u) − µ µ u≤2tn u≤tn i=1 −α −α σf u σf α ˜ . ≤ sup n Hn (u) − + sup n N (un ) − u µ µ u≤2tn u≤2tn
Define Y1 (u) = n−α
PN (unα ) i=1
(n)
(n)
fn (i)−σf u, where σf
E Y1 (t)|Fs = Y1 (s) + E Y1 (t) − Y1 (s)|Fs
= Y1 (s) +
=
P
i∈[n] fn (i)di /ℓn
(A.8)
= σf /µ+o(1) and hence
1 X (n) fn (j) 1 − exp(−dj (t − s)n2/3 ℓ−1 n ) − (t − s)σf ≤ Y1 (s). α n ′
(A.9)
j ∈V / s
Thus, (Y1 (u))u≥0 is also a super-martingale and by noting that E Y1 (0) = 0 we have, X E Y1 (t) = −E Y1 (t) = σ (n) t − n−α fn (i) 1 − exp(−tnα di ℓ−1 n ) f i∈[n]
=n
−α
X
fn (i) exp(−tn
α
di ℓ−1 n )
i∈[n]
t2 dmax = nα−2/3 1/3 2 n
1 n
P
− 1 + tn
i∈[n] fn (i)di , ℓn 2 n
30
α
di ℓ−1 n
≤n
−α t
2
2
n
2α
P
2 i∈[n] fn (i)di ℓ2n
(A.10)
where the last step follows by conditions on fn , di and the fact that α ≤ 2/3. Also, X −tnαdi −tnαdi fn2 (i) exp 1 − exp Var Y1 (t) = n−2α ℓn ℓn i∈[n] P i∈[n] fn (i)di ≤ n−2α max(fn (i))tnα ℓn i∈[n] P t i∈[n] fn (i)di = n−α max(fn (i)) . ℓn i∈[n]
(A.11)
Recalling the assumptions on fn from the statement of Proposition 29, another application of (A.6) yields ˜ n (u) − σf u = OP (an ∨ bn ) (A.12) sup n−α H µ u≤t
where an , bn are as stated in Proposition 29. Thus, (A.8) together with (A.12) and (A.7) completes the proof.
Acknowledgement This research has been supported by the Netherlands Organisation for Scientific Research (NWO) through Gravitation Networks grant 024.002.003. In addition, RvdH has been supported by VICI grant 639.033.806 and JvL has been supported by the European Research Council (ERC).
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