Percolation of averages in the stochastic mean field model: the near-supercritical regime

0.5 MB·English

by  Jian Ding

Percolation of averages in the stochastic mean field model:
the near-supercritical regime
Jian Ding∗ Subhajit Goswami∗
University of Chicago University of Chicago
5
1 November 23, 2015
0
2

v
o
N Abstract

9 For a complete graph of size n, assign each edge an i.i.d. exponential variable with mean n.
1 For λ > 0, consider the length of the longest path whose average weight is at most λ. It was
shown by Aldous (1998) that the length is of order logn for λ<1/e and of order n for λ>1/e.
]
R In this paper, we study the near- supercritical regime where λ = e−1 +η with η > 0 a small
fixed number. We show that there exist two absolute constants C ,C >0 such that with high
P √ √ 1 2
. probabilitythelengthisinbetweenne−C1/ η andne−C2/ η. Ourresultcorrectsanon-rigorous
h
prediction of Aldous (2005).
t
a
m
Key words and phrases. Combinatorialoptimization,stochasticdistancemodel,percolation.
[

3
1 Introduction
v
9
7 Let W be a complete undirected graph of n vertices where each edge is assigned an independent
n
5
exponential weight with mean n; this is referred to as the stochastic mean- field (SMF ) model.
3 n
0 For a (self-avoiding) path π = (v ,v ,...,v ), define its length len(π) and average weight A(π) by
0 1 m
.
1
0 len(π) = m, and A(π) = 1(cid:80)m W ,
m i=1 (vi−1,vi)
5
1
where W is the weight of the edge (u,v). For λ > 0, let L(n,λ) be the length of the longest
: (u,v)
v path with average weight below λ, i.e.,
i
X
r L(n,λ) = max{len(π) : A(π) ≤ λ, π is a path in SMFn model}.
a
In a non-rigorous paper of Aldous [2], it was predicted that L(n,λ) (cid:16) n(λ−e−1)β with β = 3
λ ↓ e−1. Our main result is the following theorem, which corrects Aldous’ prediction.
Theorem 1.1. Let λ = 1/e+η where η > 0. Then there exist absolute constants C ,C ,η∗ > 0
1 2
such that for all η ≤ η∗,
lim P(cid:0)ne−C1/√η ≤ L(n,λ) ≤ ne−C2/√η(cid:1) = 1. (1.1)
n→∞
∗Partially supported by NSF grant DMS-1313596.
1
The study of the object L(n,λ) was initiated by Aldous [1] where a phase transition was
discoveredatthethresholde−1. ItwasshownthatwithhighprobabilityL(n,λ)isoforderlognfor
λ < e−1 and L(n,λ) is of order n when λ > e−1. The critical behavior was established in [4], where
it was proved that with high probability L(n,λ) is of order (logn)3 when λ is around e−1 within
a window of order (logn)−2. Our Theorem 1.1 describes the behavior in the near-supercritical
regime, and in particular states that L(n,λ)/n is a stretched exponential in η with η = λ−e−1 ↓ 0.
Another interesting result proved in [4] states that L(n,λ) ≥ n1/4 in a somewhat similar regime
namely λ ≥ 1/e + β(logn)−2, where β > 0 is an absolute constant. Notice that substituting
η = C(logn)−2 in (1.1), we indeed get a fractional power of n. In fact our method should work,
subject to some technical modifications, all the way down to η = C(logn)−2 for a large absolute
constant C. However, we do not attempt any rigorous proof of this in the current paper.
A highly related question is the length for the cycle of minimal mean weight, which was studied
by Mathieu and Wilson [9]. An interesting phase transition was found in [9] with critical threshold
e−1 on the mean weight. Further results on this problem have been proved in [5]. It might
be relevant to mention here that the method used in [5] could be potentially useful for nailing
down the second phase transition detected in [4], namely the transition from η = α(logn)−2 to
η = β(logn)−2 where α,β are positive constants.
Another related question is the classical travelling salesman problem (TSP), where one mini-
mizes the weight of the path subject to passing through every single vertex in the graph. For the
TSP in the mean-field set up, W¨astlund [12] established the sharp asymptotics for more general
distributions on the edge weight, confirming the Krauth-M´ezard-Parisi conjecture [10, 11, 8]. In-
deed, it is an interesting challenge to give a sharp estimate on L(n,λ) for e−1 < λ < λ∗ (here λ∗
is the asymptotic value for TSP), interpolating the critical behavior and the extremal case of TSP.
A question of the same flavor on steiner tree is given in [13].
One can also look at the maximum size of tree with average weight below a certain threshold,
where a phase transition was proved in [1]. The extremal case of the question on the tree with
minimal average weight is the well-known minimal spanning tree problem, where a ζ(3) limit is
established by Frieze [7].
Main ideas of our proofs. A straightforward first moment computation as done in [1] implies
that lim P(cid:0)L(n,λ) = O(logn)(cid:1) = 1 when λ < 1/e (see also [4, Theorem 1.3]). For λ > 1/e,
n→∞
a sprinkling method was employed in [1] to show that with high probability L(n,λ) = Θ(n). The
author first proved that with high probability there exist a large number of paths with average
weight slightly above 1/e and then used a certain greedy algorithm to connect these paths into
a single long path with average weight slightly above 1/e. However, the method in [1] was not
able to describe the behavior at criticality. In [4] (see also [9] for the cycle with minimal average
weight), a second moment computation was carried out restricted to paths of average weight below
1/e and with the maximal deviation (defined in (3.2) below) at most O(logn), thereby yielding
that with high probability L(n,1/e) = Θ((logn)3). A crucial fact responsible for the success of the
√
second moment computation is that the length of the target path is Θ((logn)3) (cid:28) n. As such, a
straightforward adaption of this method would not be able to succeed in the regime considered by
this paper.
TSP, where one studies paths (cycles) that visit every single vertex, is in a sense analogous to
the question of finding the minimal value λ for which L(n,λ) = n with high probability. W¨astlund
[12] showed that the minimum average cost of TSP converges in probability to a positive constant
by relaxing it to a certain linear optimization problem. But it seems difficult to extend his method
to“incomplete”TSPi.e. whenthetargetobjectistheminimumcostcyclehavingatleastpnmany
edges for some p ∈ (0,1). Since our problem is in a sense dual to incomplete TSP in the regime we
are interested in, the method of [12] does not seem to be suitable for our purpose either. In the
2
current work, our method is inspired by the (first and) second moment method from [4, 9] as well
as the sprinkling method employed in [1].
In order to prove the upper bound, our main intuition is that if L(n,λ) were greater than
√
e−C2/ ηn then we would have a larger number of short and light paths (a light path refers to
a path with small average weight — at most a little above 1/e) than we would typically expect.
Formally,let(cid:96) = c1 wherec isasmallpositiveconstant,andconsiderthenumberofpaths(denoted
η 1 √
by N ) with length (cid:96) and total weight no more than λ(cid:96)−c (cid:96) for a positive constant c . We
η/c1,c2 2 2 √
call such a path a downcrossing. A straightforward computation gives EN = O(1)n(cid:96)e−c3/ η
η/c1,c2
for a positive constant c depending on c and c . Now we consider the number of paths (denoted
3 1 2
by N ) of length δ(λ)n and average weight at most λ. Such paths have two possibilities: (1) The
δ
pathcontainssubstantiallymorethanEN manydowncrossings,whichisunlikelybyMarkov’s
η/c1,c2
inequality. (2)ThepathdoesnothavesubstantiallymorethanEN manydowncrossings. This
η/c1,c2
is also unlikely for the following reasons: (a) A straightforward first moment computation gives
that ENδ = O(n)ec4δnη for a constant c4 > 0; (b) The number of downcrossings along a path of
this kind, or a random variable that is “very likely” smaller, should dominate a Binomial random
variable Bin(δn/(cid:96),c ) where c > 0 is an absolute constant (since in the random walk bridge, every
5 5
subpath of size (cid:96) has a positive chance to have such a downcrossing). If we choose δ suitably
large as in Theorem 1.1, we are suffering a probability cost for the constraint on the number of
downcrossings (probability for a binomial much smaller than its mean) and this probability cost
is of magnitude e−c6δn/(cid:96) for a constant c6 > 0 depending in c1. If we choose c1 small enough this
probability cost kills the growth of ec4δnη in EN . Therefore, paths of this kind do not exist either.
δ
The details are carried out in Section 2.
For the lower bound, our proof consists of two steps. In light of the preceding discussion, we
cannot hope to directly apply a second moment method from [4, 9] to show the existence of a light
path that is of length linear in n. As such, in the first step of our proof we prove that with high
probability there exists a linear (in n) number of disjoint paths, each of which has weight slightly
√
below λ and is of length ec7/ η for an absolute constant c7 > 0. This is achieved by two second
momentcomputations, whichareexpectedtosucceedasthelengthofthepathunderconsideration
√
is (cid:28) n (indeed remains bounded as n → ∞). In the second step, we propose an algorithm which,
withprobabilitygoingto1, stringstogetherasuitablecollectionoftheseshortlightpathstoforma
√
lightpathoflengthe−c8/ ηnforanabsoluteconstantc8 > 0. Ouralgorithmissimilartothegreedy
algorithm (or in a different name exploration process) employed in [1]. But in order to ensure that
the additional weight introduced by these connecting bridges only increases the average weight of
the final path by at most a multiple of η, we have to use a more delicate algorithm. The details
are carried out in Section 3.
Notation convention. For a graph G, we denote by V(G) and E(G) the set of vertices and edges
of G respectively. A path in a graph G is an (finite) ordered tuple of vertices (v ,v ,··· ,v ),
0 1 m
all distinct. For a path π = (v ,v ,··· ,v ), we also use π to denote the graph whose vertices
0 1 m
are v ,v ,··· ,v and edges are (v ,v ),··· ,(v ,v ). This would be clear from the context.
0 1 m 0 1 m−1 m
The weight of an edge e in W is denoted by W and we define the total weight W(π) of a path
n e
(cid:80)
π as W . The collection of all paths in W of length (cid:96) ∈ [n] is denoted as Π . We let
e∈E(π) e n (cid:96)
λ = 1/e+η where η is a fixed positive number. A path is called λ-light if its average weight is at
√
most λ, and a path is called (λ,C)-light if its total weight is at most λ(cid:96)−C (cid:96) where (cid:96) is length
of the path. For nonnegative real or integer valued variables x ,x ,··· ,x , let S be a statement
0 1 n
involving x ,x ,··· ,x . We say that S holds “for large x (given x ,··· ,x )” or “when x is large
0 1 n 0 1 n 0
(given x ,··· ,x )” if it holds for any fixed values of x ,··· ,x in their respective domains and
1 n 1 n
x ≥ a where a is some positive number depending on the fixed values of x ,··· ,x . In case
0 0 0 1 n
3
a is an absolute constant, the phrase “(given x ,··· ,x )” will be dropped. We use “for small
0 1 n
x ” or “when x is small” with or without the qualifying phrase “(given x ,x ,··· ,x )” in similar
0 0 1 2 n
situations if the statement S holds instead for 0 < x ≤ a . Throughout this paper the order
0 0
notations O(.),Θ(.),o(.) etc. are assumed to be with respect to n → ∞ while keeping all the other
involved parameters (such as (cid:96), η etc.) fixed. We will use C ,C ,... to denote constants, and each
1 2
C will denote the same number throughout of the rest of the paper.
i
Acknowledgements. We are grateful to David Aldous for very useful discussions, and we thank
an anonymous referee for a careful review of an earlier manuscript and suggesting a simpler proof
of Lemma 3.8.
2 Proof of the upper bound
Let η(cid:48) be a multiple of η by a constant bigger than 1 whose precise value is to be selected. Set
(cid:96) = (cid:98)1/η(cid:48)(cid:99) and let N be the number of “(λ,1)-light” paths of length (cid:96). We assume η < 1 so that
η(cid:48)
(cid:96) ≥ 1. As outlined in the introduction, we shall first control N .
η(cid:48)
It is clear that the distribution of the total weight of a path of length k follows a Gamma
distribution Γ(k,1/n), where the density f (z) of Gamma(k,θ) is given by
θ,k
f (z) = θkzk−1e−θz/(k−1)! for all z ≥ 0,θ > 0 and k ∈ N. (2.1)
θ,k
By (2.1) and the Stirling’s formula, we carry out a straightforward computation and get that
(cid:16) √ (cid:17)
EN = (1+o(1))×n(cid:96)+1×P Gamma((cid:96),1/n) ≤ λ(cid:96)− (cid:96)
η(cid:48)
√ √
e−(λ(cid:96)− (cid:96))/n(λ(cid:96)− (cid:96))(cid:96)
= (1+o(1))×n(cid:96)+1×
(cid:96)!n(cid:96)
= (1+o(1))C (η)αeeη/η(cid:48)(cid:112)η(cid:48)e−e/√η(cid:48)n, (2.2)
0
where C (η) → 1 as η → 0, and α is a positive constant. Furthermore the factors 1 +o(1) are
0
strictly less than 1.
We also need a bound on the second moment of N to control its concentration around EN . For
η(cid:48) η(cid:48)
(cid:80)
γ ∈ Π , define F to be the event that γ is (λ,1)-light. Then clearly we have N = 1 . In
(cid:96) γ η(cid:48) γ∈Π(cid:96) Fγ
ordertocomputeE(N )2,weneedtoestimateP(F ∩F )forγ,γ(cid:48) ∈ Π . InthecaseE(γ)∩E(γ(cid:48)) =
η(cid:48) γ γ(cid:48) (cid:96)
∅, we have F and F independent of each other and thus P(F |F ) = P(F ). In the case
γ γ(cid:48) γ(cid:48) γ γ(cid:48)
|E(γ)∩E(γ(cid:48))| = j > 0, we have
P(F |F ) ≤ P(cid:0)Gamma((cid:96)−j,1/n) ≤ λ(cid:96)(cid:1) ≤ 1 (λ(cid:96))(cid:96)−j. (2.3)
γ(cid:48) γ ((cid:96)−j)! n(cid:96)−j
Further notice that if |E(γ)∩E(γ(cid:48))| = j, then |V(γ)∩V(γ(cid:48))| is at least j +1 as γ ∩γ(cid:48) is acyclic.
So given any γ ∈ Π , the number of paths γ(cid:48) such that |E(γ) ∩ E(γ(cid:48))| = j is at most O(n(cid:96)−j).
(cid:96)
Altogether, we obtain that
EN2 = (cid:80) P(F ∩F ) = (cid:80) P(F )(cid:80) P(F |F )
η(cid:48) γ,γ(cid:48)∈Π γ γ(cid:48) γ∈Π γ γ(cid:48)∈Π γ(cid:48) γ
(cid:96) (cid:96) (cid:96)
≤ (cid:80) P(F )(cid:16) (cid:88) P(F )+ (cid:88) (cid:88) 1 (λ(cid:96))(cid:96)−j(cid:17)
γ∈Π(cid:96) γ γ(cid:48) ((cid:96)−j)! n(cid:96)−j
γ(cid:48):E(γ(cid:48)∩γ)=∅ 1≤j≤(cid:96)γ(cid:48):|E(γ(cid:48)∩γ)|=j
= (cid:80) P(F )(cid:16) (cid:88) P(F )+ (cid:88) O(n(cid:96)−j)(λ(cid:96))(cid:96)−j(cid:17)
γ∈Π(cid:96) γ γ(cid:48) ((cid:96)−j)! n(cid:96)−j
γ(cid:48):E(γ(cid:48)∩γ)=∅ 1≤j≤(cid:96)
(cid:16) (cid:17) (cid:16) (cid:17)
≤ (cid:80) P(F ) EN +O(1) = EN EN +O(1) . (2.4)
γ∈Π γ η(cid:48) η(cid:48) η(cid:48)
(cid:96)
4
Since EN = Ω(1) as implied by (2.2), (2.4) yields that
η(cid:48)
EN2 = (EN )2(1+o(1)). (2.5)
η(cid:48) η(cid:48)
As a consequence of Markov’s inequality (applied to |N −EN |2), we get that
η(cid:48) η(cid:48)
P(cid:0)N ≥ 2EN (cid:1) = o(1). (2.6)
η(cid:48) η(cid:48)
Next, we set out to show that any long λ-light path should have a large number of subpaths
which are (λ,1)-light. Let π be a path of length δn for some δ > 0. Denote its successive edge
weights by X ,X ,...X and let S = (cid:80)k X for 1 ≤ k ≤ δn. Probabilities of events involving
1 2 δn k i=1 i
edgeweightsofπ, unlessspecficallymentioned, willbeassumedtobeconditionedon“{A(π) ≤ λ}”
throughouttheremainderofthissection. Nowdivideπ intoedge-disjointsubpathsoflength(cid:96)(with
the last subpath of length possibly less than (cid:96) in the case (cid:96) does not divide δn) and denote the k-th
subpath by bπ for 1 ≤ k ≤ δn/(cid:96). Call any such subpath a downcrossing if it is (λ,1)-light. Let
k
Dπ be the event that bπ is a downcrossing. The following well-known result about exponential
k,η(cid:48),n k
random variables (see, e.g., [3, Theorem 6.6]) will be very useful.
Lemma 2.1. Let W ,W ,...,W be i.i.d. exponential random variables with mean 1/θ, and let
1 2 N
S = (cid:80)k W for 1 ≤ k ≤ N. Then the random vector (W1,..., WN−1) follows Dirichlet(1 )
k i=1 i SN SN N
distribution, S follows Gamma(N;θ) distribution, and they are independent of each other. Here
N
1 is the N-dimensional vector whose all entries are 1.
N
We will also require the following simple lemma which we prove for sake of completeness.
Lemma 2.2. Let Z ,Z ,...,Z be i.i.d. exponential random variables with mean 1 and let S =
1 2 N N
(cid:80)N S . Then
i=1 N
√
P(S ≥ N +α) ≤ e−α2/4N for all 0 < α ≤ (2− 2)N , (2.7)
N
P(S ≤ N −α) ≤ e−α2/2N , for all α > 0. (2.8)
N
Proof. By Markov’s inequality, we get that for any α > 0 and 0 < θ < 1,
P(S ≥ N +α) = P(eθSN ≥ eθ(N+α)) ≤ e−θN−θα.
N (1−θ)N
√
When θ ≤ 1−1/ 2, the right hand side is bounded above by eNθ2−αθ. So setting θ = α/2N yields
√
(2.7) as long as 0 < α/2N ≤ 1−1/ 2. One can prove (2.8) in the same manner.
As hinted in the introduction, let us begin with the intent to prove that the number of down-
crossings along the first half of π (or any fraction of it) dominates a Binomial random variable
Bin(δn/2(cid:96),p) for some positive, absolute constant p. So essentially we need to prove that a sub-
path bπ can be a downcrossing with probability p regardless of the first (k −1)(cid:96) edges of π that
k
precedeit. NowconditionaldistributionofX ,X ,··· ,X givenX ,X ,··· ,X
(k−1)(cid:96)+1 (k−1)(cid:96)+2 δn 1 2 (k−1)(cid:96)
and A(π) ≤ λ is essentially the the distribution of X ,X , ··· ,X conditioned on
(k−1)(cid:96)+1 (k−1)(cid:96)+2 δn
(cid:80)δn X ≤ λδn − S . On the other hand we get from Lemma 2.1 that conditional
i=(k−1)(cid:96)+1 i (k−1)(cid:96)
mean and variance of W(bπ) given S − S = µ(δn − (k − 1)(cid:96)) are µ(cid:96) and µ2(cid:96)(1 + o(1))
k δn (k−1)(cid:96)
respectively for all µ > 0 and k ≤ δn/2. Hence it is plausible to expect that probability of the
√
event {W(bπ) ≤ Λ ((cid:96)−C (cid:96))} conditional on any set of values for X ,X ,··· ,X is bounded
k k 1 2 (k−1)(cid:96)
away from 0 for large (cid:96) and n, where Λ = Λπ = (S −S )/(δn−(k −1)(cid:96)) and C is some
k k δn (k√−1)(cid:96)
positive number. Let us denote the event {W(bπ) ≤ Λ ((cid:96)−C (cid:96))} by AC,π . Thus it seems more
k k k,η(cid:48),δ,n
5
immediate to prove the stochastic domination for number of occurrences of AC,π ’s which, for the
k,η(cid:48),n
time being, can be treated as a “proxy” for the number of downcrossings. The formal statement is
given in the next lemma where we use 6 as the value of C since this allows us to avoid unnecessary
named variables and also suits our specific needs for the computations carried out at the end of
this section.
Lemma 2.3. Let Nπ be the number of occurrences of events Aπ = A6,π for 1 ≤ k ≤ δn/2(cid:96).
η(cid:48),n k,η(cid:48),n k,η(cid:48),n
Then for any 0 < η(cid:48) < η where η is a positive, absolute constant and any 0 < δ < 1 there
0 0 0
exists a positive integer n = n (δ ,η(cid:48)) and an absolute constant c > 0 such that for all δ ≥ δ and
d d 0 0
n ≥ n the conditional distribution of Nπ given {A(π) ≤ λ} stochastically dominates the binomial
d η(cid:48),n
distribution Bin(δn/2(cid:96),c).
Proof. Notice that it suffices to prove that there exist positive absolute constants (cid:96) ,c such that
0
uniformly for µ > 0, (cid:96) ≥ (cid:96) and large L (given (cid:96))
0
√
P(S ≤ SL((cid:96)−6 (cid:96))|S = µL) ≥ c.
(cid:96) L L
To this end, we see that for L > (cid:96)
√ √ √
P(S ≤ SL((cid:96)−6 (cid:96))|S = µL) = P(S(cid:96) ≤ ((cid:96)−6 (cid:96))/L|S = µL) = P(S(cid:96) ≤ ((cid:96)−6 (cid:96))/L), (2.9)
(cid:96) L L SL L SL
where the last equality follows from Lemma 2.1. Since distribution of S(cid:96) does not depend on the
SL
mean of the underlying X ’s, we can in fact assume that X ’s are i.i.d. exponential variables with
j j
mean 1 for purpose of computing (2.9). By (2.8), we have
√
P(cid:0)S /L ≤ 1−1/(2 (cid:96))(cid:1) ≤ e−L/8(cid:96).
L
√
So for (cid:96)−6 (cid:96) > 0, we get
√ √
P(S ≤ SL((cid:96)−6 (cid:96))) ≥ P(S ≤ (cid:96)−6.5 (cid:96))−e−L/8(cid:96). (2.10)
(cid:96) L (cid:96)
√
By central limit theorem there exist absolute numbers (cid:96) ,c(cid:48) > 0 such that P(S ≤ (cid:96)−6.5 (cid:96)) ≥ c(cid:48)
0 (cid:96)
for (cid:96) ≥ (cid:96) . Hence from (2.10) it follows that for any (cid:96) ≥ (cid:96) there exists L = L ((cid:96)) such that the
0 0 0 0
right hand side of (2.9) is at least c = 0.99c(cid:48) for L ≥ L .
0
Now what remains to show is that the number of downcrossings N˜π along π is bigger than
η(cid:48),n
Nπ with high probability. Notice that the occurrence of Aπ \Dπ implies that Λ must
η(cid:48),n k,η(cid:48),n k,η(cid:48),n k
be “significantly” above λ. But that can only be caused by a substantial drop in S for some
k
1 ≤ k ≤ δn/2, an event that occurs with small probability.
√
Lemma 2.4. Denote by Eπ the event that Λ is more than λ+ η(cid:48) for some 1 ≤ k ≤ δn. Then
η(cid:48),n k 2(cid:96)
for any 0 < η(cid:48) < 1/4 and 0 < δ < 1 there exists a positive integer n = n (δ ,η(cid:48)) such that,
0 s s 0
P(Eπ |A(π) ≤ λ) ≤ 2ne−δnη(cid:48)/16 for all δ ≥ δ and n ≥ n . (2.11)
η(cid:48),n 0 s
√
Proof. For 1 ≤ k ≤ δn/2(cid:96), let (cid:96) = (k−1)(cid:96), n = (cid:100)2(cid:96)/δ (cid:101) and Eπ = {Λ ≥ λ+ η(cid:48)}. Assume
k s 0 k,η(cid:48),n k
n ≥ n so that δn/2(cid:96) ≥ 1. On Eπ , we have
s k,η(cid:48),n
√ √
S (cid:96) S /δn− η(cid:48)(δn−(cid:96) ) (cid:96) η(cid:48)(δn−(cid:96) )
(cid:96)k ≤ k δn k ≤ k − k ,
S S δn δn
δn δn
6
where the last inequality holds since we are conditioning on S ≤ λδn and λ ≤ 1 when η(cid:48) < 1/4
δn
(recall that η < η(cid:48)). Therefore, we get
P(Eπ |A(π) ≤ λ) ≤ P(S ≤ Sδn(cid:0)(cid:96) −(cid:112)η(cid:48)(δn−(cid:96) ))) (2.12)
k,η(cid:48),n (cid:96)k δn k k
Now we evaluate the right hand side of (2.12). Analogous to (2.9) in the proof of Lemma 2.3, we
can assume without loss of generality that X ,X ,...X are i.i.d. exponential variables with mean
1 2 L
1. It is routine to check that
(cid:112) (cid:0) (cid:112) (cid:1) (cid:112)
(1+ η(cid:48)/2)× (cid:96) − η(cid:48)(δn−(cid:96) ) ≤ (cid:96) − η(cid:48)δn/4, for all 1 ≤ k ≤ δn/2(cid:96).
k k k
Thus, for all 1 ≤ k ≤ δn/2(cid:96) we get
P(cid:16)S ≤ Sδn(cid:0)(cid:96) −(cid:112)η(cid:48)(δn−(cid:96) )(cid:1)(cid:17) ≤ P(cid:16)S ≤ (cid:96) −(cid:112)η(cid:48)δn/4(cid:17)+P(cid:16)Sδn ≥ 1+(cid:112)η(cid:48)/2(cid:17)
(cid:96) k k (cid:96) k
k δn k δn
≤ e−δnη(cid:48)/16+e−δnη(cid:48)/16,
where the second inequality follows from (2.8) and (2.7) respectively. Combined with (2.12), it
gives that
P(Eπ |A(π) ≤ λ) ≤ 2e−δnη(cid:48)/16, for all 1 ≤ k ≤ δn/2(cid:96).
k,η(cid:48),n
An application of a union bound over k completes the proof of the lemma.
Proof of Theorem 1.1: upper bound. Assume that η(cid:48) < 1/4∧η where η is same as given in the
0 0
statement of Lemma 2.3. Fix a δ = δ (η(cid:48)) in (0,1) and let n = n (δ ,η(cid:48)) = n (δ ,η(cid:48))∨n (δ ,η(cid:48)),
0 0 0 0 0 d 0 s 0
where n ,n are as stated in Lemmas 2.3 and 2.4 respectively. In the remaining part of this section
d s
we will assume that n ≥ n and δ ≥ δ , so that Lemmas 2.3 and 2.4 become applicable. Now let π
0 0 √
be a path with length δn. From Lemma 2.4 we get that with large probability Λ ≤ λ+ η(cid:48) for all
k √
k between 1 and δn/2(cid:96). But it takes a routine computation to show that Aπ \{Λ ≤ λ+ η(cid:48)} ⊆
n,k,η(cid:48) k
Dπ when η(cid:48) is small. Thus N˜π ≥ Nπ except on Eπ . Consequently Lemma 2.3 allows us to
n,k,η(cid:48) η(cid:48),n η(cid:48),n η(cid:48),n
use binomial distribution to bound quantities like P(N˜π ≤ x) with a “small error term” caused
η(cid:48),n
by the rare event Eπ . Formally,
η(cid:48),n
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
P N˜π ≤ 2EN |A(π) ≤ λ ≤ P Nπ ≤ 2EN |A(π) ≤ λ +P E |A(π) ≤ λ
η(cid:48),n η(cid:48) η(cid:48),n η(cid:48) η(cid:48),n
(cid:16) (cid:17)
≤ P Nπ ≤ 2EN |A(π) ≤ λ +2ne−δnη(cid:48)/16,
η(cid:48),n η(cid:48)
where the last inequality follows from Lemma 2.4. Therefore, by Lemma 2.3, we get that
(cid:16) (cid:17) (cid:16) (cid:17)
P N˜π ≤ 2EN |A(π) ≤ λ ≤ P Bin(δn/2(cid:96),c) ≤ 2EN +2ne−δnη(cid:48)/16. (2.13)
η(cid:48),n η(cid:48) η(cid:48)
Next let us define a new event as
Ξ = (cid:83) (cid:83) (cid:8)N˜π ≥ 2EN ,A(π) ≤ λ(cid:9).
η,δ0,n k≥δ0n π∈Πk η(cid:48),n η(cid:48)
So Ξ is the event that there exists a λ-light path π with len(π) ≥ δ n and which contains
η,δ0,n 0
at least 2EN many downcrossings. Thus occurrence of Ξ implies that N ≥ 2EN which
η(cid:48) η,δ0,n η(cid:48) η(cid:48)
has small probability owing to (2.6). On the other hand if Ξ does not occur, L(n,λ) ≥ δ n
η,δ0,n 0
implies the existence of a λ-light path of length at least δ n that has no more than 2EN many
0 η(cid:48)
downcrossings. Formally,
7
P(cid:0)L(λ,n) ≥ δ n(cid:1) = P(cid:0)Ξ (cid:1)+P(cid:0){L(λ,n) ≥ δ n}\Ξ (cid:1)
0 η,δ0,n 0 η,δ0,n
(cid:16) (cid:17)
≤ P(cid:0)N ≥ 2EN (cid:1)+P (cid:83) (cid:83) (cid:8)N˜π ≤ 2EN ,A(π) ≤ λ(cid:9)
η(cid:48) η(cid:48) k≥δ0n π∈Πk η(cid:48),n η(cid:48)
(cid:16) (cid:17)
≤ o(1)+(cid:80) (cid:80) P N˜π ≤ 2EN |A(π) ≤ λ P(cid:0)A(π) ≤ λ(cid:1). (2.14)
k≥δ0n π∈Πk η(cid:48),n η(cid:48)
Now choose δ = δ (η(cid:48)) such that
0 0
δ nη(cid:48)c/4 = 2EN . (2.15)
0 η(cid:48)
Since 1/(cid:96) ≥ η(cid:48), we then get from Binomial concentration that for δ ≥ δ ,
0
(cid:16) (cid:17)
P Bin(δn/2(cid:96),c) ≤ 2EN ≤ e−δnη(cid:48)c2/16.
η(cid:48)
Plugging this into (2.13) we have
(cid:16) (cid:17)
P N˜π ≤ 2EN |A(π) ≤ λ ≤ 2ne−len(π)η(cid:48)/16+e−len(π)η(cid:48)c2/16,
η(cid:48),n η(cid:48)
whenever len(π) ≥ δ n. A straightforward computation using (2.1) yields
0
(cid:80) P(cid:0)A(π) ≤ λ(cid:1) ≤ √n eekη.
π∈Π
k 2πk
The last two displays and (2.14) together imply that
P(cid:0)L(λ,n) ≥ δ n(cid:1) ≤ o(1)+(cid:80) (2ne−kη(cid:48)/16+e−kη(cid:48)c2/16)eekη√n . (2.16)
0 k≥δ0n 2πk
Setting η(cid:48) = 32eη/c2 we get from (2.16),
P(cid:0)L(n,λ) ≥ δ n(cid:1) = o(1).
0
It remains to be checked whether δ obtained from (2.15) has the correct functional form as in
0
(1.1). To this end recall from (2.2) that
2EN ≤ 3αeeη/η(cid:48)(cid:112)η(cid:48)e−e/√η(cid:48)n,
η(cid:48)
√
where η is small enough so that C0(η) in (2.2) is less than 3/2. Hence δ0 ≤ e−C2/ η for some
absolute constant C when η is small.
2
3 Proof of the lower bound
3.1 Existence of a large number of vertex-disjoint light paths
As we mentioned in the introduction, the proof of lower bound is divided into two steps. In the
first step we split the vertices into two parts and show that there exist a large number of short
(i.e. of O(1) length) vertex-disjoint λ-light paths containing vertices from only one part. In the
second step we use vertices in the other part as “links” to connect a subcollection of the short
paths obtained from step 1 into a long (i.e. of Θ(n) length) and light path. The current and next
subsections are devoted to these two steps in respective order.
8
In light of the preceding discussion, let us first select a complete subgraph W∗ of W containing
n n
n = n = (1−ζ η)n vertices where η,ζ ∈ (0,1). To be specific we can order the vertices of
∗ ∗;η,ζ1 1 1
W in some arbitrary way and define W∗ as the subgraph induced by “first” n vertices. It will be
n n ∗
shown that there are substantially many short and light paths that can be formed with the vertices
in V(W∗). We will in fact require slightly more from a path than just being λ-light. For π ∈ Π
n (cid:96)
and some ζ > 0, define
2
(cid:110) √ (cid:111)
G = G = λ(cid:96)−1 ≤ W(π) ≤ λ(cid:96),M(π) ≤ (ζ / η).(W(π)/λ(cid:96)) , (3.1)
π π;η,ζ2 2
where M(π) is the maximum deviation of π away from the linear interpolation between the
starting and ending edges, formally given by
M(π) = sup | (cid:80)k W − kW(π) | . (3.2)
i=1 ei (cid:96)
1≤k≤(cid:96)
A similar class of events were considered in [4, 9] in order for second moment computation. As the
authors mentioned in these papers, the factor W(π)/λ(cid:96) provides some technical ease in view of the
following property which is a consequence of Lemma 2.1:
√
P(M(π) ≤ (ζ / η).(W(π)/λ(cid:96)) | W(π) = w) ≡ constant for all w > 0. (3.3)
2
Call a path π ∈ Π good if G occurs. Since we are only interested in good paths whose vertices
(cid:96) π
come from V(W∗), we need some related notations. For (cid:96) ∈ N, denote by Π∗ = Π∗ the set
n (cid:96) (cid:96);η,ζ1
of all paths of length (cid:96) in W∗ and by N∗ = N∗ the total number of good paths in Π∗, i.e.,
N∗ = (cid:80) 1 . In ordernto carry ou(cid:96)t secon(cid:96);dη,ζm1,ζo2ment analysis of N∗ we need to contr(cid:96)ol the
(cid:96) π∈Π∗ Gπ (cid:96)
(cid:96)
correlation between 1 and 1 where π, π(cid:48) ∈ Π∗. It is plausible that such correlation depends
Gπ Gπ(cid:48) (cid:96)
on the number of common edges between π and π(cid:48) and in fact bounding the correlation in terms of
the number of common edges was sufficient for proving (2.5) in Section 2. But in this case we need
an additional measurement instead of just |E(π)∩E(π(cid:48))|. This is discussed in detail in [4, 9] and
some of their results will be used. Let π be a path in Π∗ and S ⊆ E(π). A segment of π is called
(cid:96)
an S-component or a component of S if it is a maximal segment of π whose all edges belong to S.
Notice that S-components can be defined solely in terms of S. For two paths π and π(cid:48), define a
functional θ(π,π(cid:48)) to be the number of S-components where S = E(π)∩E(π(cid:48)). As π and π(cid:48) are
self-avoiding, θ(π,π(cid:48)) is basically the number of maximal segments shared between π and π(cid:48). We
refer the readers to Figure 1 for an illustration.
The following result ([4, Lemma 2.9]) relates cardinality of V(S), the union of all endpoints of
edges in S = E(π)∩E(π(cid:48)), to θ(π,π(cid:48)) and |S|.
|V(S)| = |S|+θ(π,π(cid:48)).
Thepair(cid:0)θ(π,π(cid:48)),|E(π)∩E(π(cid:48))|(cid:1)turnsouttobesufficientforboundingthecorrelationbetween
1 and 1 from above. Consequently it makes sense to partition Π∗ based on the value of this
Gπ Gπ(cid:48) (cid:96)
pair. More formally for π ∈ Π∗ and integers i ≤ j, define the set A as
(cid:96) i,j
A ≡ A (π) = {π(cid:48) ∈ Π∗ : θ(π,π(cid:48)) = i,|E(π)∩E(π(cid:48))| = j}. (3.4)
i,j i,j (cid:96)
We need a number of lemmas from [4].
Lemma 3.1. ([4, Lemma 2.10]) For any 1 ≤ (cid:96) ≤ n and any π ∈ Π∗, we have that for any positive
∗ (cid:96)
integers i ≤ j
|A (π)| ≤ (cid:0)(cid:96)+1(cid:1)(cid:0)n∗−i−j(cid:1)2i((cid:96)+1−j)! ≤ (cid:96)3in(cid:96)+1−i−j.
i,j 2i (cid:96)+1−i−j ∗
9
Figure 1 – Components of the set of edges common to two paths. In this figure
the sequences of vertices v ,v ,v ,v ,v ,v ,v ,v ,v and v(cid:48),v ,v(cid:48),v ,v ,v(cid:48),v ,v ,v ,v(cid:48)
1 2 3 4 5 6 7 8 9 1 2 3 3 4 5 6 7 8 9
definethepathsπ andπ(cid:48) respectively. ThedarkedgesbelongtoS =E(π)∩E(π(cid:48)). Here
θ(π,π(cid:48))=2 with the segments (v ,v ) and (v ,v ,v ) being the two S-components.
3 4 6 7 8
Lemma 3.2. ( [4, Lemma 2.3]) Let Z be i.i.d. exponential variables with mean θ > 0 for 1 ≤ i ≤ (cid:96).
i
For 1/4 ≤ ρ ≤ 4, consider the variable
M = sup | (cid:80)k Z −ρk | . (3.5)
(cid:96) i=1 i
1≤k≤(cid:96)
Then there exist absolute constants c∗, C∗ > 0 such that for all r ≥ 1 and (cid:96) ≥ r2,
e−C∗(cid:96)/r2 ≤ P(M ≤ r | (cid:80)(cid:96) Z = ρ(cid:96)) ≤ e−c∗(cid:96)/r2.
(cid:96) i=1 i
Lemma 3.3. [4, Lemma 3.2] Let Z be i.i.d. exponential variables with mean θ > 0 for i ∈ N.
√ i
Consider 1 ≤ r ≤ (cid:96) and the integer intervals [a ,b ],[a ,b ],··· ,[a ,b ] such that 1 ≤ a ≤ b ≤
1 1 2 2 m m 1 1
a ≤ ··· ≤ a ≤ b ≤ (cid:96) and q = (cid:80)m (b −a +1) ≤ (cid:96)−1. Let 1/4 ≤ ρ ≤ 1 and M be defined as
2 m m i=1 i i (cid:96)
in the previous lemma. Also write A = ∪m [a ,b ]∩N and p = P(M ≤ r|(cid:80)(cid:96) Z = ρ(cid:96)). Then
i=1 i i (cid:96) (cid:96) i=1 i
for all z such that
j
(cid:80)bi z −ρ(b −a +1) ≤ 2r,
j=ai j i i
we have
P(M ≤ r|(cid:80)(cid:96) Z = ρ(cid:96),Z = z for all j ∈ A) ≤ C r(cid:112)q∧((cid:96)−q)p 10100mreC∗q/r2, (3.6)
(cid:96) i=1 i j j 3 (cid:96)
where C∗ is the constant from Lemma 3.2 and C > 0 is an absolute constant.
3
Remark 3.4. (1) Notice that the bounds in Lemma 3.2 and 3.3 do not depend on the particular
mean of Z ’s due to Lemma 2.1. (2) Although the bounds on p in Lemma 3.2 do not contain any
i (cid:96)
ρ (as it was restricted to a bounded interval), p actually depends on r only through the ratio r/ρ.
(cid:96)
This follows from an application of Lemma 2.1 with little manipulation. (3) Lemma 3.3 is same
as Lemma 3.2. in [4] except that in the latter q is restricted to be at most (cid:96)−10r. But we can
easily extend this to all q ≤ (cid:96)−1. To see this assume (cid:96)−1 ≥ q ≥ (cid:96)−10r. Then the right hand
side in (3.6) becomes at least C p eC∗(cid:96)/r2e−10C∗/r. Now from Lemma 3.2 we get p eC∗(cid:96)/r2 ≥ 1. So
3 (cid:96) (cid:96)
the right hand side in (3.6) is bigger than C e−10C∗ whenever (cid:96)−1 ≥ q ≥ (cid:96)−10r. Increasing C if
3 3
necessary we can make this number bigger than 1 and thus Lemma 3.3 follows.
10