Expansion of percolation critical points for Hamming graphs

0.67 MB·

by  Lorenzo Federico

EXPANSION OF PERCOLATION CRITICAL POINTS
FOR HAMMING GRAPHS
LORENZO FEDERICO, REMCO VAN DER HOFSTAD, FRANK DEN HOLLANDER,
AND TIM HULSHOF
7
1 Abstract. The Hamming graph H(d,n) is the Cartesian product of d complete graphs on
0 nvertices. Letm=d(n−1)bethe degree andV =nd bethenumberof vertices of H(d,n).
2 Let p(d) be the critical point for bond percolation on H(d,n). We show that, for d∈N fixed
c

n and n→∞,
a 1 2d2−1 1
p(d) = + +O(m−3)+O(m−1V−1/3),
J c m 2(d−1)2m2

9 which extends the asymptotics found in [10] by one order. The term O(m−1V−1/3) is the
widthofthecriticalwindow. Ford=4,5,6wehavem−3 =O(m−1V−1/3),andsotheabove
]
R formula represents the full asymptotic expansion of p(d). In [16] we show that this formula is
c
P a crucial ingredient in the study of critical bond percolation on H(d,n) for d=2,3,4. The
. proof uses a lace expansion for the upper bound and a novel comparison with a branching
h
random walk for the lower bound. The proof of the lower bound also yields a refined
t
a asymptotics for the susceptibility of a subcritical Erdo˝s-R´enyi random graph.
m
[

MSC 2010. 60K35, 60K37, 82B43.
1
Keywords and phrases. Hamming graph, percolation, critical point, critical window, lace expansion.
v
9
9
0 1. Introduction and main result
2
0 1.1. Percolation on the Hamming graph. The Hamming graph H(d,n) is the Cartesian
.
1 product of d complete graphs on n vertices (e.g., H(3,7) = K ×K ×K ). Bernoulli bond
7 7 7
0
percolation is the model where, given a graph, each edge is retained independently with
7
the same probability p. In this paper we study the location of the critical point of bond
1
: percolation on H(d,n) for the phase transition in the size of the largest connected component
v
i when d is fixed and n → ∞.
X Formally, we define the Hamming graph H(d,n) for d,n ∈ N as the graph with vertex set
r V := {0,1,...,n−1}d and edge set
a
E := {(v,w) : v,w ∈ V, v (cid:54)= w for exactly one j}. (1.1)
j j
Thus, H(d,n) is a transitive graph on V := nd vertices with degree m := d(n−1). Bernoulli
bond percolation is synonymous with the probability space (Ω,P ), where Ω := {0,1}E and
p
P is the measure such that
p
P (ω) = (cid:89)(cid:0)(1−p)δ +pδ (cid:1) ∀ω ∈ Ω, (1.2)
p 0,ω(e) 1,ω(e)
e∈E
Date: January 10, 2017.
1
2 L.FEDERICO,R.VANDERHOFSTAD,F.DENHOLLANDER,ANDT.HULSHOF
where δ is the Kronecker delta. When ω(e) = 1 we say that the edge e is open, when
x,y
ω(e) = 0 we say that the edge e is closed. Given a vertex x ∈ V, we write C(x) for the graph
whose vertex set consists of all vertices that can be reached from x through a path of open
edges, and whose edge set consists of all open edges between these vertices. We call C(x)
the connected component of x, or cluster of x, and write |C(x)| for its number of vertices.
We write C for the cluster C(x) with the largest cardinality |C(x)| (using some tie-breaking
1
rule). Two of the main objects of study in percolation are |C(x)| and |C |, the cardinalities of
1
C(x) and C . For percolation on infinite graphs G it is often observed that the critical point
1
of the percolation phase transition on G, defined by
pG := inf{p ∈ [0,1]: P (|C(x)| = ∞) > 0}, (1.3)
c p
is non-trivial, i.e., pG ∈ (0,1) (see for example Grimmett [20]) for most infinite graphs (an
c
exception being Z1). Moreover, Aizenman and Barsky [1] and independently Menshikov [34]
proved that on transitive graphs,
pG = sup{p ∈ [0,1]: E [|C(x)|] < ∞}. (1.4)
c p
Since we consider percolation on H(d,n) with d,n finite and P is a product measure, any
p
event that is measurable with respect to P has a probability that is a polynomial in p, and
p
therefore is continuous in p: the finite model cannot undergo a non-trivial phase transition
in p as described above. Nevertheless, it does make sense to study the percolation phase
transition on finite graphs in the limit as n → ∞. To see why, let us give a rough sketch of
an important related problem: the emergence of the giant component in the Erd˝os-R´enyi
Random Graph (ERRG).
1.2. Giant component. TheErdo˝s-R´enyirandomgraphisthecommonnameforpercolation
on the complete graph K . Erd˝os and R´enyi [14] proved that in the limit as n → ∞, if
n
p = p(n) < n−1, then |C | = Θ(logn) w.h.p., 1 while if p > n−1, then |C | = Θ(n) w.h.p.
1 1
Moreover, zooming in on the transition point n−1 by choosing p = (1+ε )n−1 for a sequence
n
(εn)n∈N such that limn→∞εn = 0, Bollob´as [7] showed that 2
(cid:66) |C | = Θ(ε−2log(ε3n)) w.h.p. when ε3n → −∞ (subcritical),
1 n n n
(cid:66) |C | = Θ(n2/3) w.h.p. when ε n3 → a ∈ R (critical),
1 n
(cid:66) |C | = Θ(ε n) w.h.p. when ε n3 → +∞ (supercritical).
1 n n
What this shows is that the size of the largest component undergoes a sharp transition
around n−1. As mentioned above, there is no critical point for a finite graph, but the
transition occurs in a slice of the parameter space with a width of order n−4/3, which is
asymptotically vanishing with respect to the center of the window located around n−1. This
behaviour inspired the notion of critical window: to indicate that the transition of the ERRG
occurs around n−1 in a range of width n−4/3, we use the short-hand notation 3
pKn = n−1+O(n−4/3). (1.5)
c
1Given a sequence of random variables (Xn)n∈N, we write Xn =Θ(f(n)) w.h.p. (with high probability) if
there exist constants C ≥c>0 such that P (cf(n)≤X ≤Cf(n))→1 as n→∞.
p n
2Subsequent results in [3,33,36,40] are much sharper and comprehensive than what is summarized here,
and there is an extensive body of literature on the problem.
3Given three sequences (a ),(b ),(c ), we write that a =b +O(c ) when there exists a constant K <∞
n n n n n n
such that |a −b |≤Kc for all n.
n n n
EXPANSION OF PERCOLATION CRITICAL POINTS FOR HAMMING GRAPHS 3
Erd˝os and Spencer [15] conjectured that if we replace K by a more “geometric” graph
n
sequence (their primary candidate was H(d,2), the d-dimensional hypercube, with d → ∞),
then the critical behaviour should remain largely intact. In fact, it turned out that to a large
extent the picture is the same for a large class of graph sequences with “sufficiently weak”
geometries.
Of particular interest to us here are the papers by Borgs et al. [9–11], demonstrating that
graph sequences satisfying the so-called triangle condition (which serves as an indicator of
what is meant by sufficiently weak geometry; see e.g. [2,9–11,21]) have a phase transition
that strongly resembles that of the ERRG, and that both (H(d,2))d∈N and (H(d,n))n∈N
satisfy the triangle condition. More precisely, consider a sequence (Gm)m∈N = (Vm,Em)m∈N of
vertex transitive graphs of degree m, and write V := |V |. Write x ←→ y for the event that
m m
y ∈ C(x), and define the two-point function τ (x−y) := P (x ←→ y) and the susceptibility
p p
χ(p) := E [|C(x)|] = (cid:80) τ (x−y) (note that χ(p) does not depend on x by transitivity
p y∈Vm p
and that τ (x−y) depends on the relative difference of x and y only because the graphs under
p
consideration are tori). The triangle condition is satisfied for percolation on (G ) if for all p
m
such that χ(p)3/V ≤ β for some sufficiently small β , and for all x,y ∈ V , we have 4
m 0 0 m
(cid:88) χ(p)3
∇ (x,y) := τ (x−u)τ (u−v)τ (v−y) = δ +10 +O(m−1). (1.6)
p p p p x,y
V
u,v
Borgs et al. prove that the triangle condition holds for a class of models that includes
(H(d,n))n∈N for any fixed d ≥ 2 (see [10, Theorem 1.3]). An alternative proof, applying to
e.g. Hamming graphs and hypercubes, was given by van der Hofstad and Nachmias [26,27].
1.3. Critical window. Fix some θ ∈ (0,∞) and define pGm(θ) as the unique solution of the
c
equation
χ(p (θ)) = θV1/3. (1.7)
c
Borgs et al. [9,10] prove that if we consider percolation on a sequence (G ) that satisfies the
m
triangle condition (1.6) with p = p (θ)(1+ε ) and ε → 0, then we see subcritical behaviour
c m m
when ε3V → −∞ and critical behaviour when ε3V → a ∈ R, just as in the ERRG. Sharper
results about mean-field supercritical behaviour of percolation models when ε3V → ∞ were
derived later by van der Hofstad and Nachmias [26], who investigate the supercritical phase
and thus establish that (1.7) really constitutes the critical window for several high-dimensional
tori including the hypercube and Hamming graphs. Moreover, it was shown in [9, Theorem
1.1] that the critical window satisfies
p (θ) = m−1+O(m−2)+O(m−1V−1/3), (1.8)
c
and that p (θ )−p (θ ) = O(m−1V−1/3) for any θ ,θ > 0, i.e., any choice of θ yields the
c 1 c 2 1 2
same critical window.
Compare (1.8) with the critical window of the ERRG in (1.5), and note that K has
n
O(m−1V−1/3) = O(n−4/3) because m = n−1 and V = n. Thus, by that analogy, the second
error term above corresponds to the width of the critical window, while the first error term
can be viewed as a “correction” in m−1 to p itself. In this interpretation, (1.8) describes the
c
criticalwindowasymptotically precisely forthetwo-dimensionalHamminggraphH(2,n), since
4Hereandbelowwewillfrequentlysuppresssub-andsuperscriptswhentheirpresenceisclearfromcontext.
Likewise, we do not always stress that we are considering asymptotic results for sequences.
4 L.FEDERICO,R.VANDERHOFSTAD,F.DENHOLLANDER,ANDT.HULSHOF
in this case m = 2(n−1) and V = n2, so that the correction term m−2 is vanishingly small
compared to m−1V−1/3. Moreover, (1.8) is also asymptotically precise for H(3,n) because
the two O-terms coincide.
1.4. Expansion of the critical point. This brings us to the main result of our paper. We
write p(d)(θ) for the critical value of percolation on H(d,n) defined in (1.8), and compute the
c
second term of p(d)(θ) for all d ≥ 2:
c
Theorem 1.1 (Critical window for percolation on H(d,n)). For all θ ∈ (0,∞) and all d ≥ 2,
2d2−1
p(d)(θ) = m−1+ m−2+O(m−3)+O(m−1V−1/3), (1.9)
c 2(d−1)2
where the constants in the error terms may depend on θ.
Observe that for d ≥ 4, the correction term of order m−2 is asymptotically larger than
the width of the critical window, and that when d = 4,5,6 the above expansion is again
asymptotically precise, since we have m−3 = O(m−1V−1/3).
To see the relevance of Theorem 1.1, we compare it with other expansions of p in the
c
literature. The van der Hofstad and Slade [29] proved that for percolation on G, with G either
the infinite lattice Zd with nearest-neighbour edges or the hypercube H(d,2), as d → ∞, pG
c
can be expanded up to three terms as
7
pG = m−1+m−2+ m−3+O(m−4), (1.10)
c
2
where in both cases m denotes the degree of the graph G. Moreover, they [28] also proved
that, for any N ∈ N,
N N
pZd = (cid:88)a (2d)−k +O((2d)−N−1), pH(d,2)(θ) = (cid:88)b d−k +O(d−N−1), (1.11)
c k c k
k=1 k=1
where (a ),(b ) are rational coefficients. The critical window of the hypercube has width
k k
O(d−12−d/3), so we believe that the expansion cannot be asymptotically precise, regardless of
Zd
the choice of N. Furthermore, it was conjectured that the expansion for p , although it may
c
exist, is divergent for all d as N → ∞ (in the sense that the power series z (cid:55)→ (cid:80)∞ a zk has
k=1 k
radius of convergence 0). We conjecture that the expansion for the Hamming graph is very
different. We believe that for any d ≥ 2 there exist coefficients (c (d)) such that
k
(cid:98)d/3(cid:99)
2d2−1 (cid:88)
p(d)(θ) = m−1+ m−2+ c (d)m−k +O(m−1V−1/3), (1.12)
c 2(d−1)2 k
k=3
i.e., we conjecture that p(d) has an asymptotically precise expansion in m−1 of order (cid:98)d/3(cid:99)
c
for all d. Heydenreich and van der Hofstad state the conjecture in (1.12) as [22, Open
Problem 15.4].
Theorem 1.1 in [9] confirms this conjecture for d = 2,3, and our current work confirms it
for d = 4,5,6. The argument of van der Hofstad and Slade [28] establishing (1.11) for the
lattice and the hypercube crucially uses the fact that a ball of a radius r restricted to a d(cid:48)
dimensional subspace has the same shape for all d ≥ d(cid:48), so that we can express each coefficient
in terms of events that happen on a fixed subgraph. Balls in the Hamming graph instead
grow very rapidly when n increases. Each coefficient is obtained as a limit and it will be
EXPANSION OF PERCOLATION CRITICAL POINTS FOR HAMMING GRAPHS 5
more involved to prove the existence of this limit. Hence we do not have significant evidence
suggesting that all coefficients in (1.12) have to be rational.
We note that the existence of a finite asymptotically precise expansion makes the proof of
the critical window of the Hamming graph more challenging than for the hypercube. Roughly
speaking, because the critical window of the hypercube is exponentially narrower than any of
H(d,2)
the expansion terms, we can approximate p up to any fixed order by a value p that is
c
in fact subcritical, by choosing a negative coefficient for the error term. This allows one to
exploit the fact that χ(p) is poly-logarithmic in V, which simplifies the analysis considerably.
In our case, the approximating p will be much closer to p(d), and so we need a much more
c
refined analysis. We will explain this in more detail in Section 5.5.
1.5. Scaling limit of largest cluster sizes. Besides offering an interesting comparison
with other graphs with sufficiently weak geometry, the expansion of p(d) also has another
c
motivation. The Hamming graph is an excellent example to investigate the universality class
of the ERRG, since it has a non-trivial geometry yet is highly mean field. See [17,24,25,35]
for a small sample of the literature from this perspective. A crucial motivation for the present
paper is that it serves as a companion paper to [16], where we establish the scaling limit of
the cluster sizes of the largest clusters within the critical window. More precisely, writing C
j
for the j-th largest cluster, we prove that for any fixed N ∈ N and for d = 2,3,4 the largest
critical clusters of Hamming graph percolation satisfy
(cid:0)V−2/3|C |(cid:1) −d→ (X ) (1.13)
j j≥1 j j≥1
for a certain sequence of θ-dependent continuous random variables (Xi)i∈N supported on
[0,∞). Aldous [3] proved this scaling limit for the ERRG. Since then, many other random
graph models have been shown to have the same (or at least a similar) scaling limit. See for
instance [5,6,32,38] and the references therein. The above result for the Hamming graph,
however, is the first indication that the same scaling occurs for models with an underlying
high-dimensional geometry. Moreover, it is the most precise determination to date of the
critical behaviour of percolation on a finite transitive graph (other than the ERRG scaling
limit of Aldous). The proof of (1.13) and various other results in [16] crucially rely on the
asymptotically precise determination of the critical window that we give here.
1.6. Alternative definition of the critical point. It is worth noting that a disadvantage
of the definition p in (1.7) is that it imposes an ad hoc relation between p and V1/3, which
c c
is known not to hold in general and believed to be associated with “high-dimensional” models.
In other words, (1.7) is possibly only a valid definition of p for percolation models in the
c
universality class of the ERRG. Nachmias and Peres in [37] observed that it would be desirable
to have a definition of p that applies more generally, and they proposed
c
d χ (p)
p˜G := argmaxdp G (1.14)
c
χ (p)
p∈(0,1) G
as a definition of the critical point for any graph G. Their motivation for this definition is
that Russo’s formula [41] implies that p = p˜G is the point where a small change in p has the
c
greatest impact on the relative size of the connected components, i.e., χ(p) changes most
dramatically at p˜G. A serious downside of this definition appears to be that p˜G may be very
c c
difficult to compute. Thus far, the only non-trivial determination of p˜G is given in recent work
c
6 L.FEDERICO,R.VANDERHOFSTAD,F.DENHOLLANDER,ANDT.HULSHOF
by Janson and Warnke [31]. They determine that, for the ERRG, |p˜Kn −1/n| = O(n−4/3),
c
so p˜Kn is a point inside the critical window (1.5), that χ(p)−1 d χ(p) around p˜Kn describes
c dp c
the critical window (1.5) as well, and that, interestingly, p˜Kn does not equal either 1/n or
c
1/(n−1). It would be interesting to see whether their methods can be applied to the current
setting of percolation on H(d,n).
1.7. Susceptibility of the subcritical ERRG. In Section 2 we prove Theorem 1.1, and
also derive refined asymptotics for the susceptibility of a subcritical ERRG, its second moment,
and its surplus: given a connected graph G, let Sp(G) := |E(G)|−|V(G)|+1 denote the
number of surplus edges in G. Besides being interesting in their own right, these will be
crucial for proving the lower bound on p(d), because the restriction of critical percolation on
c
H(d,n) to a one-dimensional subspace of H(d,n) is equivalent to a subcritical ERRG. To
prove the lower bound of Theorem 1.1 we rely on the following asymptotics, which, to the
best of our knowledge, are sharper than results in the literature:
Theorem 1.2 (Second order asymptotics for susceptibility of the subcritical ERRG). Let
G = G(n,p) be the ERRG with p = λ and 0 < λ < 1. Then as n → ∞,
n−1
1 2λ2−λ4
χ (p) = E [|C(v)|] = − n−1+O(n−2), (1.15)
G p 1−λ 2(1−λ)4
1
E [|C(v)|2] = +O(n−1), (1.16)
p (1−λ)3
λ3
E [Sp(C(v))] = n−1+O(n−2). (1.17)
p 2(1−λ)2
The second-order coefficient computed in (1.15) improves the result by Durrett in [13,
Theorem 2.2.1], which states that χ (p) = (1−λ)−1 −O(n−1), while (1.16) provides the
G
matching lower bound to well-known upper bound derived with the usual branching process
domination. To achieve the sharper asymptotics we need a new way to encode the usual
breadth-first search in the ERRG with the help of a branching random walk. We believe
that there exists an infinite polynomial expansion of χ (p) in powers of p for all p = λ
G n−1
with 0 < λ < 1. There is substantial literature related to (1.17), see e.g. the classic book on
random graphs by Bollob´as [8, Section 5.2] as well as the seminal paper by Janson, Knuth
L(cid:32) uczak and Pittel [30] computing generating functions of components having various cycle
structures. As far as we are aware, the second order asymptotics in (1.17) is new.
1.8. Outline. We prove Theorem 1.1 by separately proving a lower bound and an upper
bound on p(d). In Section 2 we prove Theorem 1.2. This theorem is used in Section 3 to prove
c
the lower bound in Theorem 1.1 with the help of an exploration process that uses the fact
that the restriction of critical bond percolation on H(d,n) to a one-dimensional subspace has
the same distribution as a subcritical ERRG. This is used to obtain a sharp enough branching
process upper bound on the susceptibility. In Section 4 we estimate connection probabilities
and estimate bubble, triangle and polygon diagrams. In Section 5 we prove the upper bound
in Theorem 1.1 with the help of the lace expansion. Perhaps surprisingly, these disparate
methods yield compatible bounds, due to the fact that both methods are asymptotically
sharp. The lace expansion method may be improved to prove Theorem 1.1, but this would be
EXPANSION OF PERCOLATION CRITICAL POINTS FOR HAMMING GRAPHS 7
more difficult than our current proof and less interesting. We do not see how the exploration
process proof could be improved to also prove the upper bound in Theorem 1.1.
2. Susceptibility of the subcritical Erdo˝s-Re´nyi Random Graph
In this section we prove Theorem 1.2. To give our estimate of the expected size of a
subcritical cluster, we couple a breadth-first exploration process of the cluster to a process
related to a Branching Random Walk (BRW). The breadth-first exploration exploration
process is defined in Section 2.1, the branching random walk exploration in 2.2. The proof of
the susceptibility asymptoticis is given in Section 2.3.
2.1. Breadth-first/surplus exploration. We start by defining a version of the breadth-
first (BF) exploration. This is a very standard tool in the study of the ERRG (see e.g. [23,
Section 5.2.1]). In a nutshell, a breadth-first exploration is a process that, starting from a
vertex v, “discovers” its adjacent edges, “activating” the direct neighbours of v in some fixed
order, and then explores those vertices, discovering their adjacent edges and activating any
unexplored, unactivated neighbours, and so on, always choosing the vertex that was activated
the longest time ago as the next vertex to explore from. The BF exploration keeps track of
which vertices have been explored (the “dead” set), which vertices have been activated but
not explored (the “active” set), and the time at which a vertex was activated or explored.
Crucially, the “traditional” BF exploration will only explore a vertex once, so the process
terminates once all vertices are explored, and the edges associated with newly activated
vertices describe a subtree of the component of v, but the process provides little information
about the surplus, i.e., the discovered edges that do not activate new vertices (also sometimes
referred to as the “tree excess” of the graph). For our purposes it is important that we also
know about the surplus, so we consider the following modification of the BF exploration:
Definition 2.1 (BF exploration process of a graph). Given a graph G = (V,E) and a vertex
v ∈ V we define the breadth-first/surplus (BF) exploration process as the sequence of dead,
active and surplus sets (D(t),A(t),Sp(t)) as follows:
t≥0
(cid:66) Initiation. Initiate the exploration with the dead, active and surplus sets at time
t = 0 as
D(0) := ∅, A(0) := {v}, Sp(0) := ∅, (2.1)
and at time t = 1 as
D(1) := {v},
A(1) := {w : {v,w} ∈ E}, (2.2)
Sp(1) := ∅.
(cid:66) Time t ≥ 2. Choose the vertex vt ∈ A(t−1) that minimizes min{i : vt ∈ A(i)},
breaking ties according to an arbitrary but predetermined rule.5 Update the active,
5An example of such a rule: Fix an order on the vertex set V. If at step t−1 we have explored and/or
activated a total of k vertices, and we activate (cid:96) more at step t, then we assign to these (cid:96) newly explored
vertices the labels k+1 through k+(cid:96), according to the order on V. At time s+1 we explore from the active
vertex with the smallest label.
8 L.FEDERICO,R.VANDERHOFSTAD,F.DENHOLLANDER,ANDT.HULSHOF
dead and surplus sets as follows:
D(t) := D(t−1)∪{vt},
A(t) := (A(t−1)\{vt})∪{w ∈/ A(t−1)∪D(t−1) : {vt,w} ∈ E}, (2.3)
Sp(t) := Sp(t−1)∪{{vt,w} ∈ E : w ∈ A(t−1)}.
(cid:66) Stop. Terminate the exploration when A(t) = ∅. Set T = t.
NotethatD(t)andA(t)aresubsetsofV, whereasSp(t)isasubsetofE. WhenA(t) = ∅, this
means that we have completely explored the connected component C(v) and T = |D(T)| =
|C(v)|. IntheBFwefindanewedgeeverytimeweactivateavertex(excepttheinitialvertexv)
orwediscoveranedgebetweenactivevertices. Itfollowsthat|E(C(v))| = |D(T)|−1+|Sp(T)|.
We conclude that that |Sp(T)| = |E(C(v))|−|D(T)|+1 = Sp(C(v)).
2.2. The branching random walk exploration. The subtree generated by a “traditional”
BF exploration is often studied through a comparison to a branching process (see e.g. [13,23]).
To study our BF exploration, we define a suitable extension, the branching random walk
(BRW) exploration, in which we randomly embed a branching process in the graph, and keep
track of its self-intersections.6 This is made precise in the following definition:
Definition 2.2 (Branching random walk). Given an m-regular graph G = (V,E) and
m
p ∈ [0,1], we define the p-branching random walk (p-BRW) on G started at v ∈ V as the
m
pair (T,φ ), where T is a Bin(m,p) Galton-Watson tree, and φ is a random mapping of T
v v
into the vertex set V whose law satisfies: (1) φ maps the root ρ of T to v; (2) given any node
v
x ∈ T and its set of children C(x) ⊂ T, the marginal law of φ (C(x)) is the same as that of
v
|C(x)| distinct neighbours of φ (x) in G chosen uniformly at random, independently for all
v m
x ∈ T. (Here, for a set A ⊂ T and a mapping φ : T → G, we define φ (A) = ∪ φ (a), and
v v a∈A v
by convention set φ (∅) = ∅.)
v
Next, we define a process that explores a p-BRW and keeps track of any self-intersections.
Briefly, the idea is that we explore the p-BRW by exploring the tree T in a breadth-first
fashion from the root upward. If the p-BRW intersects its own trace, then we declare the
particle that intersected, and all its offspring, to have become “ghosts”. We differentiate
between particles that became ghosts through intersecting with active and dead vertices.
In Proposition 2.4 below we prove that this exploration process can be coupled to a BF
exploration of a percolation cluster:
Definition 2.3 (BRW exploration process). Given an m-regular graph G = (V,E), a
m
vertex v ∈ V, and a p-BRW (T,φ ) on G , we define the BRW exploration process
v m
(A(t),D(t),PA(t),PD(t))T as the sequence of dead, active, active ghost and dead ghost sets as
t=0
follows:
(cid:66) Initiation. Initiate the exploration with the dead, active, active ghost and dead
ghost sets at time t = 0 as
D(0) := ∅, A(0) := {ρ}, PA(0) = ∅, PD(0) = ∅, (2.4)
6From now on the term nodes will refer to elements of GW trees, while vertices will refer to elements of
graphs. Moreover, the progeny of a node x will indicate the set of vertices whose path to the root ρ passes
through x, while the children of x are only the vertices for which x is the first vertex encountered on such a
path. We write C(x) for the set of children of x in T.
EXPANSION OF PERCOLATION CRITICAL POINTS FOR HAMMING GRAPHS 9
and at time t = 1 as
D(1) := {ρ}, A(1) := {y ∈ C(ρ)}, PA(1) := ∅, PD(1) := ∅. (2.5)
(cid:66) Time t ≥ 2. Choose the node xt ∈ A(t−1) that minimizes min{i : xt ∈ A(i)}, breaking
ties according to an arbitrary but predetermined rule, and update the exploration as
follows:
D(t) := D(t−1)∪{xt},
A(t) := (A(t−1)\{xt})∪(cid:8)y ∈ C(xt) : φ (y) ∈/ φ (cid:0)D(t−1)∪A(t−1)(cid:1)(cid:9),
v v
(2.6)
PA(t) := PA(t−1)∪(cid:8)y ∈ C(xt) : φ (y) ∈ φ (cid:0)A(t−1)(cid:1)(cid:9),
v v
PD(t) := PD(t−1)∪(cid:8)y ∈ C(xt) : φ (y) ∈ φ (cid:0)D(t−1)(cid:1)(cid:9).
v v
(cid:66) Stop. If A(t) = ∅, then terminate the exploration. Set T = t.
Using the BRW exploration, we define the subgraph C˜(v) as the graph traced out by a
p-BRW where the particles are killed when they intersect with the active set. More precisely,
we let T˜ be the subtree in T induced by D(T)∪PA(T), and define
C˜(v) := (cid:0)φ (D(T)),{{φ (x),φ (y)} : {x,y} ∈ T˜}(cid:1). (2.7)
v v v
Note that, by Definition 2.3, φ (D(T)∪PA(T)) = φ (D(T)), so C˜(v) is indeed a subgraph of
v v
G = (V,E).
m
We now show that C˜(v) has the same law as C(v), the connected component of v in an
ERRG, by coupling the BF and BRW explorations:
Proposition 2.4 (Coupling of BF and BRW explorations). Consider percolation on an
m-regular graph G with parameter p. Consider the BF exploration on the percolated graph
m
G (p) and the p-BRW exploration processes on G , both starting from the vertex v (and
m m
using the same tie-breaking rule). Then C(v) with respect to P has the same law as C˜(v).
p
Proof. WeshowinductivelythatwecancoupleeachstepoftheBRWandoftheBFexploration
in such a way that C˜(v) = C(v) almost surely. We start by showing that there exists a
coupling such that for all t ≥ 0,
D(t) = φ (D(t)),
v
A(t) = φ (A(t)),
v
(2.8)
Sp(t) = (cid:91)(cid:8){φ (xs),w} : w ∈ φ (PA(s)\PA(s−1))(cid:9).
v v
s≤t
We start with the inductive base. At time t = 0, by Definitions 2.1 and 2.3,
D(0) = ∅ = φ (D(0)),
v
A(0) = {v} = φv({ρ}) = φv(A(0)), (2.9)
Sp(0) = ∅ = (cid:8){φ (x0),w} : w ∈ φ (PA(0))(cid:9).
v v
Next, we prove the inductive step: the induction hypothesis is that the relations in (2.8) holds
for all r < t. We extend the coupling so that it also holds at time t. Our assumption is that
we use the same tie-breaking rule for both explorations, so by the induction hypothesis we
choose vt = φ (xt).
v
10 L.FEDERICO,R.VANDERHOFSTAD,F.DENHOLLANDER,ANDT.HULSHOF
Given φ (xt), fix a set Ut = {u1,u2,...,uk} of k neighbours of φ (xt). By Definition 2.2,
v k v
the mapping φ is such that |C(xt)| neighbours of φ (xt) are distinct neighbours chosen
v v
uniformly at random, so
P(cid:0)φ (C(xt)) = Ut(cid:1) = P(|C(xt)| = k)P(cid:0)φ (C(xt)) = Ut | |C(xt)| = k(cid:1)
v k v k
(cid:18)m(cid:19) (cid:18)m(cid:19)−1 (2.10)
= pk(1−p)m−k = pk(1−p)m−k.
k k
Next,considertheBFexplorationattimet. Given(D(s),A(s),Sp(s))t−1,wecandetermine
s=0
vt. For t ≥ 0, let N(t) denote an independent set-valued random variable that contains
the vertex w with probability p, independently for all w such that {vt,w} ∈ E, so that
P(cid:0)N(t) = Ut(cid:1) = pk(1−p)m−k. For every set Ut of neighbours of vt we have P(cid:0)N(t) = Ut(cid:1) =
k k k
P(cid:0)φ (C(xt)) = Ut(cid:1), and so there exists a trivial coupling of N(t) and φ (C(xt)) such that
v k v
P(cid:0)N(t) = φ (C(xt)(cid:1) = 1.
v
Consider an edge {vt,w}. Observe that if w ∈/ D(t − 1), then {vt,w} has not been
discovered by the exploration, so it is open in the percolation conditionally independently with
probability p, while if w ∈ D(t−1), then {vt,w} has been discovered in the BF exploration,
so its status can be determined from (D(s),A(s),Sp(s))t−1. Let X(t) denote the vertices
s=0
that are end-points of edges that are discovered in the t-th step, i.e.,
X(t) := (cid:8){vt,w} : {vt,w} ∈ C(v)\{{vt,u} : u ∈ D(t−1)}(cid:9). (2.11)
Note that if w ∈ X(t), then either w becomes activated at time t or w ∈ A(t−1). By the
above observation, we can couple X(t) to N(t) such that X(t) = N(t)\D(t−1) almost surely,
conditionally on (D(s),A(s),Sp(s))t−1.
s=0
Consider henceforth the setting in which X(t), N(t), φ (C(vt)), and (D(s),A(s),Sp(s))t−1
v s=0
aresimultaneously coupledaccordingtotheabovedescription. (SincebothN(t)andφ (C(xt))
v
are essentially independent p-random subsets, it is easy to make this coupling explicit; we
leave those details to the reader.) Using this coupling, the induction hypothesis (2.8), and
Definitions 2.1 and 2.3, we derive
D(t)\D(t−1) = {vt} = φ(xt) = φ (D(t))\φ (D(t−1)), (2.12)
v v
and
A(t)\A(t−1) = X(t)\A(t−1)
= N(t)\(D(t−1)∪A(t−1))
(2.13)
= (φ (C(xt))\(φ (D(t−1))∪φ (A(t−1)))
v v v
= φ (A(t))\φ (A(t−1)),
v v
and
Sp(t)\Sp(t−1) = {{vt,w} : w ∈ X(t)∩A(t−1)}
= {{vt,w} : w ∈ N(t)∩A(t−1)}
(2.14)
= {{φ (xt),w} : w ∈ φ (C(xt))∩φ (A(t−1))}
v v v
= {{φ (xt),w} : w ∈ φ (PA(t)\PA(t−1))}.
v v
Since φ (xt) = vt, we obtain that (2.8) holds also at time t almost surely, and thus, by
v
induction, for all t ∈ {0,1,...,T}, almost surely.