Critical window for the vacant set left by random walk on random regular graphs

0.52 MB·English

by  Jiri Cerny

CRITICAL WINDOW FOR THE VACANT SET LEFT BY RANDOM
WALK ON RANDOM REGULAR GRAPHS
JIRˇ´I CˇERNY´1 AND AUGUSTO TEIXEIRA2
Abstract. We consider the simple random walk on a random d-regular graph with n
vertices, and investigate percolative properties of the set of vertices not visited by the walk
until time un, where u > 0 is a fixed positive parameter. It was shown in [CˇTW11] that
this so-called vacant set exhibits a phase transition at u=u : there is a giant component
(cid:63)
if u<u and only small components when u>u . In this paper we show the existence of
(cid:63) (cid:63)
a critical window of size n−1/3 around u . In this window the size of the largest cluster is
(cid:63)
1 of order n2/3.
1
0
2

n
Introduction
1.
a
J
The study of percolative properties of the vacant set left by a random walk on finite
0
1 graphs was initiated by Benjamini and Sznitman [BS08] for the case of random walk on a
high-dimensional discrete torus (Z/NZ)d. In [BS08] it is proved that if the random walk
]
R runs up to time uNd, where u is a small constant, the vacant set has a giant component
P with volume of order Nd, asymptotically as N grows. On the other hand, if instead we
.
h consider a large constant u, all components of the vacant set have a volume of order at most
at logλN (for some λ > 0) as was proved in [TW10]. This shows the existence of two distinct
m
phases for the connectivity of the vacant set on the torus as u varies. However, the above
[ mentioned works leave several open questions, such as whether the transition between these

1 two phases happens abruptly at a given critical threshold and, if this is the case, how does
v the vacant set behave at the critical point?
8
Such problems are much better understood when instead of the torus one considers ran-
7
9 dom d-regular graphs or d-regular large-girth expanders on n vertices. In this case, when
1
random walks runs up to time un, it is known that the vacant set exhibits a sharp phase
.
1 ˇ
transition [CTW11]: the size of the largest connected component of the vacant set drops
0
1 abruptly from order n to order logn at a computable critical value u(cid:63). In this paper we
1 explore more closely this phase transition, in particular we prove that the size of the largest
:
v component of the vacant set exhibits a double-jump, similar to that observed in Erd˝os-R´enyi
i
X random graphs.
r Let us now give a precise definition of the model. Let Gn,d be the set of all non-oriented
a
d-regular simple graphs with n vertices (here and later we tacitly assume that nd is even).
Let Pn,d be the uniform probability distribution on Gn,d. For any graph G = (V,E) let PG
denote the canonical law on the Skorokhod space D([0,∞),V) of a continuous-time simple
random walk on G started from the uniform distribution. We use (X ) to denote the
t t≥0
Date: 10 January 2011.
1 Department of Mathematics, ETH Zurich, Raemistrasse 101, 8092 Zurich, Switzerland.
2 Department of Mathematics and their Applications, Ecole Normale Sup´erieure, 45 rue d’Ulm, F-75230
Paris, France.
This research was partially supported by the AXA Research Fund Fellowship. A.T. thanks the
Forschungsinstitut fu¨r Mathematik (FIM) of ETH Zurich, where a part of this work has been completed,
for its hospitality.
1
canonical coordinate process. For a fixed parameter u ≥ 0, we define the vacant set as the
set of all vertices not visited by the random walk before the time u|V|,
(1.1) Vu = Vu = V \{X : 0 ≤ t ≤ u|V|}.
G t
We use Cu to denote the maximal connected component of the subgraph of G induced
max
by Vu. The vacant set and in particular its maximal connected component are the main
objects of investigation in this paper.
ˇ
As proved in [CTW11], the phase transition in the connectivity of the vacant set occurs
at the value u given by
(cid:63)
d(d−1)ln(d−1)
(1.2) u = ,
(cid:63) (d−2)2
and can be described as follows: Let Gn be a graph distributed according to Pn,d. Then
with Pn,d-probability tending to one as n → ∞:
Super-critical phase: For any u < u and σ > 0 there exist ρ and c depending on u,
(cid:63)
σ, and d, such that
(1.3) PGn[|Cu | ≥ ρn] ≥ 1−cn−σ.
max
Sub-critical phase: For any u > u and σ > 0 there exist K and c depending on u,
(cid:63)
σ, and d, such that
(1.4) PGn[|Cu | ≤ Klogn] ≥ 1−cn−σ.
max
In this paper we study the behaviour of the vacant set in the vicinity of the critical point.
The main results of this paper are the following two theorems. In their statement we use
P to denote the annealed measure
n,d
(cid:90)
(1.5) Pn,d(·) = PG(·)Pn,d(dG).
We say that an event A occurs P -asymptotically almost surely (or simply P -a.a.s.) if
n,d n,d
lim P (A) = 1.
n→∞ n,d
Theorem 1.1 (Critical window). Let (u ) be a sequence satisfying
n n≥1
(1.6) |n1/3(u −u )| ≤ λ < ∞ for all n large enough.
n (cid:63)
Then for every ε > 0 there exists A = A(ε,d,λ) such that for all n large enough
(1.7) P [A−1n2/3 ≤ |Cun | ≤ An2/3] ≥ 1−ε.
n,d max
If u is not in the critical window, then the maximal connected component behaves
n
differently:
Theorem 1.2. (a) When (u ) satisfies
n n≥1
(1.8) u −u −n−→−∞→ 0, and ω := n1/3(u −u ) −n−→−∞→ ∞,
(cid:63) n n (cid:63) n
then for v = 2n2/3ω d−2 e−u(cid:63)(d−2)/(d−1) and for every ε > 0
n n(d−1)2
(cid:12)|Cun | (cid:12)
(1.9) (cid:12) max −1(cid:12) ≤ ε P -a.a.s.
(cid:12) (cid:12) n,d
v
n
(b) When (u ) satisfies
n n≥1
(1.10) u −u −n−→−∞→ 0, and ω := n1/3(u −u ) −n−→−∞→ −∞,
(cid:63) n n (cid:63) n
2
then for every ε > 0 there exists B = B(ε) > 0, such that for all n large enough
(cid:2) (cid:3)
(1.11) P |Cun | ≤ Bn2/3|ω |−1/2 ≥ 1−ε.
n,d max n
The above theorems confirm that the maximal connected component of the vacant set
behaves similarly as the largest connected cluster of the Bernoulli percolation on random
regular graphs, see [ABS04, NP10, Pit08]. Remark that the upper bound in part (b) of
Theorem 1.2 seems to be non-optimal. The result is however sufficient to confirm that the
width of the window is n−1/3. To improve such a statement, it is necessary to obtain better
estimate in Theorem 3.2(ii) that we quote below.
The methods of this paper are largely inspired by the recent article by Cooper and Frieze
[CF10], where the authors develop a new technique to prove (1.3) and (1.4). This technique
ˇ
is very specific to deal with random regular graphs, in contrast with the results of [CTW11]
which hold for a more general class of graphs, including e.g. large girth expanders. In
the present article we extend the methods in [CF10] to the critical case. We believe that
ˇ
obtaining such an extension from the techniques in [CTW11] should be rather difficult.
The crucial observation of [CF10], allowing for a very elegant proof of (1.3), (1.4) for
random regular graphs is the following. Under the annealed measure (1.5), given the infor-
mation about the graph discovered by the random walk up to time u|V|, the subgraph of
G induced by the vacant set Vu is distributed as a random graph uniformly chosen within
the set of graphs with a given (random) degree sequence, see Proposition 3.1 below.
The paper [CF10] further uses the fact that the behaviour of the uniform random graphs
Gd with a given degree sequence d : V → N is sufficiently well known. More precisely, as
follows from [MR95], there exists a single parameter Q = Q(d) (see (3.6) below) such that
Q < 0 implies that G is typically sub-critical (i.e. has only small components), and Q > 0
d
implies that G is supercritical (i.e. has a giant component). Moreover, the recent paper
d
[HM10] establishes the existence of an intermediate regime (the so-called critical window)
when Q(d) converges to zero at a certain rate as n tends to infinity, see also [JL09]. The
results mentioned in this paragraph that will be useful in this paper are summarised in
Theorem 3.2 below.
The principal contribution of this paper is thus to obtain sufficiently sharp estimates on
the random degree sequence of the vacant set Vu, and consequently on the value of the
parameter Q, see Theorems 5.1, 5.2 and (6.3) below. Weaker estimates of this type were
shown in [CF10], which combined with [MR95], allowed them to deduce (1.3), (1.4).
We should remark that [CF10] contains also a statement on the critical behaviour. More
precisely, Theorem 2(iii) of [CF10] states that for some u = u (1 + o(1)) (which might
n (cid:63)
be random), the size of Cmunax is n2/3+o(1), Pn,d-a.a.s. Our results considerably improve this
statement.
Notealsothatmuchmoreisknownabouttherandomgraphswithagivendegreesequence,
seeforinstancetheresultsin[FR09]and[JL09]. Oftenthehypothesisoftheseresultscanbe
shown to hold true for Vu, using Theorems 5.1 and 5.2. If this is the case, their conclusions
will also apply to Vu (P -a.a.s), providing us with more information on the geometry of
n,d
the vacant set.
As an example of such application, we obtain the following improvement on the statement
(1.3) about the super-critical behaviour of the vacant set.
Theorem 1.3. Let u < u . Then there is ρ = ρ(u,d) ∈ (0,1) such that for every ε > 0
(cid:63)
(1.12) n−1|Cu | ∈ (ρ−ε,ρ+ε) P -a.a.s.
max n,d
3
Intheabovestatement,thevalueofρcanbeexplicitlycalculated,see(6.9)below. Remark
also that to obtain the above theorem, our precise estimates on Q are not necessary, in fact
the precision obtained in [CF10] would have been sufficient.
Finally, let us briefly describe Theorems 5.1 and 5.2. The former, establishes an estimate
on the expected degree distribution of Vu, by approximating the probability that a random
walk visits a neighbourhood of a given vertex x ∈ V before time un. For this, we make use
of the well known relation between random walks on graphs and discrete potential theory, as
well as the pairing construction introduced by Bolloba´s, which we detail in Section 2. Then
in Theorem 5.2 we prove that with high probability the degree sequence of Vu concentrates
around its expectation. This is done using a standard concentration inequality, together
with the fast mixing properties of the random walk on a random regular graph.
This paper is organised as follows. In Section 2 we introduce some of the notation needed
in the paper and the pairing construction of random regular graphs. In Section 3, we recall
the results of [CF10, HM10, JL09] needed later. Section 4 contains precise estimates on
the behaviour of the simple random walk on random regular graphs. In Section 5, we give
the estimates on the degree sequence of the vacant set. Theorems 1.1–1.3 are proved in
Section 6. The Appendix summarises some general facts concerning random walks on finite
graphs.
Notation and definitions
2.
2.1. Basic notation. We now introduce some basic notation. Throughout the text c or
c(cid:48) denote strictly positive constants only depending on d, with value changing from place
to place. Dependence of constants on additional parameters appears in the notation. For
instance cu denotes a positive constant depending on u and possibly on d. We write N =
{0,1,...} for the set of natural numbers, and [d] for the set {1,...,d}. For a set A we
denote by |A| its cardinality. For any sequence of probability measures P and events A
n n
we say that A holds P -a.a.s. (asymptotically almost surely), when lim P [A ] = 1.
n n n→∞ n n
In this paper, the term graph stands for a finite simple graph, that is a graph without
loops or multiple edges. Sometimes we intentionally allow the graph to have loops and/or
multiple edges and in this case we use the term multigraph. For arbitrary (multi)graph
G = (V,E), we use dist(·,·) to denote the usual graph distance and write B(x,r) for the
closed ball centred at x with radius r, that is B(x,r) = {y ∈ V : dist(x,y) ≤ r}.
We use G (resp. M ) to denote the set of all d-regular graphs (resp. multigraphs)
n,d n,d
with vertex set Vn = {1,...,n}. Given a degree sequence d : Vn → N, we use Gd to denote
the set of graphs for which every vertex x ∈ V has the degree d = d(x). Similarly, M
n x d
stands for the set of such multigraphs; here loops are counted twice when considering the
degree. Pn,d and Pd denote the uniform distributions on Gn,d and Gd respectively.
2.2. Pairing construction. In order to study properties of random regular graphs, Bol-
loba´s (see e.g. [Bol01]) introduced the so-called pairing construction, which allows to gen-
erate such graphs starting from a random pairing of a set with dn elements. The same
construction can be used to generate a random graph chosen uniformly at random from
G . Since this pairing construction will be important in what follows, we give here a short
d
overview of it.
(cid:80)
From now on, whenever we consider a sequence d : Vn → N, we suppose that x∈Vndx is
even. Given such a sequence, we associate to every vertex x ∈ V , d half-edges. The set of
n x
half-edges is denoted by H = {(x,i) : x ∈ V ,i ∈ [d ]}. We write H for the case d = d
d n x n,d x
for all x ∈ V . Every perfect matching M of H (i.e. partitioning of H into |H |/2 disjoint
n d d d
4
pairs) corresponds to a multigraph G = (V ,E ) ∈ M with
M n M d
(cid:8) (cid:8) (cid:9) (cid:9)
(2.1) E = {x,y} : (x,i),(y,j) ∈ M for some i ∈ [d ], j ∈ [d ] .
M x y
We say that the matching M is simple, if the corresponding multigraph G is simple, that
M
¯
is GM is a graph. With a slight abuse of notation, we write Pd for the uniform distribution
on the set of all perfect matchings of H , and also for the induced distribution on the set
d
of multigraphs Md. It is well known (see e.g. [Bol01] or [McD98]) that a P¯d distributed
multigraph G conditioned on being simple has distribution Pd, that is
(2.2) P¯d[G ∈ ·|G ∈ Gd] = Pd[G ∈ ·],
and that, for d constant, there is c > 0 such that for all n large enough
(2.3) c < P¯n,d[G ∈ Gn,d] < 1−c.
¯
These two claims allow to deduce Pn,d-a.a.s. statements directly from Pn,d-a.a.s. statements.
The main advantage of dealing with matchings is that they can be constructed sequen-
tially: To construct a uniformly distributed perfect matching of H one samples without
d
replacements a sequence h ,...,h of elements of H in the following way. For i odd, h
1 |Hd| d i
can be chosen by an arbitrary rule (which might also depend on the previous (h ) ), while
j j<i
if i is even, h must be chosen uniformly among the remaining half-edges. Then, for every
i
1 ≤ i ≤ |H |/2 one matches h with h .
d 2i 2i−1
It is clear from the above construction that, conditionally on M(cid:48) ⊆ M for a (partial)
matching M(cid:48) of H , M \M(cid:48) is distributed as a uniform perfect matching of H \{(x,i) :
d d
(x,i) is matched in M(cid:48)}. Since the law of the graph G does not depend on the labels ‘i’
M
of the half-edges, we obtain for all partial matchings M(cid:48) of H
d
(2.4) P¯d[GM\M(cid:48) ∈ ·|M ⊃ M(cid:48)] = P¯d(cid:48)[GM ∈ ·],
where d(cid:48) is the number of half-edges incident to x in H that are not yet matched in M(cid:48),
x (cid:12) d (cid:12)
that is d(cid:48)x = dx − (cid:12){{(y1,i),(y2,j)} ∈ M(cid:48) : y1 = x,i ∈ [dx]}(cid:12), and GM\M(cid:48) is the graph
corresponding to a non-perfect matching M \M(cid:48), defined in the obvious way.
2.3. Random walk notation. For an arbitrary multigraph G = (V,E), we use PG to
x
denote the law of canonical continuous-time simple random walk on G started at x ∈ V,
that is of the Markov process with generator given by
(cid:88)
(2.5) Lf(x) = pxy(f(y)−f(x)), for f : V → R,x ∈ V.
y∈V
Here p = n /d , n is the number of edges connecting x and y in G, and d is the degree
xy xy x xy x
of x; the loops are counted twice in n and d .
xx x
We write PG,(cid:96) for the restriction of PG to D([0,(cid:96)],V) and PG,(cid:96) for the law of random walk
x x xy
bridge, thatisforPG,(cid:96) conditionedonX = y. WewriteEG,EG,(cid:96),EG,(cid:96) forthecorresponding
x (cid:96) x x xy
expectations. The canonical shifts on D([0,∞),V) are denoted by θ . The time of the n-th
t
jump is denoted by τ , i.e. τ = 0 and for n ≥ 1, τ = inf{t ≥ 0 : X (cid:54)= X }◦θ +τ .
n 0 n t 0 τn−1 n−1
The process counting the number of jumps before time t is denoted by N = sup{k : τ ≤ t}.
t k
Note that, when G is simple, under PxG, (Nt)t≥0 is a Poisson process on R+ with intensity
one. We write Xˆn, n ∈ Z+, for the discrete skeleton of the process Xt, that is Xˆn = Xτn.
Given A ⊂ V, we denote by H and H˜ the respective entrance and hitting time of A
A A
(2.6) H = inf{t ≥ 0 : X ∈ A}, and H˜ = H ◦θ +τ .
A t A A τ1 1
We denote by π the stationary distribution for the simple random walk on G, which is
uniform if G is d-regular (even if G is not simple). PG stands for the law of the simple
5
random walk started at π and EG for the corresponding expectation. For all real valued
functions f,g on V we define the Dirichlet form
1 (cid:88) (cid:88)
(2.7) D(f,g) = (f(x)−f(y))(g(x)−g(y))π p = − Lf(x)g(x)π .
x xy x
2
x,y∈V x∈G
The spectral gap of G is given by
(cid:8) (cid:9)
(2.8) λ = min D(f,f) : π(f2) = 1,π(f) = 0 .
G
From [SC97], p. 328, it follows that for d-regular graphs,
(2.9) sup |P [X = y]−π | ≤ e−λGt, for all t ≥ 0.
x t y
x,y∈V
It is also a well known fact (see e.g. [Fri08]) that there exist α > 0 such that
¯
(2.10) λG > α, both Pn,d-a.a.s. and Pn,d-a.a.s.
Preliminaries
3.
3.1. Distribution of the vacant set. Recall the notation Vu = Vu for the vacant set of
G
the random walk on the graph G = (V,E) at level u, (1.1). For the purpose of this paper,
it is suitable to define a closely related object, the vacant graph Vu = (V,Eu) where
(3.1) Eu = {{x,y} ∈ E : x,y ∈ Vu}.
G
It is important to notice that the vertex set of Vu is V and not Vu, in particular Vu is not
the graph induced by Vu in G. Observe however that the maximal connected component
of the vacant set C (defined before in terms of the graph induced by Vu in G) coincides
max
with the maximal connected component of the vacant graph Vu (except when Vu is empty,
but this difference can be ignored in our investigations).
We use Du : V → N to denote the (random) degree sequence of Vu, and write Qu for
n,d
the distribution of this sequence under the annealed measure P¯ , defined by P¯ (·) :=
n,d n,d
(cid:82) PG(·)P¯n,d(dG).
The following proposition from Cooper and Frieze [CF10] allows us to reduce questions
on the properties of the vacant set Vu of the random walk on random regular graphs to
questions on random graphs with given degree sequences.
Proposition 3.1 (Lemma 6 of [CF10]). For every u ≥ 0, the distribution of the vacant
graph Vu under P¯n,d is given by P¯d where d is sampled according to Qun,d, that is
(cid:90)
(3.2) P¯n,d[Vu ∈ ·] = P¯d[G ∈ ·]Qun,d(dd).
Although a proof of Lemma 3.1 can be found in [CF10], we provide a proof here for the
sake of completeness.
Proof. Let M be a P¯n,d-distributed pairing of Hn,d and let X be a random walk on G = GM.
Define M ⊂ M to be the set of all pairs of half-edges incident to a vertex X with s ≤ t,
t s
(cid:8) (cid:9)
(3.3) M = {(x,i),(y,j)} ∈ M : x ∈ {X : s ≤ t},i ∈ [d] .
t s
It is easy to see that the edges of the vacant graph Vu correspond exactly to the pairs in
M\M , thatisVu = G . Inparticular, Du(x)isthenumberofthehalf-edgesincident
un M\Mun
to x not matched in M . Denoting by F the σ-algebra generated by ((X ,M ),s ≤ un),
un u s s
the above implies that Du is F -measurable.
u
6
It follows from (2.4) that, conditionally on F , the distribution of G only depends
u M\Mun
¯
on the sequence of half-edges that are not matched in Mun, and is given by PDu. More
precisely,
(3.4) P¯n,d[Vu ∈ ·|Fu] = P¯n,d[GM\Mun ∈ ·|Du] = P¯Du[G ∈ ·],
and thus
(cid:90)
(3.5) P¯n,d[Vu ∈ ·] = P¯n,d(cid:2)P¯n,d[Vu ∈ ·|Fu](cid:3) = P¯n,d(cid:2)P¯Du[G ∈ ·](cid:3) = P¯d[G ∈ ·]Qun,d(dd),
where the last equality follows from the definition of Qu . This concludes the proof of
n,d
(cid:3)
Proposition 3.1.
3.2. Behaviour of random graphs with a given degree sequence. Wenowsummarise
the results about the behaviour of random graphs with a given degree sequence which will
be used in this paper. For a degree sequence d : Vn → N we define
(cid:80)n d2
(3.6) Q(d) = x=1 x −2,
(cid:80)n d
x=1 x
and set n (d) to be the number of x ≤ n with d = i,
i x
(cid:12) (cid:12)
(3.7) n (d) = (cid:12){x ∈ V : d = i}(cid:12).
i n x
For any graph G we use C (G) to denote the maximal connected component of G.
max
The following theorem summarises the results of [MR95, JL09, HM10] needed later.
Theorem 3.2. Let (dn)n≥1, dn : Vn → N, be a sequence of degree sequences. We assume
that the degrees are uniformly bounded (i.e. max{dn : n ≥ 1,x ≤ n} ≤ ∆ < ∞), and that
x
n (dn) ≥ ζn for a ζ > 0.
1
(i) (critical window) If |Q(dn)| ≤ λn−1/3 for all n ≥ 1, then for every ε > 0 there exists
A = A(ζ,λ,ε,∆) such that for all n large enough
P¯dn[A−1n2/3 ≤ |Cmax(G)| ≤ An2/3] ≥ 1−ε.
(ii) (below the window) If lim n1/3Q(dn) = −∞ and lim Q(dn) = 0, then for
n→∞ n→∞
every ε > 0 exists B = B(ζ,ε,∆) < ∞ such that for all n large enough
(cid:104) (cid:112) (cid:105)
P¯dn |Cmax(G)| < B n/|Q(dn)| > 1−ε.
(iii) (above the window) Let lim n1/3Q(dn) = +∞ and lim Q(dn) = 0. In addi-
n→∞ n→∞
tion, assume that
n (dn)
(3.8) lim i = p , for all 0 ≤ i ≤ ∆,
i
n→∞ n
for some probability distribution (p ) on {0,...,∆}, and set λ = (cid:80)∆ ip ,
i 0≤i≤∆ i=0 i
β = (cid:80)∆ i(i−1)(i−2)p , and v = 2nλ2β−1Q(dn). Then, for every ε
i=0 i n
(cid:12)|C (G)| (cid:12)
(cid:12)(cid:12) max −1(cid:12)(cid:12) < ε, P¯dn-a.a.s.,
v
n
(iv) (super-critical regime) Let lim Q(dn) = Q > 0 and assume that (3.8) holds.
n→∞ ∞
Let g be the generating function of (p ), g(x) = (cid:80)∆ p xi. Then there exists a unique
i i=0 i
solution ξ to g(cid:48)(x) = λx in (0,1), and for ρ = 1−g(ξ) and any ε > 0
(cid:12)|C (G)| (cid:12)
(cid:12)(cid:12) max −ρ(cid:12)(cid:12) ≤ ε, P¯dn-a.a.s.
n
7
Proof. Parts (i), (ii) correspond to Theorems 1.1 and 1.2 of [HM10], where these statements
are proved under more general assumptions. In particular, [HM10] does not require the
uniform upper bound ∆ on the maximal degree. The restriction to the uniformly bounded
degree sequences implies that the constant R(dn) used in [HM10] satisfies c < R(dn) < c−1
for all n large enough and is therefore immaterial for our purposes.
Parts (iii), (iv) are taken from Theorems 2.3 and 2.4 of [JL09]. When reading those
theorems it is useful to realise that the (p )-distributed random variable D used in [JL09]
i
satisfies E[D] = λ and that E[D(D−2)] = λQ∞ in our notation.
Remark however that neither [HM10], or [JL09] consider degree sequences with vertices
of degree zero, that is with n (dn) > 0. It can however be seen easily, that if n (dn) does
0 0
not exceed ζ(cid:48)n, ζ(cid:48) < 1 (which is implied by the assumptions of the theorem), the vertices
of degree zero do not have any influence on the existence of the giant cluster, they only
change the constants A, B in (i), (ii). For (iii), (iv), when n (dn)/n → p (cid:54)= 0, one applies
0 0
the theorem for the modified sequence (d¯n) where all vertices of degree 0 are omitted. The
new degree sequences d¯n are functions on V , with n¯ = n(1 − p ) + o(1). They satisfy
n¯ 0
n (d¯n)/n¯ −n−→−∞→ p /(1−p ) =: p¯. Therefore, denoting by the letters with bars the quantities
i i 0 i
related to the distribution (p¯), we obtain Q(d¯n) = Q(dn), λ¯ = λ/(1−p ), β¯ = β/(1−p ),
i 0 0
g¯(x) = (g(x)−p )/(1−p ), and ξ¯= ξ. This implies that v¯ = v and ρ¯n¯ = ρn, confirming
0 0 n¯ n
that zero-degree vertices have no influence on the asymptotics of the size of the maximal
(cid:3)
connected component. This completes the proof.
Random walk estimates
4.
This section contains estimates on the random walk on random regular multigraphs which
will be useful later in order to estimate the typical degree sequence of the vacant graph Vu.
We start by introducing some notation. We use Td to denote the infinite d-regular tree
with root ∅. For a d-regular multigraph G = (V,E), a map φ from Td → V is said to be a
covering of G from x ∈ V, if φ(∅) = x, and for every y ∈ Td, φ maps the d neighbours of y
inTd totheneighboursofφ(y)inG, includingthemultiplicitiesandtheloops. Ford-regular
multigraphs constructed by the pairing construction this means that the neighbours of y
are sent by φ to the vertices which are paired with (φ(y),i), i ∈ [d].
d
In agreement with our previous notations, PT denotes the law of the continuous-time
y
simple random walk on Td starting from y. It is important to notice that fixing a covering
φ from x, the image by φ of a random walk in Td with law PTd is distributed as PG.
∅ x
For every finite connected A ⊂ Td and z ∈ A we now define the escape probabilities
(4.1) e (z) = PTd[H˜ = ∞],
A z A
which can be calculated explicitly in practical examples using the fact that
d−2
(4.2) PTd[H = ∞] = ,
y ∅ d−1
for every neighbour y of ∅. This comes from a standard calculation for a one-dimensional
simple random walk with drift, see for instance [Woe00], proof of Lemma (1.24).
We use zi ∈ Td, i ∈ [d] to denote the neighbours of ∅ listed in some predefined order. For
x ∈ V, let φ be a covering of G from x. From now on, if G was obtained by the pairing
x
construction, we require that φ (z ) is the vertex matched with (x,i). Otherwise φ can be
x i x
chosen arbitrarily, since our statements will not depend on which particular choice of φ is
x
picked. For any D ⊂ [d], we define the sets
(4.3) BD = {∅}∪{zi : i ∈ D} and Bx,D = {x}∪{φx(zi) : i ∈ D}.
8
φ(z1)
z1 φ
x
∅
z3 φ(z2)
z2 φ(z3)
Figure 1. A covering φ of a regular graph from x.
The sets B will be used later in the calculation of the degree distribution of the vacant
x,D
set: using an inclusion-exclusion formula, we can express the event {the degree of x in Vu
is k} in terms of events {B ⊂ Vu}, for D ⊂ [d].
x,D
We first prove a technical lemma describing the graph V after removing the set B .
x,D
Lemma 4.1. For K > 0, we say that the graph G is K-good, if there is no x ∈ V and
D ⊂ [d] such that:
(a) B(x,3) is a tree, and
(b) either V \B is disconnected or diam(V \B ) ≥ Klogn.
x,D x,D
Then, there is K > 1 such that G is K-good, P¯n,d-a.a.s.
Proof. To prove this claim we use Lemma 2.14 of [Wor99] which states that P¯n,d-probability
that there exists a (s,j)-separating set in G, that is a set S ⊂ V with |S| = s such that
G\S contains a component of exactly j vertices, satisfies
(cid:16)j +s(cid:17)j(d−1)
(4.4) P¯n,d[G has an (s,j)-separating set] ≤ 32+s/d 2 n2s(j +s)23s.
n
(This formula is used in [Wor99] to prove that G is Pn,d-a.a.s. d-connected. However, the
d-connectedness cannot be used directly to prove our claim, since |B(x,1)| = d+1.)
Observe first that it is sufficient to consider D = [d]. Indeed, if x is such that B(x,3) is
a tree and the graph is connected after removing B , then it is connected after removing
x,[d]
B for any D ⊂ [d]. In addition, obviously diam(G\B ) ≤ diam(G\B )+2.
x,D x,D x,[d]
Let now D = [d], that is B = B(x,1). Assume that G\B(x,1) is disconnected. Then
x,D
removing the set B(x,1) \ {x} from G, divides the graph into at least three components.
One of them is {x}, and the other ones are are contained in G\B(x,1). Since we require
that B(x,3) is a tree and since G is d-regular, the size of these other components is at least
4. We thus apply (4.4) with s = d and j ≥ 4: The P¯n,d-probability that there is x ∈ V such
that B(x,3) is a tree and G\B(x,1) is disconnected is bounded from above by
n/2
(cid:88)
¯
(4.5) Pn,d[G has an (s,j)-separating set].
j=4
The largest term in this sum, corresponding to j = 4, is O(n4−32d) = o(1). All the remaining
terms are much smaller. Actually, as in the proof of Theorem 2.10 of [Wor99], it can be
shown that (4.5) tends to 0 as n → ∞. Hence, P¯n,d-a.a.s. there is no x ∈ V such that B(x,3)
is a tree and G\B(x,1) is disconnected.
To bound the diameter of G\B(0,1) we use the fact that P¯n,d-a.a.s. diam(G) ≤ 2logd−1n,
see [BF82]. We may also assume that G\B(x,1) is connected, which as we proved occurs
¯
Pn,d-a.a.s.
9
We now claim that
removing one vertex v of degree d in an arbitrary graph while keeping it
(4.6)
connected can increase the diameter of the graph at most by a factor of 3d−1.
To prove this claim we first consider the removal of an edge: Removing one edge e from a
graph while keeping in connected can increase the diameter of the graph at most by factor 3.
To see this it is sufficient to consider the shortest path µ in G\{e} connecting the vertices
of e (such path must exist since G\{e} is connected, and cannot be longer than 2diamG),
and to replace the edge e by the path µ in every geodesic of G that contains e.
Having understood the removal of edges, we can analyse the removal of a vertex v. We
first remove all but one of the edges of G incident to v, in this procedure the diameter of
the graph is multiplied by at most 3d−1. Removing v together with the last edge linking it
to G\{v} yields the claim (4.6).
The claim (4.6) and |B(x,1)| ≤ d+1 then imply that P¯n,d-a.a.s
(4.7) diam(G\B(x,1)) ≤ 3(d−1)(d+1)diamG ≤ 3(d−1)(d+1) ·2log n.
d−1
(cid:3)
This completes the proof of the lemma.
We now start controlling how the random walk visits the sets B defined in (4.3). In
x,D
the lemma below, we show that the probability of escaping from B for a large time can
x,D
be approximated by the escape probability on the infinite tree, defined in (4.1).
Lemma 4.2. For every x ∈ V , D ⊂ [d] and i ∈ D
n
(cid:104)(cid:12) (cid:12)(cid:105) clog4n
(4.8) E¯n,d (cid:12)(cid:12)PφGx(zi)[H˜Bx,D > log2n]−eBD(zi)(cid:12)(cid:12) ≤ n .
and
clog2n
(4.9) P¯ [H˜ ≤ log2n] ≤ .
n,d Bx,D n
Proof. To simplify the notation we write B, B, z and φ(z) for BD, Bx,D, zi and φx(zi). Using
the fact that φx maps a random walk on Td to a random walk on G, we can write
PG [H˜ ≤ log2n] = PTd[H˜ ≤ log2n]
(4.10) φ(z) B z φ−x1(B)
= PTd[H˜ ≤ log2n]+PTd[H˜ ≤ log2n,H˜ > log2n].
z B z φ−x1(B)\B B
Therefore, the left-hand side of (4.8) can be bounded from above by
(cid:12) (cid:12) (cid:104) (cid:105)
(4.11) (cid:12)(cid:12)PzTd[H˜B > log2n]−eB(z)(cid:12)(cid:12)+E¯n,d PzTd[H˜φ−x1(B)\B ≤ log2n] .
Using the Markov property at time log2n and the definition of e (z), the first term equals
B
PTd[log2n < H˜ < ∞]
z B
(4.12) (cid:104) d−2 (cid:105) (cid:104) (cid:105)
≤ PzTd d(∅,Xlog2n) ≤ 2d log2n + sup PuTd H∅ < ∞ .
u:d(u,∅)>d2−d2log2n
Since d(Xt,∅) under PzTd is a random walk on N with expected drift given by (d − 2)/d,
both terms above are bounded by cexp{−c(cid:48)log2n}.
To bound the second term in (4.11), note that if at time t the random walk on Td started
from z visits a point in φ−x1(Bx,D)\B, then the trajectory of the image walk on G together
10