Percolation for the stable marriage of Poisson and Lebesgue with random appetites

0.14 MB·English

by  Daniel Andrés D\'\iaz Pachón

Percolation for the stable marriage of Poisson and
0
1 Lebesgue with random appetites
0
2

g Daniel D´ıaz∗
u
A
August 20, 2010

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1

]
R Abstract
P
LetΞbeasetofcenterschosenaccordingtoaPoissonpointprocessin
.
h Rd. Letψ be anallocationofRd toΞ inthe senseofthe Gale-Shapley
t
a marriageproblem, with the additional feature that every center ξ ∈Ξ
m
has a random appetite αV, where α is a nonnegative scale constant
[ andV isanonnegativerandomvariable. Generalizingpreviousresults

by Freire, Popov and Vachkovskaia ([3]), we show the absence of per-
3
colation depending on certain characteristics of the moment of V.
v
5 Key words: Poisson process, percolation, stable marriage, random
2 appetite.
3
5
. 1 Introduction
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0
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Let Ξ be a set of centers chosen according to a Poisson point process of
0
: law P and expectation E with intensity λ. Let V be a nonnegative random
v variable (r.v.) with law P and expectation E. Consider also a nonnegative
i
X constant α ∈ R.
r The product αV will be called appetite. Every center ξ ∈ Ξ will have
a
a random appetite in such a way that appetites corresponding to different
centers are independent of each other and identically distributed. The ap-
petites are also independent of Ξ. Keeping in mind the independence (with
a little abuse of notation) we will denote the appetite of a center ξ by αV ,
ξ
when it does not lead to confusion.
The elements x of Rd will be called sites. Let ψ be a function of Rd to
Ξ∪{∞,∆}; we will call ψ an allocation. The goal of ψ is to allocate every
site to a center. If ψ cannot warrant that allocation, then the site will be
∗Instituteof Mathematics and Statistics - Sao Paulo University. [email protected]
1
sent to the element ∞. Also, there is a null set of sites under the Lebesgue
measure L which will be sent to ∆. Given a center ξ, its territory will be
given by ψ−1(ξ). The allocation ψ satisfies the restriction L ψ−1(ξ) ≤ αV
ξ
(i.e., the volume of the territory of each center is bounded by its appetite).
(cid:2) (cid:3)
A center ξ will be called unsated if L ψ−1(ξ) < αV ; we will say that ξ is
ξ
sated if L ψ−1(ξ) = αV . Whenever ψ(x) = ξ for some ξ ∈ Ξ\∆ we will
ξ
(cid:2) (cid:3)
say that x is claimed; if ψ(x) = ∞ we will call it unclaimed.
(cid:2) (cid:3)
Consider a site x and a center ξ such that ψ(x) ∈/ {ξ,∆}. We will say
that the site x desires the center ξ if |x−ξ| < |x−ψ(x)| or if ψ(x) = ∞
(here and henceforth |·| is the Euclidean norm). If there exists x′ ∈ ψ−1(ξ)
such that |x−ψ(ξ)| < |x′ −ψ(ξ)|, or if ξ is unsated, we will say that the
center ξ covets the site x. A pair (x,ξ) will be unstable if x desires ξ and
ξ covets x. If ψ does not produce any unstable pair it will be called stable.
Finally, call C the closure of the set of claimed sites.
Hoffman,Holroyd,andPeresgaveanexplicitconstructionofthefunction
ψ for the case of constant appetite (see Theorem 1 in [7]) that can be easily
extended to the general case of random appetites so we will omit it here.
Suchconstruction usestheGale-Shapley stablemarriage algorithm (see [4]).
Informally, it can be thought as all the centers growing a ball at the same
linear speed, where each center will claim for itself all the sites in the ball
not belonging to the territory of any other center. The process continues
for each center ξ up to the point in which ξ is sated (has claimed for itself
a territory with volume αV ). If its appetite is never satisfied, the ball will
ξ
continue growing to ∞ trying to find unclaimed sites and the center ξ will
be unsated. Some nice graphics for the model with constant appetite can
be found in [7].
A formal construction for the probability space for this model can be
found in [2], based on the construction proposed by Meester and Roy for
the Boolean model of continuum percolation (see [10], pp. 16–17). Suffice
it to say that it warrants the independence between appetites and centers,
and the independence among the appetites for different centers. Besides, it
also warrants the application of some ergodic properties needed below.
We denote the model by (Ξ,αV,ψ) with joint law P and expectation E.
Since ψ is translation-equivariant (i.e., ψ (x) = ξ implies that ψ (Tx) =
Ξ TΞ
Tξ, for every translation T ∈ Rd and where ψ is an allocation to the
Ξ
support [Ξ] of Ξ), we have that P is translation invariant. Its Palm version
(Ξ∗,αV∗,ψ∗), with law P∗ and expectation operator E∗, is conditioned to
have a center on the origin. Note that since α is independent of Ξ, then α∗
and Ξ∗ are also independent and α = α∗.
Thorissonshowedin[12](whichwasthebeginningofthecurrentresearch
2
in allocations) that Ξ and Ξ∗ can be coupled so that almost surely one is a
translation oftheother. Therefore,ψΞ∗ isdefinedP∗-a.s. Finally, remember
thatsinceΞisaPoissonprocess,Ξ∗ isaPoissonprocesswithanaddedpoint
at the origin. (See a similar reasoning in [7] p. 1256 and [9] pp. 8–9).
2 Previous results
Besides the existence of the stable allocation for a discrete set of centers,
in [1] there were some generalizations of previous results in [7]. Here we
mention, without proof, some relevant versions for our purpose:
Theorem 2.1 (Almost sure uniqueness). Let Ξ be a Poisson point pro-
cess with law P and finite intensity λ, where each center ξ chooses its ap-
petite with law P, then there exists an L-a.e. unique stable allocation ψ to
Ξ, P-a.s.
Theorem 2.2 (Phase transitions). Let Ξ be a Poisson point process with
law P and intensity λ ∈ (0,∞).
• If λαEV < 1 (subcritical), then all the centers are sated but there
exists an infinite volume of unclaimed sites, P-a.s..
• If λαEV = 1 (critical), then all the centers are sated and L-a.e. site
is claimed, P-a.s.
• If λαEV > 1 (supercritical), then there are unsated centers but L-
a.e. site is claimed, P-a.s.
Proposition 2.1 (Monotonicity on the centers). Let Ξ ,Ξ be two set
1 2
of centers chosen according to a Poisson processes such that Ξ ⊆ Ξ . Take
1 2
{V } , a set of iid r.v. distributed like V. Fix {v } , for a realization
ξ ξ∈Ξ2 ξ ξ∈Ξ2
of {V } , as the appetites of the centers Ξ , and take {v : ξ ∈ Ξ ∩Ξ }
ξ ξ∈Ξ2 2 ξ 1 2
for a realization of the appetites of Ξ . Let ψ and ψ be their respective
1 1 2
(L-a.e. unique) stable allocations, then |x−ψ (x)| ≥ |x−ψ (x)| for L-a.e.
1 2
x ∈Rd, P-a.s.
Proposition 2.2 (Monotonicity on the appetites). Let Ξ be a set of
centers, let {v } be a realization of {V } . Let ψ and ψ be two stable
ξ ξ∈Ξ ξ ξ∈Ξ 1 2
allocations with appetites α v and α v , respectively, where α ≤ α , then
1 ξ 2 ξ 1 2
|x−ψ (x)| ≥ |x−ψ (x)|, for L-a.e. x ∈ Rd, P-a.s.
1 2
3
Remark 1. Theorems 2.1 and 2.2 have more general statements in
[7] (and [1]). Theorem 2.1 was proved for every translation-invariant point
process, and Theorem 2.2 was proved for every point process ergodic under
translations. For our interest, the versions so stated will suffice.
Remark 2. For the two last propositions, C ⊆ C , where C and C
1 2 1 2
are the closure of the claimed sites for the stable allocations ψ and ψ ,
1 2
respectively.
Remark 3. Proposition 2.2 admits another way to generalize it (also
found in [1]): Let V and V be two independent nonnegative random vari-
1 2
ables such that V is stochastically dominated by V . Then there exists a
1 2
coupling (Vˆ ,Vˆ ) of V and V such that C ⊆ C , for the stable allocations
1 2 1 2 1 2
ψ and ψ , respectively, where the appetites of the centers are chosen ac-
1 2
cording to the law of Vˆ for the allocation ψ , and with the law of Vˆ for the
1 1 2
allocation ψ .
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3 Statement of results
Definition 3.1 (Percolation). There is percolation if there exists an un-
bounded connected subset of C.
Let us denote A the event {0 belongs to a connected unboundedsubset
α
of C in the d-dimensional model with appetites given by αV }.
i
Definition 3.2 (Critical point).
α (d) := sup{α :P[A ]= 0}.
p α
It is noted immediately that, by Theorem 2.2, there is percolation of
C whenever α ≥ (λEV )−1, since L-a.e. site is claimed, P-a.s. Therefore,
ξ
regarding percolation, one of the mostinteresting questions is whether there
exists α (d) > 0 (i.e., whether there exits a positive α such that there is no
p
percolation of C).
Take a small δ ∈ R+ and consider a random variable V′ to [δ ,∞) with
1 1
law P′, mean µ, variance σ2 and such that V is stochastically dominated by
V′. DenotebyP′ andbyE′ theprobabilityandtheexpectation,respectively,
of the model (Ξ,αV′,ψ). The result to be proved in this article is:
Theorem 3.1. (i) Let {V′} be a collection of nonnegative iid r.v.’s dis-
ξ
tributed like V′; suppose that there exists δ ∈ R+ such that v2+δdP(v) <
v≥0
∞. Let D be the diameter of the connected component of C containing the
R
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origin. Then, for every s, there is a small enough α such that EDs < ∞.
Thus, there is not percolation of C.
(ii) Under the same conditions for small enough α there is percolation
of Rd\C (i.e., there is an unbounded connected component of the unclaimed
sites).
The r.v. V′ is chosen because of a proof technicality. However, given
Remark 3 and Theorem 3.1, there exists a coupling (V,V′) of V and V′ such
thatthereisnotpercolation oftheclaimed sitesbythemodelwithappetites
given by αV for small enough α.
In order to prove Theorem 3.1 , we willrely on a strongresultby Gou´er´e
on subcritical regimes of Boolean models. It was based on previous develop-
mentsbythesameauthor(see[5])inwhichheimprovedthephasetransition
results for the Poisson Boolean model (see [10]). Before state it, we define
some concepts.
Let Π be a point process on Rd such that its law is translation-invariant
on Rd and its intensity is locally finite. Let ρ be a r.v. with locally finite
law ν on (0,∞). The Boolean model can now be defined: The points of Π
will be centers of circumferences with random radii given by ρ. Denote this
model by (Π,ρ). Call the joint probability of the Boolean model P and its
ν
expectation E .
ν
Theorem 3.2. Let S be the connected component of the Boolean model
containing the origin. Also, define M := sup kxk and let C > 0. Now
x∈S
consider the following four properties:
P1. For all r > 0 and all x ∈ Rd \B(0,Cr), these two point process are
independent:
Π∩B(0,r)×(0,r] and Π∩B(x,r)×(0,r]
P2.
suprdν([r,∞)) ≤ H;
r>0
P3.
βdν(dβ) < ∞;
Z[1,∞)
P4. Given s, a positive real,
βd+sν(dβ) < ∞;
Z[1,∞)
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where H is positive constant which depends only on the dimension d and
the constant C. Thus, if properties P1–P3 are satisfied, then S is P -a.s.
ν
bounded. If properties P1–P4 are satisfied, then E [Ms] is finite.
ν
The proof of Theorem 3.2 is quite involved and can be found in [6]. The
next two paragraphs present a sketch to understand the first part of the
proof (that S is P -a.s. bounded). This sketch will allow us to use some of
ν
theses concepts for the proof of Theorem 3.1 (ii).
β
Let (Π,ρ ) be the Boolean model restricted to the balls with radii in
γ
the closed interval [γ,β], where γ > 0, β > 0. Say that the event G(x,γ,β)
occurs if and only if there is a connected component from B(x,β) to Rd \
β
B(x,2β) using only balls of (Π,ρ ).
γ
Very roughly, to prove the first part of Theorem 3.2 it suffices to show
that P [M > β] → 0 when β → ∞. To do this, Gou´er´e proves that P [M >
ν ν
β] is bounded above by pG(β) := lim P [G(0,γ,β)] plus an error term
ν γ→0 ν
that is bounded and that goes to 0 when β goes to ∞:
P [M > β]≤ pG(β)+D˜ rdν(dr),
ν ν
Z[β,∞)
where D˜ is a positive constant. However,
pG(β) → 0 when β → ∞, (1)
ν
which proves the result.
4 Proofs
Theproof of Theorem 3.1 (i) presented hereis based on the work by Gou´er´e
in [6]. The part (ii) is proved with an idea developed by Freire, Popov and
Vachkovskaia in [3]. However, part (i) of Theorem 3.1 can be proven using
fractal percolation, as in [3]; nonetheless, the methodology proposed in [6]
allows a more general result (the finiteness of EMs for all positive s) and is
simpler.
Let Ξ beaPoisson point process in Rd whoseintensity measureis λ. For
every ξ ∈ Ξ we define R(ξ,Ξ) as follows:
R(ξ,Ξ) := inf r > 0: αV′ ≤ π (ξ)rd ,
 ζ d 
 ζ∈BX[ξ,2r] 
 
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inf∅ = ∞. Here π (ξ)rd is the volume of the d-dimensional ball centered at
d
ξ with radius r; B[ξ,r] is the closed ball with center ξ and radius r. Note
that there is at least one center in the closed ball B[ξ,2R] and its appetite
would be at least αδ ; therefore, the minimum value of R is (αδ′π−1)1/d and
1 d
we will denote it by b (here and henceforth π is the volume of a ball in d
d
dimensions with unit radius). We can now construct a Boolean model with
its centers being Ξ and the the radii of every ξ ∈ Ξ given by R(ξ,Ξ).
The idea behind this concept was first used in [8] to prove some tail
bounds and after in [3] to prove the absence of percolation; both of these
cases related to the stable marriage of Poisson and Lebesgue with constant
appetites.
4.1 Domination
Lemma 4.1. Let Σ be the union of the balls of the Boolean model previously
constructed. If α is small enough, then C is contained in Σ, P′-a.s.
Proof. Let ξ ∈ Ξ and remember that R(ξ,Ξ) is finite. To prove the Lemma
we only need to check that ψ−1(ξ) ⊂ B[ξ,R], where R = R(ξ,Ξ). We have:
αV′ = π (ξ)Rd.
ζ d
ζ∈B[ξ,2R]
X
Take ε > 0 such that there are no points of Ξ in B[ξ,2R +2ε]\B[ξ,2R].
Then
αV′ < π (ξ)(R+ε)d.
ζ d
ζ∈B[ξ,2R+2ε]
X
Therefore,
L ψ−1(Ξ∩B[ξ,2R+2ε]) < π (ξ)(R+ε)d.
d
Then, there exists(cid:2)x ∈ B[ξ,R+ε] such tha(cid:3)t ψ(x) ∈ Ξ∪{∞}\B[ξ,2R+2ε].
If ψ(x) ∈Ξ,
|x−ψ(x)| > R+ε and |x−ξ|≤ R+ε.
Thus, x desires ξ. If ψ(x) = ∞, x, by definition, desires ξ. Since ψ is
stable, ξ does not covet x. As a consequence, ψ−1(ξ) ⊂ B[ξ,|x − ξ|] and
then ψ−1(ξ) ⊂ B[ξ,R+ε]. Taking ε > 0 arbitrarily small, we obtain that
ψ−1(ξ) ⊂ B[ξ,R].
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4.2 Large deviations
We need a classical result of Nagaev on large deviations of sums of random
variables (Corollary 1.8 of [11]) and also the Chernoff bounds for Poisson
random variables. Those results will be used to bound the the probability
of the event {R(ξ,Ξ) > r/2}, where r is a constant.
Lemma 4.2 (Large deviations of sums). Let S = n Z , where {Z }
n i=1 i i
is a sequence of iid r.v. with 0-mean and variance σ2. Define A+ :=
P 2+δ
u2+δdP(u) <∞ (δ as defined on Theorem 3.1). Then
u≥0
R
4+δ 2+δ −2e−(2+δ)x2
P[S > x]≤ nA+ x−(2+δ) +exp
n 2+δ 2+δ (4+δ)2nσ2
( )
(cid:18) (cid:19)
Remark 4. In [11], Lemma 4.2 was more general: the r.v.’s did not
need to be identically distributed and the integral utdP(u) existed for
u≥0
some t ≥ 2.
R
Remark 5. Although there are other Nagaev results for moments less
than 2, those results are notusefulfor our purposes,as will beexplained be-
low. Indeed, weneed something else to the finitenessof thesecond moments
—we need the finiteness of the “moment” A+ .
2+δ
Lemma 4.3 (Poisson large deviations). Let N be a r.v. with Poisson
distribution and mean 0 < λ < ∞ (denoted N ∼ P(λ)), then
P[N ≥ a]≤ e−λg(λ/a) for a > λ
where g(x) = (x−1−logx)/x and g(x) → ∞ when x → 0.
Lemma 4.4. Given the same set of conditions of Theorem 3.1, for r > 0,
then ν((r,∞]) ≤cr−d(1+δ), where c∈ R+.
Proof. Let r > 0. By definition of ν and Ξ we have:
ν((r,∞]) = E′(|{Ξ∩[0,1]d :R(ξ,Ξ) > r}|)
= E′ 1
{R(ξ,Ξ)>r}
 
ξ∈ΞX∩[0,1]d
≤ λP′[R(ξ,Ξ)> r/2] 
≤ λP′ V′ > π (ξ)rd2−dα−1 .
 ζ d 
ζ∈B[ξ,2r]
X
 
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Forb ∈ (1,r)letK = λπ rdbd andN ∼ P(λπ rdbd),thiswayN dominates
d d
Ξ∩B[ξ,r]. Let Vˆ = V′−µ and x = π (ξ)rd2−dα−1−Kµ (note that x has
i (cid:6)i (cid:7) d
order rd). With these considerations we have:
N
ν((r,∞]) ≤ λP′ V′ > π (ξ)rd2−dα−1
k d
" #
k=1
X
N
≤ λ P′ V′ > π (ξ)rd2−dα−1|N ≤ K P′[N ≤ K]+P′[N > K]
k d
( " # )
k=1
X
K
≤ λ P′ Vˆ > x +P′[N > K]
k
( " # )
k=1
X
Thus, using Lemma 4.2 for the first term in the braces and Lemma 4.3 for
the second one, we obtain for large enough r that
ν((r,∞]) ≤ λ c r−d(1+δ) +e−c2rd +e−c1(λ)rd
3
≤ cλnr−d(1+δ), o
for suitable constants c ,c and c, and where c (λ) → ∞ when λ → 0. The
3 2 1
last inequality is obtained for a small λ and because the first term, which is
decreasing slower than the other terms, has order r−d(1+δ) by the definition
of K. Note that the last expression goes to 0 when λ → 0.
4.3 Proof of Theorem 3.1
Given Lemma 4.1, we only need to check that our Boolean model satisfies
the conditions of Theorem 3.2. Now we proceed in the same way as in [6]:
P1: Let r > 0. For all ξ ∈ Ξ define
R˜(ξ,Ξ) = inf s ∈ [0,r] : αV′ ≤ π (ξ)sd
 ζ d 
 ζ∈BX[ξ,2s] 
And R˜(ξ,Ξ) = r if there isno such s. Note that, for all ξ ∈Ξ, if R(ξ,Ξ) < r
or R˜(ξ,Ξ) < r, then R˜(ξ,Ξ) = R(ξ,Ξ). So,
{ξ ∈ Ξ: R(ξ,Ξ) <r} = {ξ ∈ Ξ : R˜(ξ,Ξ) < r}.
Thus, for every r > 0, we have that {ξ ∈ Ξ∩B(x,r) : R(ξ,Ξ) < r} only
depends on Ξ∩ B(x,3r). By the independence of the Poisson process we
9
obtainthatifx ∈ Rd\B(0,6r), thenthethepointprocesses{ξ ∈ Ξ∩B(0,r) :
R(ξ,Ξ) < r}and{ξ ∈Ξ∩B(x,r): R(ξ,Ξ)< r}areindependent. Therefore,
C = 7 satisfies the property.
P2: Using first that V′ is upper bounded and using after Lemma 4.4,
we obtain
suprdν([r,∞)) = suprdν([r,∞))
r>0 r>b
≤ suprdcλr−d(1+δ)
r>b
= cλsupr−dδ
r>b
= cλb−dδ
for small enough λ. Note that r−dδ is decreasing, this is why we needed the
“moment” A+ and V′ > δ —to make the last expression upper bounded.
2+δ 1
P3 and P4. By Lemma 4.4 we obtain: rd+sν(dr) < ∞ for every
(0,∞)
s, given small enough λ.
R
Therefore, given small enough λ, we can use Theorem 3.2 to obtain that
EMs < ∞, for every s and by Lemma 4.1, we get that EDs < ∞. Thus,
rescaling Theorem 3.1 (i) is proved for fixed λ and small enough α.
To prove Theorem 3.1 (ii) consider the following sequence of events:
{every connected subset in Rd\C is bounded}
⊂{for any bounded K ⊂R2 there is a contour around K in C ∩R2}
⊂{for any bounded K ⊂R2 there is a contour around K in (Ξ,R)∩R2}
⊂{G(0,γ,β) occurs for every β ∈ R},
where (Ξ,R) is the Boolean model with centers given by Ξ and radii given
by R. By (1) and the Borel-Cantelli lemma, the last event has probability
0 when α is small enough. Thus, Theorem 3.1 (ii) is proved.
5 Acknowldegments
The author is grateful to Serguei Popov, his Ph.D. advisor, for his guidance
in the development of this work. He is also grateful to Ang´elica Mar´ıa
Vega for her timely and valuable comments. The author acknowledges the
support of CAPES during his research. Finally, he wants to thank Carolina
Hern´andez and Juan Sa´enz for their independent proof-reading and style
suggestions
10