Contact process on a dynamical long range percolation M. SeilerA. Sturm

2025-05-02 0 0 868.89KB 31 页 10玖币

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Contact process on a dynamical long

range percolation

M. Seiler∗A. Sturm†

November 27, 2023

In this paper we introduce a contact process on a dynamical long range percolation

(CPDLP) deﬁned on a complete graph (

V, E

). A dynamical long range percolation is a

Feller process deﬁned on the edge set

, which assigns to each edge the state of being

open or closed independently. The state of an edge

is updated at rate

and is open

after the update with probability

and closed otherwise. The contact process is then

deﬁned on top of this evolving random environment using only open edges for infection

while recovery is independent of the background. First, we conclude that an upper

invariant law exists and that the phase transitions of survival and non-triviality of the

upper invariant coincide. We then formulate a comparison with a contact process with

a speciﬁc infection kernel which acts as a lower bound. Thus, we obtain an upper bound

for the critical infection rate. We also show that if the probability that an edge is open is

low for all edges then the CPDLP enters an immunization phase, i.e. it will not survive

regardless of the value of the infection rate. Furthermore, we show that on

and under suitable conditions on the rates of the dynamical long range percolation the

CPDLP will almost surely die out if the update speed converges to zero for any given

infection rate λ.

Keywords — Contact process, evolving random environment, dynamical random graphs, interacting

particle systems, long range percolation

1 Introduction

The classical contact process on a ﬁxed graph describes the spread of an infection over time and

space. It has been studied intensively and many variations have been considered, see Section 3.1 for

more background. In this article we study a contact processes on a dynamical long range percolation

(CPDLP), in which infections over any distance are possible depending on whether the corresponding

edge is present, which also changes dynamically.

We assume that the underlying graph

= (

V, E

) is a connected and transitive graph with

bounded degree. The graph distance of

is denoted by

(

·,·

) and the set of all possible edges by

{x, y}

x, y ∈V, x ̸

. From now on we consider the complete graph (

V, E

) which we

also equip with the original graph distance d(·,·).

There are several notions of transitivity for graphs in the literature. Thus, we specify the notion

brieﬂy. Here a graph

is called transitive if for any pair of vertices

x1, x2∈V

and respectively for

any pair of edges

e1, e2∈E

, there exists a graph automorphism

which maps

and

A graph automorphism is a permutation on

which preserves the graph structure, i.e.

{x, y} ∈ E

iﬀ {ϕ(x), ϕ(y)} ∈ E. In the literature this is sometimes called vertex and edge transitivity.

The CPDLP (C

B) = (C

)

t≥0

is a Markov process on

(

)

× P

(

), where

(

) and

(

)

denote the power sets of

and

. We equip the space

(

)

× P

(

) with the topology induced

∗

Frankfurt Institute for Advanced Studies, Ruth-Moufang-Straße 1, 60438 Frankfurt am Main, Germany

E-mail: seiler@ﬁas.uni-frankfurt.de

†

Institute for Mathematical Stochastics, Georg-August-Universit¨at G¨ottingen, Goldschmidtstr. 7, 37077

G¨ottingen, Germany E-mail: anja.sturm@mathematik.uni-goettingen.de

arXiv:2210.08907v2 [math.PR] 23 Nov 2023

by pointwise convergence, i.e. (

Cn, Bn

)

∈ P

(

)

× P

(

) converges to (

C, B

) if

1{(x,e)∈(Cn,Bn)}→

1{(x,e)∈(C,B)}

for all (

x, e

)

∈ P

(

)

×P

(

)

Note that

(

)

×P

(

) is a partially ordered space with

respect to “

⊂

”. Furthermore, we denote by “

⇒

” weak convergence of probability measures on

P(V)× P(E). As usual we denote by |A|the cardinality of a set A.

We call Cthe infection process, which takes values in

(

). If

x∈

then we call

infected at

time

. The process Bdescribes an evolving edge random environment and takes values in

(

Thus, we call Bthe background process. If

e∈

we call

open at time

and closed otherwise.

Furthermore, we assume that Bevolves autonomously of C. Given that Bis currently in state

the transitions of the infection process Ccurrently in state Care for all x∈V,

C→C∪ {x}at rate λ· |{y∈C:{x, y} ∈ B}| and

C→C\{x}at rate r, (1)

where

λ >

0 denotes the infection rate and

r >

0 the recovery rate. We write C=C

when C

For the background dynamics we consider a dynamical long range percolation. Let (

ˆpe

)

e∈E ⊂

and (

ˆve

)

e∈E ⊂

,∞

) be sequences of real numbers such that

ˆp{x,y}

ˆp{x′,y′}

and

ˆv{x,y}

ˆv{x′,y′}

(

x, y

) =

(

x′, y′

). We exclude the trivial case that

ˆpe

= 0 for all

e∈ E.

Now the dynamical long

range percolation Bcurrently in state Bhas transitions

B→B∪ {e}at rate ˆveˆpeand

B→B\{e}at rate ˆve(1 −ˆpe)(2)

for all

e∈ E

. As initial distribution we choose B

0∼π

, where

is the invariant law of Bwhich

means that the events ({e∈Bt})e∈E are independent and P(e∈Bt) = ˆpefor all e∈ E and t≥0.

We will in particular be interested in the behavior of our process when we scale the percolation

and speed kernels. For this we will assume that they are of the form

ˆpe= ˆpe(q) := qpeand ˆve= ˆve(γ) := γve(3)

for all

e∈ E

for some

γ >

0 and

q∈

1] and ﬁxed kernels (

)

e∈E ⊂

1] and (

)

e∈E ⊂

,∞

A long range percolation model assigns to every edge

e∈ E

independently the state of being open

with probability

ˆpe

and otherwise closed. The term ”dynamical” means that we update the state

of every edge

as time evolves. This is done independently for every edge

{x, y}

at update

speed ˆvewhich depends only on the length d(x, y) of that edge. This yields a translation invariant

background dynamic, where we use that the graph Gis transitive.

Since we are in a long range setting we need some assumptions regarding the ﬂip rates of the

background process to ensure that the CPDLP is well-deﬁned.

In order to ensure that the transition rates of the infection process are not inﬁnite we need that

at any given time

the neighborhood of any vertex

remains ﬁnite. Therefore, we assume that the

sequence (ˆpe)e∈E and (ˆve)e∈E satisfy

y∈V\{x}

ˆv{x,y}ˆp{x,y}<∞and X

y∈V\{x}

ˆv−1

{x,y}<∞for all x∈V. (4)

Note that if the kernels are of the form (3) then (

)

e∈E ⊂

1] and (

)

e∈E ⊂

,∞

) satisfy these

assumptions iﬀ (ˆpe)e∈E ⊂[0,1] and (ˆve)e∈E ⊂(0,∞) do.

Remark 1.1. The assumptions in (4) imply that

ˆv{x,y}ˆp{x,y}→

0 and

ˆv{x,y}→ ∞

(

x, y

)

→ ∞

Since we also have that

ˆve>

0 for all

e∈ E

it follows that

infy:y̸=xˆv{x,y}>

0, where

does

not depend on the choice of

x∈V

due to translation invariance. This implies together with

(4)

that

0< C X

y∈V\{x}

ˆp{x,y}≤X

y∈V\{x}

ˆv{x,y}ˆp{x,y}<∞,

i.e. that the sequence (ˆp{x,y})y∈V\{x}is summable.

Thus, as a consequence of the assumptions in (4) the probability that a long edge is open, i.e. an

edge connecting two vertices over a long distance, becomes exceedingly small. Broadly speaking

this means that a successful infection over a long distance is getting more and more unlikely as

the distance increases. The second part of the assumption can be seen as assuming that all edges

attached to an arbitrary vertex are updated after a ﬁnite time. This might seem a bit unintuitive,

but we need this assumption for technical reasons. We discuss the necessity of this rate assumption

brieﬂy in Section 3 right after Problem 3.

In Section 5 we will explicitly construct the CPDLP via a graphical representation and then

show that under these assumptions the resulting process is in fact a well-deﬁned Feller process (see

Proposition 5.7) with state space

(

)

× P

(

) and that

t|<∞

almost surely for all

t≥

0 if

|C0|<∞, even if the background is started in E(see Proposition 5.6).

We are interested in the survival behavior of the CPDLP as the parameter

varies, and later on

also as

γ >

0 and

q∈

1) vary for percolation and speed kernels of the form (3). In the general

setting, we denote by

θ(λ, C) := P(CC

t̸=∅ ∀t≥0)

the survival probability of a CPDLP with infection parameter

and initial state C

and B

0∼π

(and all other parameters ﬁxed).

We denote the critical infection rate for survival by

λc:= inf{λ≥0 : θ(λ, {x})>0},

where

x∈V

is chosen arbitrary. Note that by translation invariance it follows that

(

λ, {x}

) =

(

λ, {y}

) for all

x, y ∈V

. Furthermore, this together with the additivity of the infection process C

implies that

(

λ, C

)

0 for some

C⊂V

with 0

<|C|<∞

then this is true for all such sets. Thus,

the deﬁnition of the critical rate

λc

does not depend on the choice of the set of initially infected

vertices as long as it is ﬁnite and non-empty.

Furthermore, by standard methods and using the monotonicity of the system, see also Remark 5.3,

we get the existence of the upper invariant law

, which is the weak limit of the process started

with (C

) = (

V, E

). (Whenever relevant we will indicate the dependence of

on the parameters

of the model with subscripts.) The upper invariant law is the largest invariant law according to

the stochastic order, i.e. if

is an invariant law of the CPDLP, then

ν⪯ν

, where ”

⪯

” denotes the

stochastic order. Of course, there exists the trivial invariant law

δ∅⊗π

. This poses the question for

which parameters it holds that

δ∅⊗π

. Since it is not diﬃcult to see that

δ∅⊗π

is the smallest

invariant law possible,

δ∅⊗π

is equivalent to ergodicity of the system, i.e. that there exists a

unique invariant law which is the weak limit of the process. We deﬁne the critical infection rate for

non-triviality of the upper invariant law by

λ′

c:= inf{λ > 0 : νλ̸=δ∅⊗π}.

2 Main results

Our ﬁrst result is that the critical infection rate of survival and the critical infection rate for a

non-trivial upper invariant law are the same.

Theorem 2.1. The two critical infection rates coincide, i.e. λ′

c=λc.

The next result provides a coupling of the CPDLP with a contact process that has a general

infection kernel. As a consequence we obtain that if this contact process survives then this implies

survival of the CPDLP, and thus this leads to a suﬃcient criterion for a positive survival probability

of the CPLDP. We ﬁrst deﬁne the contact process Xwith an infection kernel (

)

e∈E ⊂

,∞

) and

recovery rate

r >

0 on the complete graph (

V, E

). We additionally assume that

a{x,y}

a{x′,y′}

d(x, y) = d(x′, y′) and X

y∈V\{x}

a{x,y}<∞(5)

for all x∈V. If Xis currently in the state Cit has the transitions

C→C∪ {x}at rate X

y∈C

a{x,y}and

C→C\{x}at rate r.

(6)

This process can again be constructed via a graphical representation and it is a well known fact that

(5)

is satisﬁed then Xis a well-deﬁned Feller process on the state space

(

), see for example

[

Lig12

, Propostion I.3.2] and [

Swa09

]. As usual we indicate the initial conﬁguration

C⊂V

adding a superscript XC, i.e. XC

0=C.

Theorem 2.2. Let

C⊂V

and (C

)

t≥0

be a CPDLP with parameter

λ, q

and

. Then there

exists a contact process (XC

t)t≥0with XC

0=Cand with infection kernel

ae(λ) := 1

2λ+ ˆve−p(λ+ ˆve)2−4λˆveˆpe≥0

for all e∈ E such that XC

t⊂CC

tfor all t≥0. Thus, in particular

λc≤λc,

where

λc

inf{λ >

0 :

(

X{x}

t̸

∅ ∀t≥

}

is the critical infection rate for survival of

(which is again independent of x∈V).

The following results are concerned with the behavior of survival as we scale percolation probability

and speed with the parameters

and

. Thus, we assume that the percolation and speed kernels are

as in (3) and consider the survival probability

and the critical infection parameter

λc

as functions

of γand q.

Remark 2.3. Note that in this setting a CPDLP with rates

λ, r, γ, q

has the same dynamics as a

CPDLP with rates

λ/r,

, γ/r, q

when time is sped up by a factor of

, and the survival probabilities

are the same. Thus, the survival behavior of a CPDLP with rates

λ, r, γ, q

can be deduced from the

survival behavior of a CPDLP with rates

λ,

, γ, q

. In other words, it is not necessary to explicitly

study the dependence on r.

As a consequence of Theorem 2.2 we obtain the following result for fast update speed.

Corollary 2.4. Let X

∞

be a contact process with infection kernel (

λqpe

)

e∈E

and denote the

corresponding critical infection rate by

λ∞

c(q) = inf{λ > 0 : Pλ,q (X{x},∞̸=∅ ∀t≥0) >0}.

Then we have

lim sup

γ→∞

λc(γ, q)≤lim

γ→∞ λc(γ, q) = λ∞

c(q)<∞.

The next result shows that for any ﬁxed speed if we have overall a low probability that an edge of

any length is open, i.e. for

small enough, we are in the immunization region for the CPDLP. This

means that the critical infection rate is inﬁnite and so no matter how large the infection rate is the

CPDLP will die out almost surely

Theorem 2.5. For any ﬁxed

γ >

0there exists

(

)

∈

1] such that Cdies out almost

surely for all

q < q0

, regardless of the choice of

λ >

0, i.e.

λc

(

γ, q

) =

∞

for all

q < q0

, and such

that

λc

(

γ, q

)

<∞

for all

q > q0

. Moreover, the function

γ7→ q0

(

)is monotone non-increasing on

(0,∞).

As a corollary we can also get more insight into the behavior of the critical infection rate

λc

(

γ, q

)

as a function of

and in particular into its asymptotic behavior as

γ→

0. Note that while it is clear

due to monotonicity that q7→ λc(γ, q) is a monotone non-increasing function, see also Remark 5.3,

this is not so clear for the function

γ7→ λc

(

γ, q

). Nonetheless, it can be shown (see Proposition 5.4)

that

λc

(

γ, q

) can at most increase linearly in

which implies that there exists a

γ0

(

) such that

λc

(

γ, q

) =

∞

for all

γ < γ0

(

) and

λc

(

γ, q

)

<∞

for all

γ > γ0

(

). Note that

γ0

(

) must be ﬁnite for

any

q >

0 due to Corollary 2.4 but that it may be 0. However, the following corollary states that for

small enough

we have

γ0

(

)

0 such that a nontrivial immunization phase exists. For this we now

set

q1:= sup

γ∈(0,∞)

q0(γ) = lim

γ→0q0(γ),(7)

where we have used the monotonicity of

γ7→ q0

(

) stated in Theorem 2.5. This means that

by Theorem 2.5 for every

q < q1

there exists a

γ >

0 such that

q < q0

(

)

< q1

which implies

λc(γ, q) = ∞and thus also γ0(q)>0. In summary, we have the following statement:

Corollary 2.6. For every

q∈

1] there exists a

γ0

(

)

∈

,∞

)such that

λc

(

γ, q

) =

∞

for all

γ < γ0

and

λc

(

γ, q

)

<∞

for all

γ > γ0

. Furthermore, there exists a

q1∈

1], see

(7)

, so that for

every

q < q1

we have

γ0

(

)

0while for every

q > q1

we have

γ0

(

)=0

This implies in particular

for every q < q1that limγ→0λc(γ, q) = ∞.

For general countable vertex sets

we can only determine that

λc

(

γ, q

)

→ ∞

when

γ→

0 if

q < q1

is small enough. But in the special case

and

{{x, y} ⊂ Z

|x−y|

= 1

}

, i.e. when

= (

V, E

) is the 1-dimensional integer lattice we can conclude that this is the case for all

q <

under some further assumptions. In fact, these assumptions even guarantee that for any ﬁxed

cannot have survival if the update speed γis small enough.

Theorem 2.7. Consider

to be the 1-dimensional integer lattice. Let

q <

1be ﬁxed and

C⊂V

non-empty and ﬁnite. Furthermore, assume that the sequences (pe)e∈E and (ve)e∈E satisfy

y∈N

yv{0,y}p{0,y}<∞and X

y∈N

yv−1

{0,y}<∞.(8)

Then, for every

λ >

0there exists

γ∗

(

λ, q

)

0such that C

dies out almost surely for all

γ≤γ∗, i.e. θ(λ, γ, q, C) = 0 for all γ≤γ∗. Thus, in particular limγ→0λc(γ, q) = ∞.

Note that

(8)

is a stronger assumption than

(4)

, and thus already implies the latter assumption.

2.1 Outline

The rest of this paper is organized as follows. In Section 3 we discuss some related literature in

order to put our results into context with the current state of research. Then, we state and discuss

some open problems and possible directions for future research.

Since we will use on several occasions a comparison with an independent long range percolation

model we introduce this type of model in Section 4 and state some conditions which imply the

absence of an inﬁnite connected component.

In Section 5 we construct the CPDLP via a graphical representation. Furthermore, in Subsection 5.1

we show that this construction yields a well-deﬁned Feller process. In Subsection 5.2 we describe the

construction of a dual infection process, which yields a self-duality relation. We then use this relation

to prove Theorem 2.1. In Subsection 5.3 we use the graphical representation to show Theorem 2.2

and Corollary 2.4.

In Section 6 we compare the dynamical long range percolation blockwise with an independent long

range percolation model and deﬁne a new infection process, which dominates the original one. We

use this newly deﬁned process to show Theorem 2.5 in Subsection 6.1. Lastly, we show Theorem 2.7

in Subsection 6.2.

3 Discussion

3.1 Related literature

The contact process was ﬁrst introduced almost half a century ago by Harris [

Har74

] on

. Since

then this process and many variations of it have been studied intensively, mostly on bounded

degree graphs. To the best of our knowledge the ﬁrst to introduce a long range variation of the

contact process, where there is no intrinsic bound on the distance between two vertices for which

a transmission of an infection can take place, was Spitzer [

Spi77

]. He studied so-called nearest

particles systems. Bramson and Gray [

BG81

] studied in particular the phase transition of similar

systems. See also [Lig12, Chapter VII] for more results on nearest particles systems.

Swart [

Swa09

] studied a contact process with general infection kernel (

)

e∈E

as in

(5)

and

(6)

Contact process on a dynamical long range percolation M. SeilerA. Sturm

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