THE TRIANGLE CONDITION FOR CONTACT

PROCESSES ON HOMOGENEOUS TREES

Roberto H. Schonmann*

Department of Mathematics, UCLA

Abstract. We complete work of C. C. Wu, by showing that for contact processes on homogeneous trees with degree at least 3 the triangle condition is satisfied below the second critical point. In particular it holds at the first critical point and therefore at this critical point the contact process has mean-field critical exponents.

1. Introduction and main results.

The contact process with infection parameter λ > 0 on a countable graph G of bounded degree is a continuous time Markov process with state space {0, 1}^{V_{G}}, where V_{G} is the set of vertices (also called sites) of the graph. Elements of this state space are called configurations. When the configuration at a given site is 1 one says that there is a particle there or that the site is occupied or that the site is infected. Otherwise one says that the site is vacant or healthy. The contact process evolves according to the following local prescription.

(i) A particle at a site gives birth to new ones at each neighboring vacant site at rate λ.

(ii) Particles die at rate 1.

For a more precise description of the model, its construction and background material, we refer the reader to [Lig1] and [Dur].

In this paper we will consider the contact process on the homogeneous tree of degree d + 1, denoted by 𝕋_{d}. The case in which d = 1 corresponds to the linear chain ℤ, and will not be considered here, so that we assume that d ≥ 2. A great deal of attention has been given to contact processes on such trees, and a substantial amount of information on its behavior is available from the papers [Pem], [MS], [MSZ], [DS], [Wu], [Zha], [Lig2], [Sta], [Lig3], [LS], [SS], and [Lal].

We denote by (ξ_{t}^{η})_{t≥0} the contact process started at time 0 from the configuration η. The configuration with a single particle at site x is, in an abuse of notation, denoted by x. An arbitrary site of the tree is called its root, denoted by 0. Configurations are identified with the set of sites where they take the value 1, so that

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 60K35.

Key words and phrases. Contact process, homogeneous trees, critical behavior, critical exponents, triangle condition.

*Work partially supported by the N.S.F. through grant DMS-9703814.

in particular ∅ is the configuration with no particles. The probability of (global) survival is given by

ρ(λ) = ℙ(ξ_{t}^{0} ≠ ∅, for all t ≥ 0).

Similarly, the probability of local survival (sometimes referred to as probability of recurrence) is given by

ρ_{local}(λ) = ℙ(ξ_{t}^{0}(0) = 1, for a unbounded set of values of t).

Interest on the contact process on homogeneous trees with d ≥ 2 stems to a great extent from the fact that there are two distinct critical points. 0 < λ_{1} < λ_{2} < ∞, defined as follows.

λ_{1} = inf{λ : ρ(λ) > 0},

and

λ_{2} = inf{λ : ρ_{local}(λ) > 0}.

In this paper we are concerned with the critical behavior of the process in the vicinity and at the threshold for global survival, λ_{1}. In [Wu], under the technical assumption that d ≥ 5, this behavior was shown to be of mean-field character, i.e, critical exponents take the branching-process value. We will build on that work and complete it, by removing that technical condition. To state the precise results, we need to introduce a few more definitions.

We suppose that the contact process on 𝕋_{d}, with infection parameter λ > 0 is constructed in the usual graphical additive fashion, by means of Poisson death marks (at rate 1 for each site) and Poisson arrows (at rate λ for each oriented pair (x, y) such that {x, y} is an edge of the graph. We will use ℙ_{λ} to denote the probability measure corresponding to the graphical construction, and 𝔼_{λ} to denote the corresponding expectation. Given two space-time points, (x, s), (y, t) ∈ V_{𝕋_{d}} × ℝ_{+}, with s < t, we say that there is a path from (x, s) to (y, t) if there is a sequence of times s = t_{0} < t_{1} < ... < t_{n} < t_{n+1} = t and spatial locations x = x_{0}, x_{1}, ..., x_{n} = y so that for i = 1, 2, ..., n there is an arrow from x_{i−1} to x_{i} at time t_{i} and the vertical segments {x_{i}} × (t_{i}, t_{i+1}) for i = 0, 1, ..., n do not contain any death mark. Given a configuration η we set

ξ_{t}^{η} = {y ∈ V_{𝕋_{d}} : there is a path from (x, 0) to (y, t) for some x ∈ η}.

In this fashion {ξ_{t}^{η} : t ≥ 0} is a version of the contact process on 𝕋_{d} started from η. We will use the notation {(x, s) → (y, t)} for the event that there is a path from (x, s) to (y, t) in the graphical construction.

The distance between two sites x and y in V_{𝕋_{d}} will be measured by the minimal length of the paths along neighboring sites which join x to y, and will be denoted by ||x − y||. We will abbreviate ||x − 0|| = ||x||. The distance between two points (x, s) and (y, t) in V_{𝕋_{d}} × ℝ_{+} will be measured by ||(x, s) − (y, t)|| = ||x − y|| + |t − s|. The open triangle diagram is now defined by

▽(λ; R) = sup{▽(λ; (z, s)): (z, s) ∈ V_{𝕋_{d}} × ℝ_{+}, ||(z, s)|| > R},

with

▽(λ; (z, s)) = ∑_{x,y∈V_{𝕋_{d}}} ∫_{0}^{∞} dt_{1} ℙ_{λ}((0, 0) → (x, t_{1}))

× ∫_{t_{1}∨s}^{∞} dt_{2} ℙ_{λ}((x, t_{1}) → (y, t_{2}))ℙ_{λ}((z, s) → (y, t_{2})).

Theorem 1. For the contact process on the homogeneous tree 𝕋_{d} with d ≥ 2, if λ < λ_{2}, then

lim_{R→∞} ▽(λ; R) = 0.

Theorem 1 will be proved in Section 2. From the paper [BW] we know that this theorem implies that certain critical exponents take their mean field value, as stated in Corollary 1 below. The susceptibility χ(λ) which appears there is defined as

χ(λ) = 𝔼_{λ} ∫_{0}^{∞} dt ∑_{x∈V_{𝕋_{d}}} ξ_{t}^{0}(x).

The notation f(λ) ≈ g(λ) used below means that there exist positive finite constants C_{1} and C_{2} such that

C_{1}g(λ) ≤ f(λ) ≤ C_{2}g(λ).

Corollary 1. For the contact process on the homogeneous tree 𝕋_{d} with d ≥ 2 we have the following mean-field power laws:

ρ(λ) ≈ (λ − λ_{1})^{+1} for λ ≥ λ_{1},
χ(λ) ≈ (λ_{1} − λ)^{−1} for λ < λ_{1}.

Our proof of Theorem 1 in Section 2 will give rise to some by-products which will be presented in Section 3.

2. Proof of Theorem 1.

Before we can start the proof of Theorem 1, we need to recall some definitions and results from [Lig3], [LS] and [Lal]. Denote as site 1 one of the neighbors of the site 0; then denote as site 2 one of the neighbors of site 1 which is different from site 0; successively denote as site n + 1 one of the neighbors of site n which is different from site n − 1. Note that site n is at distance n from the origin and that the sites 0, 1, ... are the sites of a subgraph of 𝕋_{d} which is isomorphic to the semi-infinite linear chain ℤ^{+}. Set

u_{n}(λ) = ℙ_{λ} (ξ_{t}^{0}(n) = 1 for some t ≥ 0) .

From the strong Markov property, used at the first time that the site m becomes infected, in conjunction with the property of attractiveness of the contact process one obtains the inequality

u_{n+m}(λ) ≥ u_{n}(λ)u_{m}(λ).

From this inequality it follows that the following limit exists.

lim_{n→∞}(u_{n}(λ))^{1/n} = sup{(u_{n}(λ))^{1/n}: n ≥ 1} = β(λ).  (2.1)

Note that

u_{n}(λ) ≤ (β(λ))^{n},  (2.2)

for n ≥ 0.

Similarly we have the existence of the limit

lim_{t→∞}(ℙ_{λ}(ξ_{t}^{0}(0) = 1))^{1/t} = sup{(ℙ_{λ}(ξ_{t}^{0}(0) = 1))^{1/t}: t > 0} = η(λ),  (2.3)

and the inequality

ℙ_{λ}(ξ_{t}^{0}(0) = 1) ≤ (η(λ))^{t},  (2.4)

for t ≥ 0.

From [LS] and [Lal], we know that for λ < λ_{2},

β(λ) < 1/\sqrt{d}  and  η(λ) < 1.  (2.5)

We also need to introduce the following object.

ū_{n}(λ) = ∫_{0}^{∞} ℙ_{λ} (ξ_{t}^{0}(n) = 1) dt = ∫_{0}^{∞} ℙ_{λ} ((0, 0) → (n, t)) dt.

Note that

χ(λ) = ū_{0}(λ) + ∑_{n≥1}(d + 1)d^{n−1}ū_{n}(λ).  (2.6)

Lemma 1. Given ϵ > 0, there exists C < ∞ such that for all λ ∈ [ϵ, λ_{2} − ϵ] and n ≥ 0,

ū_{n}(λ) ≤ C(n + 1)u_{n}(λ) ≤ C(n + 1)(β(λ))^{n}.

Proof. Let a_{ϵ} be the probability for the contact process with infection parameter ϵ, that between time 0 and 1 there is no death mark at the origin, and there is an arrow from the origin to its neighbor 1. Clearly a_{ϵ} = e^{−1}(1 − e^{−ϵ}) > 0, and for λ ≥ ϵ

u_{n}(λ) ≥ (a_{ϵ})^{n}.  (2.7)

Clearly also

ℙ_{λ}(ξ_{t+n}^{0}(0) = 1) ≥ (a_{ϵ})^{n} ℙ_{λ}(ξ_{t}^{0}(n) = 1).  (2.8)

Comparing this with (2.4) yields, in case λ ≤ λ_{2} − ϵ,

ℙ_{λ}(ξ_{t}^{0}(n) = 1) ≤ (a_{ϵ})^{−n}(η(λ_{2} − ϵ))^{t+n} ≤ (a_{ϵ})^{−n}(η(λ_{2} − ϵ))^{t}.

Therefore, for arbitrary K > 0,

ū_{n}(λ) = ∫_{0}^{∞} ℙ_{λ}(ξ_{t}^{0}(n) = 1)dt ≤ u_{n}(λ)Kn + ∫_{Kn}^{∞} ℙ_{λ}(ξ_{t}^{0}(n) = 1)dt
≤Knu_{n}(λ) + (a_{ϵ})^{−n}(η(λ_{2} − ϵ))^{Kn}/ log(1/η(λ_{2} − ϵ)),

where the fact that η(λ_{2} − ϵ) < 1 was used. Using again this fact and referring to (2.7), one concludes that if K is large enough and C = 2K we have

ū_{n}(λ) ≤ 2Knu_{n}(λ) ≤ Cnu_{n}(λ) ≤ C(n + 1)u_{n}(λ),

for n ≥ 1. That for large C, we have ū_{n}(λ) ≤ C(n + 1)u_{n}(λ) also in case n = 0 is trivial, since η(λ) ≤ η(λ_{2} − ϵ) < 1. The proof is complete by referring to (2.2).

Proof of Theorem 1. We will explain how Lemma 1 can be used to adapt the proof in [Wu] to all d ≥ 2.

Suppose that λ < λ_{2}. Note that from Lemma 1 and (2.5) it follows that there exists β′ ∈ (β(λ), 1/\sqrt{d}) and C_{1} < ∞ such that

ū_{n}(λ) ≤ C_{1} (β′)^{n},  (2.9)

for all n. This inequality replaces Lemma 2 in [Wu], and confirms the conjecture in that paper stated immediately before that lemma.

Now we rewrite display (40) in [Wu]. We still use his Lemma 3 as he did, but in the last step, we use (2.9) to obtain the following.

▽(λ; (z, s))

= ∑_{x,y∈V_{T_{d}}} ∫_{0}^{∞} dt_{1} ℙ_{λ}((0, 0) → (x, t_{1}))

× ∫_{t_{1}∨s}^{∞} dt_{2} ℙ_{λ}((x, t_{1}) → (y, t_{2}))ℙ_{λ}((z, s) → (y, t_{2}))

≤ e^{4} ∑_{x,y∈V_{T_{d}}} ∫_{0}^{∞} dt_{1} ℙ_{λ}((0, 0) → (x, t_{1}))

× ∫_{t_{1}}^{∞} dt_{2} ℙ_{λ}((x, t_{1}) → (y, t_{2})) ∫_{s}^{∞} dt_{3} ℙ_{λ}((z, s) → (y, t_{3}))

= e^{4} ∑_{x,y∈V_{T_{d}}} ū_{||x||}(λ) ū_{||y−x||}(λ) ū_{||z−y||}(λ)

≤ C_{2} ∑_{x,y∈V_{T_{d}}}(β′)^{||x||} (β′)^{||y−x||} (β′)^{||z−y||}.

To see that the last expression above vanishes as z → ∞, we can refer to Lemma 1 in [Wu]. For this, the reader should compare the statement of that lemma in [Wu] with his equation (23).

At this point we know that

lim_{||z||→∞} ▽(λ; (z, s)) = 0 uniformly for s ∈ [0, ∞).

So it remains to show that for any fixed z ∈ 𝕋_{d},

lim_{s→∞} ▽(λ; (z, s)) = 0.

But using the results above, this can now be done exactly in the same way as in [Wu] (see the final part of his proof of his Theorem 1).

3. Some properties of β.

It is clear that β(λ) = 1 for λ > λ_{2}. A great deal of information about the behavior of β(λ) as a function of λ in the interval (0, λ_{2}] has been obtained in the papers [Lig3], [LS] and [Lal]. We will review these known results at the end of this section; when we combine them with some new results which will be proved here.

Before we can state these new results, we need to introduce some definitions. We will let the random variable X_{n} be the total number of arrows which point from an infected site to the site n in the process (ξ_{t}^{0}: t ≥ 0). And we also define

I_{0}(λ) =𝔼_{λ}(X_{0}) + 1,
I_{n}(λ) =𝔼_{λ}(X_{n}),  n ≥ 1.

(The extra 1 in the definition of I_{0}(λ) is needed to make certain statements below true; it may intuitively be seen as accounting for the particle initially at the origin.)

Clearly we have

u_{n}(λ) ≤ I_{n}(λ),  (3.1)

for n ≥ 0, and

I_{n}(λ) = λ(ū_{n−1}(λ) + dū_{n+1}(λ)),  (3.2)

for n ≥ 1.

We claim that also

I_{m+n}(λ) ≤ I_{m}(λ)I_{n}(λ),  (3.3)

for m, n ≥ 0. To justify (3.3), we first note that in case m = 0 or n = 0 this claim is trivially true. For the other cases we compare the contact process starting with a single particle at the origin with a multitype contact process described as follows. Particles can be of type 0, 1, 2, ..., with the particle initially at the origin being of type 0. Particles of types 1,2,... evolve as independent contact processes, independent also of the behavior of particles of type 0 (in particular particles of different type can coexist at the same site at the same time). Particles of type 0 evolve as a contact process, except for the fact that they cannot infect the site m. The first time when there is an attempt by a particle of type 0 to infect the site m, a particle of type 1 is created at site m; the second time when there is an attempt by a particle of type 0 to infect the site m, a particle of type 2 is created at site m; and so on. This multitype contact process dominates the contact process in the sense that whenever there is a contact process particle at a site, there will be some multitype contact process particle (of some type) also at that site. Let X′_{m} be the total number of times when a type 0 particle tries to infect the site m in the multitype contact process (this equals the number of other types of particles ever created in this process). Let X′′_{m+n} be the the total number of times when a particle of any type tries to infect the site m + n in the multitype contact process (due to the geometry of a tree, we are only talking about particles of types different from 0 here). Now note that

I_{m+n}(λ) ≤ 𝔼_{λ}(X′′_{m+n}) = 𝔼_{λ}(𝔼_{λ}(X′′_{m+n}|X′_{m})) = 𝔼_{λ}(X′_{m}𝔼_{λ}(X_{n}))
= 𝔼_{λ}(X′_{m})𝔼_{λ}(X_{n}) ≤ 𝔼_{λ}(X_{m})𝔼_{λ}(X_{n}) = I_{m}(λ)I_{n}(λ).

The inequality (3.3) yields

lim_{n→∞}(I_{n}(λ))^{1/n} = inf{(I_{n}(λ))^{1/n}: n ≥ 1} = β̃(λ).  (3.4)

Note that

I_{n}(λ) ≥ (β̃(λ))^{n},  (3.5)

for n ≥ 0.

Theorem 2. The function β(·) is continuous on the interval (0, λ_{2}]. For λ < λ_{2}

lim_{n→∞}(ū_{n}(λ))^{1/n} = lim_{n→∞}(I_{n}(λ))^{1/n} = lim_{n→∞}(u_{n}(λ))^{1/n} = β(λ).

Moreover, given ϵ > 0 there exists C(ϵ) ∈ (0, ∞) such that for all λ ∈ [ϵ, λ_{2} − ϵ] we have

\frac{(β(λ))^{n}}{C(ϵ)} ≤ ū_{n}(λ) ≤ C(ϵ)(n + 1)(β(λ))^{n},

(β(λ))^{n} ≤ I_{n}(λ) ≤ C(ϵ)(n + 1)(β(λ))^{n},

and

\frac{(β(λ))^{n}}{C(ϵ)(n + 1)} ≤ u_{n}(λ) ≤ (β(λ))^{n}.

Proof. Suppose that ϵ > 0 is fixed and that λ ∈ [ϵ, λ_{2} − ϵ]. Using the notation introduced in the beginning of the proof of Lemma 1, we have

ū_{n}(λ) ≥ ∫_{1}^{∞} ℙ_{λ} (ξ_{t}^{0}(n) = 1) dt ≥ ∫_{1}^{∞} a_{ϵ}ℙ_{λ} (ξ_{t−1}^{0}(n − 1) = 1) dt
= ∫_{0}^{∞} a_{ϵ}ℙ_{λ} (ξ_{t}^{0}(n − 1) = 1) dt = a_{ϵ}ū_{n−1}(λ).  (3.6)

The same argument also gives

ū_{n}(λ) ≥ a_{ϵ}ū_{n+1}(λ).  (3.7)

From (3.2), (3.6) and (3.7), we have I_{n}(λ) ≤ C′(ϵ)ū_{n}(λ), and combining this with Lemma 1 and (3.1) yields

I_{n}(λ) ≤ C′(ϵ)ū_{n}(λ) ≤ C′′(ϵ)(n + 1)u_{n}(λ) ≤ C′′(ϵ)(n + 1)I_{n}(λ).  (3.8)

This chain of inequalities and the fact that ϵ > 0 is arbitrary gives us

β̃(λ) = β(λ),  (3.9)

for λ ∈ (0, λ_{2}). Combining it also with (2.2) and (3.5) gives us all the claims in the theorem, except for the continuity of β(·) in (0, λ_{2}]. The proof will therefore be complete once we show that

β(·) is left-continuous on (0, ∞),  (3.10)

and

β(·) is right-continuous on (0, λ_{2}).  (3.11)

To show the first of these claims, note that if for T > 0 we set

u_{n}^{T} = u_{n}^{T} (λ) = ℙ_{λ} (ξ_{t}^{0}(n) = 1 for some t ∈ [0, T]) ,

then

β(λ) = sup_{n≥1} (u_{n}(λ))^{1/n} = sup_{n≥1} sup_{T ≥0} (u_{n}^{T} (λ))^{1/n}.

But u_{n}^{T} (·) is clearly a continuous function, and therefore β(·) is lower-semi-conti-nuous. Since β(·) is clearly non-decreasing, it is also left-continuous.

To show (3.11) define

ū_{n}^{T} (λ) = ∫_{0}^{T} ℙ_{λ} (ξ_{t}^{0}(n) = 1) dt.

For fixed λ, ū_{n}^{T} (λ) approaches ū_{n}(λ) as T → ∞, in a fashion which can be controlled uniformly in λ ∈ [ϵ, λ_{2} − ϵ] as follows, using (2.8) and (2.4),

0 ≤ ū_{n}(λ) − ū_{n}^{T} (λ) = ∫_{T}^{∞} ℙ_{λ} (ξ_{t}^{0}(n) = 1) dt ≤ (\frac{1}{a_{ϵ}} )^{n} ∫_{T}^{∞} ℙ_{λ} (ξ_{t+n}^{0}(0) = 1) dt

≤ (\frac{1}{a_{ϵ}} )^{n} ∫_{T}^{∞}(η(λ_{2} − ϵ))^{t+n}dt.

Since η(λ_{2} −ϵ) < 1, by (2.5), the last expression above vanishes as T → ∞, showing that ū_{n}^{T} (·) → ū_{n}(·), uniformly on [ϵ, λ_{2} − ϵ]. As a function of λ, ū_{n}^{T} (λ) is clearly continuous, hence ū_{n}(·) is also continuous on [ϵ, λ_{2} − ϵ].

From the part of the theorem which has already been proved, we know that (C(ϵ)ū_{n}(λ))^{1/n} ≥ β(λ) and lim_{n→∞}(C(ϵ)ū_{n}(λ))^{1/n} = β(λ). Hence

β(λ) = inf_{n≥1} (C(ϵ)ū_{n}(λ))^{1/n}.

Therefore β(·) is upper-semi-continuous on [ϵ, λ_{2} − ϵ]. Since β(·) is non-decreasing, it is also right-continuous on this interval. Since ϵ > 0 is arbitrary, (3.11) is assured.

Combining results in [Lig3], [LS] and [Lal] with those in the present paper we know now that the function β(·), which is identically 1 on (λ_{2}, ∞), is strictly increasing and continuous on (0, λ_{2}] and satisfies also:

lim_{λ↘0} β(λ) = 0,  (3.12)
β(λ_{1}) = 1/d,  (3.13)

and

β(λ_{2}) = 1/\sqrt{d}.  (3.14)

The strict monotonicity, as well as (3.14) are proved in [Lal]. The claim (3.13) follows, for instance, from combining the following: the continuity of β(·) at λ_{1}, the result on the critical behavior of χ(λ) near λ_{1}, as given in Corollary 1, equation (2.6), and the relation between ū(λ) and β(λ), as given in Theorem 2. Alternatively, observe that [Lig3] proved that β(λ) ≤ 1/d for λ < λ_{1} and that for λ > λ_{1} clearly β(λ) ≥ 1/d (since β(λ) < 1/d implies 𝔼_{λ}(number of sites ever to be infected) < ∞, and this implies ρ(λ) = 0); the continuity of β(·) at λ_{1} yields then (3.13). Finally (3.12) is easy to verify using the standard comparison of the contact process with a branching random walk, obtained by allowing multiple occupancy of each site by particles which create offspring and may die independently.

Acknowledgements: Thanks are given to Tom Liggett and to a referee for their comments which led to improvements to this paper.

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Mathematics Department

University of California at Los Angeles Los Angeles, CA 90095 U.S.A.

10 ROBERTO H. SCHONMANN

rhs@math.ucla.edu