An improved upper bound for critical value of the contact process on $\mathbb{Z}^d$ with $d\geq 3$

In this paper we give an improved upper bound for critical value $\lambda_c$ of the basic contact process on the lattice $\mathbb{Z}^d$ with $d\geq 3$. As a direct corollary of out result, \[ \lambda_c\leq 0.384. \] when $d=3$.


Introduction
In this paper we are concerned with the basic contact process on Z d with d ≥ 3. First we introduce some notations. For each x = (x 1 , x 2 , . . . , x d ) ∈ Z d , we use x to denote the l 1 -norm of x, i.e., For any x, y ∈ Z d , we write x ∼ y when end only when x − y = 1, i.e., x ∼ y means that x and y are neighbors on Z d . For 1 ≤ i ≤ d, we use e i to denote the ith elementary unit vector of Z d , i.e., e i = (0, . . . , 0, 1 ith , 0, . . . , 0). (1.1) We use O to denote the origin of Z d . The contact process {η t } t≥0 on Z d is a spin system with state space {0, 1} Z d (see the definition of the spin system in Chapter 3 of [4]). The flip rates function of {η t } t≥0 is given by for any (η, x) ∈ {0, 1} Z d × Z d , where λ > 0 is a constant called the infection rate. That is to say, the state of the process flips from η to η x at rate c(x, η), where Intuitively, the contact process describes the spread of an epidemic on the graph. Vertices in state 1 are infected while that in state 0 are healthy. An infected vertex waits for an exponential time with rate 1 to become healthy while an healthy one is infected at rate proportional to the number of infected neighbors.
The contact process is introduced by Harris in [2]. For a detailed survey of the study of the contact process, see Chapter 6 of [4] and Part one of [6].
In this paper we are mainly concerned with the critical value of the contact process. Assuming that η 0 (x) = 1 for any x ∈ Z d , then the critical value λ c is defined as where P λ is the probability measure of the contact process with infection rate λ. The definition of λ c is reasonable according to the following property of the contact process. For λ 1 ≥ λ 2 and t > s, conditioned on all the vertices are in state 1 at t = 0, where γ(d) > 1/2 is the probability that the simple random walk on Z d starting at O never returns to O. Both these two results lead to the conclusion that lim d→+∞ 2dλ c (d) = 1.
When d = 3, according to the well-known result that γ 3 ≈ 0.659, However, α 2 (d) < α 3 (d) for sufficiently large d according to the fact that In this paper, we will give another upper bound β(d) for the critical value λ c (d) when d ≥ 3. β(d) satisfies that β(d) < min{α 1 (d), α 2 (d)} for each d ≥ 3. For the precise result, see the next section.

Main result
In this section we will give our main result. First we introduce some notations and definitions. From now on we assume that at t = 0 all the vertices on Z d are in state 1 for the contact process, then let λ c be the critical value of the contact process defined as in Equation (1.3). We write λ c as λ c (d) when we need to point out the dimension d of the lattice. We denote by {S n } n≥0 the simple random walk on Z d , i.e., for each y that y ∼ x and n ≥ 0. We define as the probability that the simple random walk never return to O conditioned on S 0 = O. We write γ as γ d when we need to point out the dimension d of the lattice.
The following theorem gives an upper bound of λ c (d) for d ≥ 3, which is our main result.
. For d = 3, according to the well known result that γ 3 ≈ 0.659, we have the following direct corollary.
This corollary improves the upper bound of λ c (3) given by α 1 (3), which is 0.517. According to the example given in Section 3.5 of [4], We will prove Theorem 2.1 in the next section. A Markov process {ξ t } t≥0 with state space [0, +∞) Z d will be introduced as a main auxiliary tool for the proof. The definition of {ξ t } t≥0 is similar with that of the binary contact path process introduced in [1], except for some modifications in several details.

Proof of Theorem 2.1
In this section we give the proof of Theorem 2.1. Throughout this section we assume that the dimension d is fixed and at least 3, which ensures that γ > 1 2 . Our aim is to prove the following lemma, Theorem 2.1 follows from which directly. .
If we choose a = b = 1, then Lemma 3.1 gives the upper bound of λ c the same as that given in [1]. However, the best choices of a, b are a = b = 1 2−γ , which gives the following proof of Theorem 2.1.
according to Lemma 3.1.
The remainder of this paper is devoted to the proof of Lemma 3.1. From now on we assume that a, b are positive constants which satisfies Let {ξ t } t≥0 be a continuous time Markov process with state space [0, +∞) Z d and generator function given by for any ξ ∈ [0, +∞) Z d and sufficiently smooth function f on [0, +∞) Z d , where x is the partial derivative of f (ξ) with respect to the coordinate ξ(x).
If a = b = 1, then {ξ t } t≥0 is the binary contact path process introduced in [1] after a time-scaling. {ξ t } t≥0 belongs to a large crowd of continuous-time Markov processes called linear systems. For the definition and basic properties of the linear system, see Chapter 9 of [4].
According to the definition of Ω, {ξ t } t≥0 evolves as follows. For each x ∈ Z d and each neighbor y of x, ξ t (x) flips to 0 at rate 1 while flips to bξ t (x) + aξ t (y) at rate λ. Between the jumping moments of {ξ t (x)} t≥0 , ξ t (x) evolves according to the ODE That is to say, if ξ(x) does not jump during [t, t + s], then for 0 < r < s.
The linear system {ξ t } t≥0 and the contact process {η t } t≥0 have the following relationship.

Lemma 3.2.
For any x ∈ Z d and t ≥ 0, let By Lemma 3.2, from now on we assume that {η t } t≥0 and {ξ t } t≥0 are coupled under the same probability space such that η 0 (x) = ξ 0 (x) = 1 for each x ∈ Z d and η t (x) = 1 when and only when ξ t (x) > 0.
The following two lemmas about expectations of ξ t (x) and ξ t (x)ξ t (y) are important for the proof of Lemma 3.1. for any x ∈ Z d and t ≥ 0.
if x = 0 and y = e 1 , 0 otherwise and e 1 is defined as in Equation (1.1).
Note that when we say for each x ∈ Z d , as the product of finite-dimensional matrixes. The proofs of Lemmas 3.3 and 3.4 rely heavily on Theorems 9.1.27 and 9.3.1 of [4]. These two theorems can be seen as the extension of the Hille-Yosida Theorem for the linear system, which ensures that we can execute the calculation d dt for each x ∈ Z d . Since ξ 0 (x) = 1 for all x ∈ Z d , Eξ t (x) does not depend on the choice of x according to the spatial homogeneity of {ξ t } t≥0 . Therefore, As a result, Eξ t (x) ≡ Eξ 0 (x) = 1.
The following lemma shows that if λ ensures the existence of an positive eigenvector of G λ with respect to the eigenvalue 0, then λ is an upper bound of λ c , which is crucial for us to prove Lemma 3.1. , we define and G λ K = 0 according to Equations (3.8), (3.9) and the definition of G λ . As a result, by Lemma 3.5, λ ≥ λ c for any λ > 1 2d 2(a+b−1)−(a 2 +b 2 −1)−2ab (1−γ) and hence .
For any ξ ∈ (−∞, +∞) Z d , we define Furthermore, we define then W is a Banach space with norm · ∞ . By the definition of G λ , it is easy to check that there exists M > 0 that for any ξ 1 , ξ 2 ∈ W , i.e., ODE (3.3) satisfies Lipschitz condition. As a result, according to the theory of the linear ODE on the Banach space, ODE (3.3) has the unique solution that F t = e tG λ F 0 for any t ≥ 0. Since F 0 (x) = 1 for any x ∈ Z d , Since G λ (x, y) ≥ 0 when x = y, e tG λ (x, y) ≥ 0 for any x, y ∈ Z d . Therefore, by Equation (3.10), As a result, λ ≥ λ c for any λ that there exists K which satisfies inf x∈Z d K(x) > 0 and G λ K = 0.