1 Introduction and the Main Results

Oriented percolation is a time-oriented model of percolation. It is also considered as a discrete-time model for the spread of an infectious disease, known as the contact process or the SIS model. Since it became known to exhibit a phase transition and critical behavior, there have been intensive researches in both theory and applications in various fields. Recently, a possible association to the laminar-turbulent flow transition was reported in [20].

Consider the following nearest-neighbor bond oriented percolation on the space–time lattice \({\mathbb L}^d\equiv \{(x,t)\in \mathbb {Z}^d\times \mathbb {Z}_+:\Vert x\Vert _1+t\) is even}. A pair of vertices \([(x,s),(y,t)\rangle \) is called a bond if \(\Vert x-y\Vert _1=1\) and \(t=s+1\). Each bond \([(x,t),(y,t+1)\rangle \) is either occupied with probability \(p\in [0,1]\) or vacant with probability \(1-p\), independently of the other bonds. Let \({\mathbb P}_p\) be the associated probability measure. We say that \((x,s)\in {\mathbb L}^d\) is connected to \((y,t)\in {\mathbb L}^d\), denoted by \((x,s)\longrightarrow (y,t)\), if either \((x,s)=(y,t)\) or there is a sequence of occupied bonds \(\{[(v_j,j),(v_{j+1},j+1)\rangle \}_{j=s}^{t-1}\) from \(v_s=x\) to \(v_t=y\). We simply write \((x,s)\longrightarrow t\) for the event \(\bigcup _y\{(x,s)\longrightarrow (y,t)\}\), and \(s\longrightarrow (y,t)\) for the event \(\bigcup _x\{(x,s)\longrightarrow (y,t)\}\).

The major quantities we are interested in are the following. The first quantity is the survival probability up to time t, defined as

$$\begin{aligned} \theta _t={\mathbb P}_p\big ((o,0)\longrightarrow t\big ), \end{aligned}$$
(1.1)

where, and in the rest of the paper, the p-dependence is suppressed for lighter notation. Since \(\{\theta _t\}_{t\in {\mathbb N}}\) is a decreasing sequence of increasing and continuous functions in p, the limit \(\theta _\infty \equiv \lim _{t\uparrow \infty }\theta _t\) is nondecreasing and right-continuous in p. Let

$$\begin{aligned} p_\mathrm {c}=\inf \{p\in [0,1]:\theta _\infty >0\}. \end{aligned}$$
(1.2)

It is proven in [8] that \(\theta _\infty \) is also left-continuous in p. In particular, \(\theta _\infty =0\) at \(p=p_\mathrm {c}\), which has not been proven yet for unoriented percolation in full generality.

The second and third quantities are the expected number of vertices at time t connected from the origin (o, 0) and the gyration radius of those vertices, defined as

$$\begin{aligned} \chi _t=\sum _x\tau (x,t),&\xi _t=\bigg (\frac{1}{\chi _t}\sum _x|x|^2\tau (x,t)\bigg )^{1/2}, \end{aligned}$$
(1.3)

where \(\tau (x,t)\) is the two-point function:

$$\begin{aligned} \tau (x,t)={\mathbb P}_p\big ((o,0)\longrightarrow (x,t)\big ). \end{aligned}$$
(1.4)

It is first proven in [1, 12], and recently reproved in a much simpler way in [5], that the critical point is unique in the sense that

$$\begin{aligned} p_\mathrm {c}=\sup \bigg \{p\in [0,1]:\sum _{t=0}^\infty \chi _t<\infty \bigg \}. \end{aligned}$$
(1.5)

The sum \(\sum _t\chi _t\) is often called the susceptibility.

Now we briefly summarize the basic properties of those quantities readily obtained from the definition. First we note that, by the Markov property and translation invariance,

$$\begin{aligned} \theta _{s+t}\ge \theta _s\theta _t,&\chi _{s+t}\le \chi _s\chi _t. \end{aligned}$$
(1.6)

With the help of the trivial inequality \(\theta _t\le \chi _t\le (2t+1)^d\theta _t\), we can conclude that there is a common relaxation time \(\zeta \in [0,\infty ]\) such that

$$\begin{aligned} \zeta =\lim _{t\uparrow \infty }\frac{-t}{\log \theta _t} =\sup _{t\in {\mathbb N}}\frac{-t}{\log \theta _t} =\lim _{t\uparrow \infty }\frac{-t}{\log \chi _t} =\inf _{t\in {\mathbb N}}\frac{-t}{\log \chi _t}. \end{aligned}$$
(1.7)

Using the second and forth equalities, we can say that \(\zeta \) is bounded away from zero and infinity when \(p<p_\mathrm {c}\), implying exponential decay of \(\theta _t\) and \(\chi _t\) in t in the subcritical regime. This is not the case at the critical point. Moreover, \(\chi _t\) is nondecreasing in t at \(p=p_\mathrm {c}\), because, otherwise, there must be a \(t_0\in {\mathbb N}\) such that \(\chi _{t_0}<1\), which together with submultiplicativity implies exponential decay of \(\chi _t\) and convergence of the susceptibility \(\sum _t\chi _t\) at \(p=p_\mathrm {c}\), such as

$$\begin{aligned} \sum _{t=0}^\infty \chi _t=\sum _{n=0}^\infty \sum _{k=0}^{t_0-1}\chi _{nt_0+k}\le \sum _{n=0}^\infty \chi _{t_0}^n\sum _{k=0}^{t_0-1}\chi _k<\infty , \end{aligned}$$
(1.8)

which is a contradiction to the result in [2]: \(\sum _t\chi _t=\infty \) at \(p=p_\mathrm {c}\).

Let \(\rho ,\eta ,\nu \) be the critical exponents for the above quantities at \(p=p_\mathrm {c}\): as \(t\uparrow \infty \),

$$\begin{aligned} \theta _t\approx t^{-\rho },&\chi _t\approx t^\eta ,&\xi _t\approx t^\nu , \end{aligned}$$
(1.9)

where \(f\approx g\) means that \((\log \,f)/\log g\) goes to 1 in the prescribed limit, allowing corrections of slowly varying functions. In higher dimensions \(d\gg 4\) (\(d>4\) is enough for sufficiently spread-out models), the lace expansion converges and the above critical exponents take on their mean-field values \(\rho =1\), \(\eta =0\) and \(\nu =1/2\): the values for branching random walk [3, 4, 9, 10, 13, 14, 18]. In lower dimensions, on the other hand, only numerical values and predictions due to non-rigorous renormalization-group methods are available (see Table 1).

Table 1 Predicted values of the critical exponents in various dimensions (e.g., [15])
Full size table

In this paper, we prove the following theorem.

Theorem 1.1

  1. (i)

    For any \(d\ge 1\), \(p\in [0,1]\) and \(t\in {\mathbb N}\), we have

    $$\begin{aligned} \chi _t\le \frac{4}{3}(4\xi _t+1)^d\,\theta _{t/2}^2, \end{aligned}$$
    (1.10)

    which implies the hyperscaling inequality (assuming existence of \(\rho ,\eta ,\nu \))

    $$\begin{aligned} d\nu \ge \eta +2\rho . \end{aligned}$$
    (1.11)
  2. (ii)

    Let \(d=1\) and \(p=p_\mathrm {c}\). Then, there is a \(K>0\) such that, for any \(t\in {\mathbb N}\),

    $$\begin{aligned} \chi _t\ge K\xi _t\theta _t^2, \end{aligned}$$
    (1.12)

    which implies the hyperscaling equality (assuming existence of at least two among \(\rho ,\eta ,\nu \))

    $$\begin{aligned} \nu =\eta +2\rho . \end{aligned}$$
    (1.13)

Remarks

  1. 1.

    The inequality (1.10) was first derived in [19]. Since its proof is easy and short, we will show it again for convenience. It was used in [19] to prove two other hyperscaling inequalities that also involve critical exponents defined in the off-critical regime. For example, if the susceptibility \(\sum _t\chi _t\) and the relaxation time \(\zeta \) diverge as \(p\uparrow p_\mathrm {c}\) as \((p_\mathrm {c}-p)^{-\gamma }\) and \((p_\mathrm {c}-p)^{-\mu }\) respectively, then, for any \(d\ge 1\), we have

    $$\begin{aligned} (d\nu -2\rho +1)\mu \ge \gamma . \end{aligned}$$
    (1.14)

    If we replace those critical exponents in (1.11) and (1.14) by their mean-field values, then we obtain \(d\ge 4\), which is a complement to the aforementioned lace-expansion results. Therefore, the upper-critical dimension \(d_\mathrm {c}\) for oriented percolation is 4.

  2. 2.

    In general, hyperscaling inequalities are believed to be equalities below and at the model-dependent upper-critical dimension. The values in Table 1 seem to support this belief. The identity (1.13) proves that it is indeed the case for at least \(d=1\). For unoriented percolation, for which \(d_\mathrm {c}=6\), similar results are proven in 2 dimensions by Kesten [11] using the Russo–Seymour–Welsh theorem on the critical box-crossing property [16, 17, 21]. Since the known critical exponents for 2-dimensional unoriented percolation are rational numbers (e.g., \(\beta =5/36\) and \(\gamma =43/18\)), it is natural to believe that there must be some balance (i.e., hyperscaling equalities) among those critical exponents. On the other hand, since the values in Table 1 do not seem to be rational numbers, the hyperscaling equality (1.13) is even more surprising.

  3. 3.

    The main reason why the right-hand side of (1.10) is bigger than its left-hand side is due to the inequality

    $$\begin{aligned} \tau (x,t)&={\mathbb P}_p\big ((o,0)\longrightarrow (x,t)\big )\nonumber \\&\le {\mathbb P}_p\big ((o,0)\longrightarrow t/2,~t/2\longrightarrow (x,t)\big )=\theta _{t/2}^2, \end{aligned}$$
    (1.15)

    where, and in the rest of the paper, we do not care much about possibilities of, e.g., t / 2 not being an integer, since it is easy (but cumbersome) to make the argument rigorous if we introduce floor functions, etc. The last equality in (1.15) is due to reversibility: if we change the direction of each bond and redefine the connectivity in the time-decreasing direction, then we have the identity \({\mathbb P}_p(t/2\longrightarrow (x,t))=\theta _{t/2}\).

  4. 4.

    The following theorem on the critical box-crossing property is the key to show the opposite inequality to (1.15):

Theorem 1.2

[6, Theorem 1.3] Let

$$\begin{aligned} V_p(w,t)&={\mathbb P}_p\Big ([0,w]\times [0,t]\text { is crossed vertically}\Big ),\end{aligned}$$
(1.16)
$$\begin{aligned} H_p(w,t)&={\mathbb P}_p\Big ([0,w]\times [0,t]\text { is crossed from left to right}\Big ). \end{aligned}$$
(1.17)

There exist a constant \(\varepsilon \in (0,1)\) and an increasing sequence of integers \(\{w_t\}_{t\in {\mathbb N}}\) such that, for all \(t\in {\mathbb N}\),

$$\begin{aligned} \varepsilon\le & {} V_{p_\mathrm {c}}(w_t,3t)\le V_{p_\mathrm {c}}(3w_t,t)\le 1-\varepsilon , \end{aligned}$$
(1.18)
$$\begin{aligned} \varepsilon\le & {} H_{p_\mathrm {c}}(3w_t,t)\le H_{p_\mathrm {c}}(w_t,3t)\le 1-\varepsilon . \end{aligned}$$
(1.19)

We will also use (1.18)–(1.19) to control an upper bound on \(\tau (x,t)\) for \(x>jw_t\) that decays exponentially in \(j\in {\mathbb N}\) (see Lemma 2.1 below). This is a key element to show that \(w_t\) is bounded below by an \(\varepsilon \)-dependent positive multiple of \(\xi _t\).

  1. 5.

    Applying (1.10) and (1.12) to [19, (5.1)] and its reverse, respectively, we can readily show that the hyperscaling inequality (1.14) also becomes an equality for \(d=1\), i.e.,

    $$\begin{aligned} (\nu -2\rho +1)\mu =\gamma . \end{aligned}$$
    (1.20)
  2. 6.

    It is easy to show that the hyperscaling inequality (1.11) holds for other finite-range models of oriented percolation and the contact process. It should not be so difficult to prove Theorem 1.2 for the nearest-neighbor models of oriented site percolation and the contact process, hence the hyperscaling equality (1.13) for \(d=1\). However, it is not so obvious to prove a similar statement to Theorem 1.2 for longer-range models. This may be worth further investigation.

2 Proof of Theorem 1.1

Proof of Theorem 1.1(i)

It suffices to prove the inequality (1.10), as the hyperscaling inequality (1.11) immediately follows by using (1.10) at \(p=p_\mathrm {c}\) (and assuming existence of the three critical exponents). First we note that

$$\begin{aligned} \chi _t=\frac{1}{\xi _t^2}\sum _x|x|^2\tau (x,t)\ge 4\sum _{x:|x|\ge 2\xi _t}\tau (x,t), \end{aligned}$$
(2.1)

hence

$$\begin{aligned} \frac{3}{4}\chi _t\le \sum _{x:|x|\le 2\xi _t}\tau (x,t). \end{aligned}$$
(2.2)

By (1.15), the right-hand side is further bounded by \((4\xi _t+1)^d\theta _{t/2}^2\). This completes the proof of (1.10). \(\square \)

To prove Theorem 1.1(ii), we first assume the following key lemma:

Lemma 2.1

Let \(d=1\) and \(p=p_\mathrm {c}\). Let \(\varepsilon \in (0,1)\) and \(w_t\) be the same as in Theorem 1.2.

  1. (i)

    For any \(t\in {\mathbb N}\) and any \(x\in [-\frac{1}{2}w_t,\frac{1}{2}w_t]\),

    $$\begin{aligned} \tau (x,t)\ge \varepsilon ^6\theta _t^2. \end{aligned}$$
    (2.3)
  2. (ii)

    For any \(j,t,x\in {\mathbb N}\) with \(j\ge 2\) and \(jw_t<x\le (j+1)w_t\),

    $$\begin{aligned} \tau (x,t)\le \varepsilon ^{-4}\theta _t^2(1-\varepsilon )^{j-2}. \end{aligned}$$
    (2.4)

Proof of Theorem 1.1(ii) assuming Lemma 2.1

Again, it suffices to prove the inequality (1.12), as the equality (1.13) is a result of the hyperscaling inequality (1.11) for \(d=1\) and the opposite inequality \(\nu \le \eta +2\rho \) that immediately follows from (1.12).

To prove (1.12), we first note that, by (2.3),

$$\begin{aligned} \chi _t\ge 2\sum _{x=1}^{\frac{1}{2}w_t}\tau (x,t)\ge \varepsilon ^6w_t\theta _t^2. \end{aligned}$$
(2.5)

To complete the proof, it suffices to show that \(w_t\) is bounded below by a positive multiple of \(\xi _t\). However, by definition,

$$\begin{aligned} \xi _t^2=2\sum _{x=1}^\infty x^2\frac{\tau (x,t)}{\chi _t}&=2\bigg (\sum _{x=1}^{2w_t} x^2\frac{\tau (x,t)}{\chi _t}+\sum _{j=2}^\infty \sum _{x=jw_t+1}^{(j+1)w_t}x^2 \frac{\tau (x,t)}{\chi _t}\bigg )\nonumber \\&\le 2w_t^2\bigg (4+\sum _{j=2}^\infty (j+1)^2\sum _{x=jw_t+1}^{(j+1)w_t}\frac{\tau (x,t)}{\chi _t}\bigg ). \end{aligned}$$
(2.6)

Then, by using (2.4)–(2.5), we obtain

(2.7)

As a result,

$$\begin{aligned} \chi _t\ge \underbrace{\frac{\varepsilon ^6}{\sqrt{2}}\bigg (4+\varepsilon ^{-10}\sum _{j=2}^\infty (j+1)^2(1-\varepsilon )^{j-2}\bigg )^{-1/2}}_{=K}\xi _t\theta _t^2. \end{aligned}$$
(2.8)

This completes the proof of (1.12). \(\square \)

The rest of the paper is devoted to showing Lemma 2.1.

Proof of Lemma 2.1(i)

First we note that, for \(1\le x\le \frac{1}{2}w_t\), the event \((o,0)\longrightarrow (x,t)\) occurs if the following four increasing events occur:

  • \((o,0)\longrightarrow t\) in \([-w_t,w_t]\times [0,t]\),

  • \(0\longrightarrow (x,t)\) in \([x-w_t,x+w_t]\times [0,t]\),

  • \([-\frac{3}{2}w_t,\frac{3}{2}w_t]\times [0,t]\) is crossed from left to right,

  • \([-\frac{3}{2}w_t,\frac{3}{2}w_t]\times [0,t]\) is crossed from right to left.

The last two events take care of the possibility that the forward cluster from the origin (o, 0) and the backward cluster from (x, t) do not collide. Using the FKG inequality (see, e.g., [7]), translation invariance and the reversibility explained below (1.15), we obtain

$$\begin{aligned} \tau (x,t)\ge {\mathbb P}_p\Big ((o,0)\longrightarrow t\text { in }[-w_t,w_t]\times [0,t]\Big )^2 H_p(3w_t,t)^2. \end{aligned}$$
(2.9)

We further note that the event \((o,0)\longrightarrow t\) in \([-w_t,w_t]\times [0,t]\) occurs if the following three increasing events occur:

  • \((o,0)\longrightarrow t\),

  • \([0,w_t]\times [0,t]\) is crossed vertically,

  • \([-w_t,0]\times [0,t]\) is crossed vertically.

Again, by the FKG inequality, translation invariance and the monotonicity \(V_p(w_t,t)\ge V_p(w_t,3t)\), we obtain

$$\begin{aligned} {\mathbb P}_p\Big ((o,0)\longrightarrow t\text { in }[-w_t,w_t]\times [0,t]\Big )\ge \theta _tV_p(w_t, 3t)^2, \end{aligned}$$
(2.10)

hence

$$\begin{aligned} \tau (x,t)\ge \theta _t^2V_p(w_t,3t)^4H_p(3w_t,t)^2. \end{aligned}$$
(2.11)

The inequality (2.3) follows from the above inequality at \(p=p_\mathrm {c}\) and (1.18)–(1.19). \(\square \)

Proof of Lemma 2.1(ii)

Recall that \(j\ge 2\) and \(jw_t<x\le (j+1)w_t\). If \((o,0)\longrightarrow (x,t)\), then the following three independent events occur:

  • (o, 0) is connected to the boundary \(\partial B_o\) of the box \(B_o\equiv [-w_t,w_t]\times [0,t]\),

  • \([w_t,(j-1)w_t]\times [0,t]\) is crossed from left to right,

  • (x, t) is connected from the boundary \(\partial B_x\) of the box \(B_x=[(j-1)w_t,(j+2)w_t]\times [0,t]\).

By this observation and using \(H_p((j-2)w_t,t)\le H_p(w_t,t)^{j-2}\le H_p(w_t,3t)^{j-2}\), we obtain

$$\begin{aligned} \tau (x,t)\le {\mathbb P}_p\Big ((o,0)\longrightarrow \partial B_o\Big )H_p(w_t,3t)^{j-2} {\mathbb P}_p\Big (\partial B_x\longrightarrow (x,t)\Big ). \end{aligned}$$
(2.12)

However, by reversibility and monotonicity, we have

$$\begin{aligned} {\mathbb P}_p\Big (\partial B_x\longrightarrow (x,t)\Big )\le {\mathbb P}_p\Big ((o,0)\longrightarrow \partial B_o\Big ). \end{aligned}$$
(2.13)

Therefore,

$$\begin{aligned} \tau (x,t)\le {\mathbb P}_p\Big ((o,0)\longrightarrow \partial B_o\Big )^2H_p(w_t,t)^{j-2}. \end{aligned}$$
(2.14)

To bound the probability on the right-hand side by \(\theta _t\), we borrow the idea in the proof of [6, (4.7)]. First, we note that \((o,0)\longrightarrow t\) if the following three increasing events occur:

  • \((o,0)\longrightarrow \partial B_o\),

  • \([0,w_t]\times [0,t]\) is crossed vertically,

  • \([-w_t,0]\times [0,t]\) is crossed vertically.

By the FKG inequality, translation invariance and the monotonicity \(V_p(w_t,t)\ge V_p(w_t,3t)\), we obtain

$$\begin{aligned} \theta _t\ge {\mathbb P}_p\Big ((o,0)\longrightarrow \partial B_o\Big )V_p(w_t,3t)^2. \end{aligned}$$
(2.15)

To summarize the above computations at \(p=p_\mathrm {c}\), we arrived at

$$\begin{aligned} \tau (x,t)\le \bigg (\frac{\theta _t}{V_{p_\mathrm {c}}(w_t,3t)^2}\bigg )^2H_{p_\mathrm {c}}(w_t, 3t)^{j-2}\le \varepsilon ^{-4}\theta _t^2(1-\varepsilon )^{j-2}, \end{aligned}$$
(2.16)

as required. \(\square \)