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In the case of oriented percolation the following regarding your question are rigorously known in any dimension. Presumably these results are also known for non-oriented percolation in half-spaces because in this case it is also known that there is no infinite cluster at criticality (Barsky, Grimmett and Newman 1991) and furthermore similar arguments apply for a continuous time percolation model where there is no orientation (i.e. time axis, see, Bezuidenhout and Grimmett 1991).

Let $\theta(p) = P(|C| = \infty)$, where $C$ is the cluster of the origin in oriented percolation with supercritical retention parameter $p > p_{c}$. Further, let $\theta(p; \gamma) = 1 - E_{p}(e^{\gamma |C|})$, $\gamma>0$. It is known that (Bezuidenhout and Grimmett 1991 or Aizenman and Jung 1991) there exist $a,b>0$ such that \begin{equation} \theta(p; \gamma) \geq a \gamma^{1 / 2}, \mbox{ for } 0<\gamma<b \end{equation} By means of emulating Tauberian theory arguments (due to Aizenmann and Barsky (1989), see Grimmett's v. 1991 book, Proposition 10.29) and using that $\theta(p_{c}) = 0$ (Bezuidenhout and Grimmett (1990)), the last display above is tantamount to \begin{equation} P( |C| \geq m) \approx m^{-\frac{1}{\delta}},\mbox{ } \delta \geq 2, \end{equation} where $\approx$ is meant in the logarithmic sence here ($a_{m} \approx b_{m}$ denotes $\frac{\log a_{m}}{\log b_{m}} \rightarrow 1$, as $m \rightarrow \infty$). Note that this implies that $P( |C| \geq m) \geq m^{-1 / 2}$, for all $m$ sufficiently large. In dimension 2 in particular the critical exponent for the probability of $\{0 \leftrightarrow \partial B(n)\}$, that is the origin being connected to the boundary of a box of size $n$, is furthermore known. Durrett (1988) proves that $\lim_{n \rightarrow \infty} \sqrt{n}P(0 \leftrightarrow \partial B(n)) = +\infty$. I do not know if the corresponding result in higher dimensions is rigorously known (without assuming the hyperscaling relations), and actually I would greatly appreciate if anyone had a hint about this!

Hope this helps and many thanks to anyone that might be able to help me with my query.

In the case of oriented percolation the following regarding your question are rigorously known in any dimension. Presumably these results are also known for non-oriented percolation in half-spaces because in this case it is also known that there is no infinite cluster at criticality (Barsky, Grimmett and Newman 1991) and furthermore similar arguments apply for a continuous time percolation model where there is no orientation (i.e. time axis, see, Bezuidenhout and Grimmett 1991).

Let $\theta(p) = P(|C| = \infty)$, where $C$ is the cluster of the origin in oriented percolation with supercritical retention parameter $p > p_{c}$. Further, let $\theta(p; \gamma) = 1 - E_{p}(e^{\gamma |C|})$, $\gamma>0$. It is known that (Aizenman and Barsky 1987, Bezuidenhout and Grimmett 1991 or Aizenman and Jung 1991) there exist $a,b>0$ such that \begin{equation} \theta(p; \gamma) \geq a \gamma^{1 / 2}, \mbox{ for } 0<\gamma<b \end{equation} By means of emulating Tauberian theory arguments (due to Aizenmann and Barsky (1989), see Grimmett's v. 1991 book, Proposition 10.29) and using that $\theta(p_{c}) = 0$ (Bezuidenhout and Grimmett (1990)), the last display above is tantamount to \begin{equation} P( |C| \geq m) \approx m^{-\frac{1}{\delta}},\mbox{ } \delta \geq 2, \end{equation} where $\approx$ is meant in the logarithmic sence here ($a_{m} \approx b_{m}$ denotes $\frac{\log a_{m}}{\log b_{m}} \rightarrow 1$, as $m \rightarrow \infty$). Note that this implies that $P( |C| \geq m) \geq m^{-1 / 2}$, for all $m$ sufficiently large. In dimension 2 in particular the critical exponent for the probability of $\{0 \leftrightarrow \partial B(n)\}$, that is the origin being connected to the boundary of a box of size $n$, is furthermore known. Durrett (1988) proves that $\lim_{n \rightarrow \infty} \sqrt{n}P(0 \leftrightarrow \partial B(n)) = +\infty$. Hope this helps.

In the case of oriented percolation the following regarding your question are rigorously known in any dimension. Presumably these results are also known for non-oriented percolation in half-spaces because in this case it is also known that there is no infinite cluster at criticality (Basky, Grimmett and Newman 1991) and furthermore similar arguments apply for a continuous time percolation model where there is no orientation (i.e. time axis, see, Bezuidenhout and Grimmett 1991).

Let $\theta(p) = P(|C| = \infty)$, where $C$ is the cluster of the origin in oriented percolation with supercritical retention parameter $p > p_{c}$. Further, let $\theta(p; \gamma) = 1 - E_{p}(e^{\gamma |C|})$, $\gamma>0$. It is known that (Bezuidenhout and Grimmett 1991 or Aizenman and Jung 1991) there exist $a,b>0$ such that \begin{equation} \theta(p; \gamma) \geq a \gamma^{1 / 2}, \mbox{ for } 0<\gamma<b \end{equation} By means of emulating Tauberian theory arguments (due to Aizenmann and Barsky (1989), see Grimmett's v. 1991 book, Proposition 10.29) and using that $\theta(p_{c}) = 0$ (Bezuidenhout and Grimmett (1990)), the last display above is tantamount to \begin{equation} P( |C| \geq m) \approx m^{-\frac{1}{\delta}},\mbox{ } \delta \geq 2, \end{equation} where $\approx$ is meant in the logarithmic sence here ($a_{m} \approx b_{m}$ denotes $\frac{\log a_{m}}{\log b_{m}} \rightarrow 1$, as $m \rightarrow \infty$). Note that this implies that $P( |C| \geq m) \geq m^{-1 / 2}$, for all $m$ sufficiently large. In dimension 2 in particular the critical exponent for the probability of $\{0 \leftrightarrow \partial B(n)\}$, that is the origin being connected to the boundary of a box of size $n$, is furthermore known. Durrett (1988) proves that $\lim_{n \rightarrow \infty} \sqrt{n}P(0 \leftrightarrow \partial B(n)) = +\infty$. I do not know if the corresponding result in higher dimensions is rigorously known (without assuming the hyperscaling relations), and actually I would greatly appreciate if anyone had a hint about this!

Hope this helps and many thanks to anyone that might be able to help me with my query.

In the case of oriented percolation the following regarding your question are rigorously known in any dimension. Presumably these results are also known for non-oriented percolation in half-spaces because in this case it is also known that there is no infinite cluster at criticality (Barsky, Grimmett and Newman 1991) and furthermore similar arguments apply for a continuous time percolation model where there is no orientation (i.e. time axis, see, Bezuidenhout and Grimmett 1991).

Let $\theta(p) = P(|C| = \infty)$, where $C$ is the cluster of the origin in oriented percolation with supercritical retention parameter $p > p_{c}$. Further, let $\theta(p; \gamma) = 1 - E_{p}(e^{\gamma |C|})$, $\gamma>0$. It is known that (Bezuidenhout and Grimmett 1991 or Aizenman and Jung 1991) there exist $a,b>0$ such that \begin{equation} \theta(p; \gamma) \geq a \gamma^{1 / 2}, \mbox{ for } 0<\gamma<b \end{equation} By means of emulating Tauberian theory arguments (due to Aizenmann and Barsky (1989), see Grimmett's v. 1991 book, Proposition 10.29) and using that $\theta(p_{c}) = 0$ (Bezuidenhout and Grimmett (1990)), the last display above is tantamount to \begin{equation} P( |C| \geq m) \approx m^{-\frac{1}{\delta}},\mbox{ } \delta \geq 2, \end{equation} where $\approx$ is meant in the logarithmic sence here ($a_{m} \approx b_{m}$ denotes $\frac{\log a_{m}}{\log b_{m}} \rightarrow 1$, as $m \rightarrow \infty$). Note that this implies that $P( |C| \geq m) \geq m^{-1 / 2}$, for all $m$ sufficiently large. In dimension 2 in particular the critical exponent for the probability of $\{0 \leftrightarrow \partial B(n)\}$, that is the origin being connected to the boundary of a box of size $n$, is furthermore known. Durrett (1988) proves that $\lim_{n \rightarrow \infty} \sqrt{n}P(0 \leftrightarrow \partial B(n)) = +\infty$. I do not know if the corresponding result in higher dimensions is rigorously known (without assuming the hyperscaling relations), and actually I would greatly appreciate if anyone had a hint about this!

Hope this helps and many thanks to anyone that might be able to help me with my query.

In the case of oriented percolation the following regarding your question are rigorously known in any dimension. Presumably these results are also known for non-oriented percolation in half-spaces because in this case it is also known that there is no infinite cluster at criticality (Basky, Grimmett and Newman 1991) and furthermore similar arguments apply for a continuous time percolation model where there is no orientation (i.e. time axis, see, Bezuidenhout and Grimmett 1991).

Let $\theta(p) = P(|C| = \infty)$, where $C$ is the cluster of the origin in oriented percolation with supercritical retention parameter $p > p_{c}$. Further, let $\theta(p; \gamma) = 1 - E_{p}(e^{\gamma |C|})$, $\gamma>0$. It is known that (Bezuidenhout and Grimmett 1991 or Aizenman and Jung 1991) there exist $a,b>0$ such that \begin{equation} \theta(p; \gamma) \geq a \gamma^{1 / 2}, \mbox{ for } 0<\gamma<b \end{equation} By means of emulating Tauberian theory arguments (due to Aizenmann and Barsky (1989), see Grimmett's v. 1991 book, Proposition 10.29) and using that $\theta(p_{c}) = 0$ (Bezuidenhout and Grimmett (1990)), the last display is tantamount to that \begin{equation} P( |C| \geq m) \approx m^{-\frac{1}{\delta}},\mbox{ } \delta \geq 2, \end{equation} where $\approx$ is meant in the logarithmic sence here ($a_{m} \approx b_{m}$ denotes $\frac{\log a_{m}}{\log b_{m}} \rightarrow 1$, as $m \rightarrow \infty$). Note that this implies that $P( |C| \geq m) \geq m^{-1 / 2}$, for all $m$ sufficiently large. In dimension 2 in particular the critical exponent for the probability of $\{0 \leftrightarrow \partial B(n)\}$, that is the origin being connected to a box of size $n$, is furthermore known. Durrett (1988) proves that $\lim_{n \rightarrow \infty} \sqrt{n}P(0 \leftrightarrow \partial B(n)) = +\infty$. I do not know if the corresponding result in higher dimensions is rigorously known (without assuming the hyperscaling relations), and actually I would greatly appreciate if anyone had a hint about this!

Hope this helps and many thanks to anyone that might be able to help me with my query.

In the case of oriented percolation the following regarding your question are rigorously known in any dimension. Presumably these results are also known for non-oriented percolation in half-spaces because in this case it is also known that there is no infinite cluster at criticality (Basky, Grimmett and Newman 1991) and furthermore similar arguments apply for a continuous time percolation model where there is no orientation (i.e. time axis, see, Bezuidenhout and Grimmett 1991).

Let $\theta(p) = P(|C| = \infty)$, where $C$ is the cluster of the origin in oriented percolation with supercritical retention parameter $p > p_{c}$. Further, let $\theta(p; \gamma) = 1 - E_{p}(e^{\gamma |C|})$, $\gamma>0$. It is known that (Bezuidenhout and Grimmett 1991 or Aizenman and Jung 1991) there exist $a,b>0$ such that \begin{equation} \theta(p; \gamma) \geq a \gamma^{1 / 2}, \mbox{ for } 0<\gamma<b \end{equation} By means of emulating Tauberian theory arguments (due to Aizenmann and Barsky (1989), see Grimmett's v. 1991 book, Proposition 10.29) and using that $\theta(p_{c}) = 0$ (Bezuidenhout and Grimmett (1990)), the last display above is tantamount to \begin{equation} P( |C| \geq m) \approx m^{-\frac{1}{\delta}},\mbox{ } \delta \geq 2, \end{equation} where $\approx$ is meant in the logarithmic sence here ($a_{m} \approx b_{m}$ denotes $\frac{\log a_{m}}{\log b_{m}} \rightarrow 1$, as $m \rightarrow \infty$). Note that this implies that $P( |C| \geq m) \geq m^{-1 / 2}$, for all $m$ sufficiently large. In dimension 2 in particular the critical exponent for the probability of $\{0 \leftrightarrow \partial B(n)\}$, that is the origin being connected to the boundary of a box of size $n$, is furthermore known. Durrett (1988) proves that $\lim_{n \rightarrow \infty} \sqrt{n}P(0 \leftrightarrow \partial B(n)) = +\infty$. I do not know if the corresponding result in higher dimensions is rigorously known (without assuming the hyperscaling relations), and actually I would greatly appreciate if anyone had a hint about this!

Hope this helps and many thanks to anyone that might be able to help me with my query.