One-arm probability ($\pi (r)$)

The one-arm probability $\pi (r)$ is the probability that a specified vertex has an open path to distance $r$.

For independent bond percolation on the cubic lattice, define ${\tau }_p(x,y)={\mathbb{P}}_p(x\leftrightarrow y)$ and $g_n={\mathbb{P}}_p(0\leftrightarrow \partial {\Lambda }_n)$, with $g_0=1$. Split an open self-avoiding walk at its first radius-$m$ boundary vertex $y$. The remaining segment reaches distance at least $n-m$ from $y$, using disjoint edges. The BK inequality, translation invariance and the union bound yield the displayed estimate. Keeping the individual connection weights can be stronger than replacing their sum by $|\partial {\Lambda }_m|g_m$.

At $p=1/2$ for independent bond percolation on the square lattice, there exist $C<\infty$ and $\alpha >0$ with ${\mathbb{P}}_{1/2}(0\leftrightarrow \partial [-n,n{]}^2)\le C{n}^{-\alpha }$. The Russo-Seymour-Welsh theorem and the Harris-FKG inequality give a uniform positive probability of a closed dual graph cycle around each geometrically spaced square ring. The independent annular barriers for percolation give the polynomial bound and imply zero critical percolation probability. This does not determine the exact critical exponent.

Closed dual graph cycles in disjoint square rings depend on disjoint primal edge sets. In independent bond percolation their barrier events are independent. If each event has probability at least $\delta >0$, an open graph path from the origin across $k$ rings has probability at most $(1-\delta {)}^k$. A geometric sequence of radii turns this estimate into the polynomial critical one-arm upper bound.

For bond percolation on the square lattice, put $g_n={\mathbb{P}}_p(0\leftrightarrow \partial [-n,n{]}^2)$. Splitting an open self-avoiding walk at its first visit to the radius-$m$ boundary gives disjoint certificates for reaching a boundary graph vertex and then reaching radius $n$ about that graph vertex. The Van den Berg-Kesten inequality and the union bound give $g_{m+n}\le 8m{g}_m{g}_n$ for positive $m,n$. With $g_0=1$ and $|\partial {\Lambda }_0|=1$, the boundary-cardinality form holds for zero indices as well.

For bond percolation on the square lattice, the one-arm probability $g_n={\mathbb{P}}_p(0\leftrightarrow \partial [-n,n{]}^2)$ has a root limit of a sequence $\gamma (p)\in [p,1]$. The BK boundary-splitting estimate implies that $32n^2{g}_n$ is submultiplicative for positive integers. Applying the Fekete lemma to its logarithm proves existence when $p>0$; at $p=0$ the rate is zero. The same rate is the root limit of the percolation two-point connection probability along a coordinate axis, by the reflection lower bound for two-point percolation.

For independent bond percolation of density $0<p<1$ on the cubic lattice in dimension $d\ge 2$, let ${\beta }_n$ be the one-arm probability for reaching maximum-norm distance $n$. The weighted BK boundary-splitting estimate gives ${\beta }_{m+n}\le |\partial {\Lambda }_n|{\beta }_m{\beta }_n$. Since the logarithm of this boundary size is $o(n)$, the asymmetrically almost-subadditive sequence argument applies to $\log {\beta }_n$. The rate exists and lies in $[0,-\log p]$; positive rate gives exponential decay.

The Russo-Seymour-Welsh theorem and the Harris-FKG inequality imply $\pi (2r)\ge c\pi (r)$ for a scale-independent constant $c>0$.

RSW rectangle crossings and positive association join two separated one-arm events with probability bounded below uniformly over scale.