= Solution
First obtain a <polynomial critical one-arm upper bound> from part (a). Fix a sufficiently large integer radius $r$. Use rotated-coordinate boxes $B_r=[-r,r]^2$ for this geometric construction, in the axes of part (a). Surround $B_r$ by a square ring inside $B_{3r}$. It can be assembled from the four strips
$$
[-3r,3r]\times[r,3r],\quad[-3r,3r]\times[-3r,-r],\quad[-3r,-r]\times[-3r,3r],\quad[r,3r]\times[-3r,3r].
$$
The first two require horizontal dual crossings; the last two require vertical dual crossings. Their corner overlaps are squares, so the adjacent long crossings intersect there and their union contains a dual <graph cycle> surrounding the inner box. Translate the strips by the half-lattice displacement of the <planar dual graph> and adjust the endpoints by one lattice unit if needed. All aspect ratios remain bounded. Part (a) and the same overlap gluing give a uniform lower bound $c_1>0$ for each dual crossing at $1/2$. Applying the <Harris-FKG inequality> to these <decreasing events> in the primal configuration gives a closed dual circuit with <probability> at least $\delta=c_1^4>0$. We may replace $\delta$ by a smaller number in $(0,1)$.
Use the <independent annular barriers for percolation> at radii $r_j=r_0 10^j$. Their supporting <edge> sets are disjoint, so the circuit events are independent. In the original unrotated coordinates, $B_{3r}$ lies inside $\Lambda_{5r}$ and the ring still separates the origin from infinity. These fixed geometric comparisons let us bound the original $g_N$. An open <graph path> from $0$ to distance $N$ must avoid every such closed dual circuit inside $\Lambda_N$. Thus, for some constants $C<\infty$ and $\alpha>0$,
$$
g_N(1/2)\leq(1-\delta)^{\lfloor\log_{10}(N/(5r_0))\rfloor+1}\leq C N^{-\alpha},\qquad
\alpha=\frac{-\log(1-\delta)}{\log10}>0,
$$
where the first bound is used only when at least one of the indicated rings fits; increasing $C$ handles smaller $N$.
Now let $\varepsilon=p-1/2>0$ and use the <monotone coupling of Bernoulli percolation> on <edges>. The event $\{0\leftrightarrow\partial\Lambda_N\}$ only needs <edges> whose endpoints lie in $\Lambda_N$, because a first boundary hit gives an internal <graph path>. There are exactly $4N(2N+1)$ such <edges>. Each changes state between parameters $1/2$ and $p$ with <probability> $\varepsilon$. The <union bound> therefore gives the <finite-box comparison for percolation parameters>
$$
\theta(p)\leq g_N(p)\leq g_N(1/2)+4N(2N+1)\varepsilon\leq C N^{-\alpha}+12N^2\varepsilon.
$$
For sufficiently small $\varepsilon$, choose $N=\lfloor\varepsilon^{-1/(\alpha+2)}\rfloor$, so that $N$ is at least half the unrounded value. Both terms then have order $\varepsilon^{\alpha/(\alpha+2)}$. In particular,
$$
\boxed{\theta(p)\leq A(p-1/2)^\beta,\qquad\beta=\frac\alpha{\alpha+2}>0}
$$
for a finite $A$. Increase $A$ to cover the remaining compact range of parameters by $\theta(p)\leq1$. This <near-critical percolation power upper bound> requires only a positive crossing constant, not the exact value of a critical exponent. It also implies $\theta(1/2)=0$ through the critical <one-arm probability> bound.
Back to article page
