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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 stripsThe 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 $\alpha >0$,where the first bound is used only when at least one of the indicated rings fits; increasing $C$ handles smaller $N$.

Now let $\epsilon =p-1/2>0$ 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 $\epsilon$. The union bound therefore gives the finite-box comparison for percolation parametersFor sufficiently small $\epsilon$, choose $N=\lfloor {\epsilon }^{-1/(\alpha +2)}\rfloor$, so that $N$ is at least half the unrounded value. Both terms then have order ${\epsilon }^{\alpha /(\alpha +2)}$. In particular,for a finite $A$. Increase $A$ to cover the remaining compact range of parameters by $\theta (p)\le 1$. 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.

The strips are nested. Every connection permitted inside $T_k$ is also permitted inside $T_{k+1}$, so $q_k(n)\le q_{k+1}(n)$ for every fixed separation. ConsequentlyThe preceding nonnegative bound makes this a decreasing sequence bounded below. By monotone convergence of real sequences,There is no assertion that this limit of a sequence is strictly positive: positivity at each fixed width need not survive an increasing-width limit of a sequence.

One can also identify the limit of a sequence. Let $q(n)$ be the percolation two-point connection probability in the whole square lattice. Every finite connecting graph path has a bounded vertical extent, so $q_k(n)\uparrow q(n)$. Commuting infima gives the strip approximation to the planar connection decay rate:The same positive-association argument identifies the final infimum with the whole-plane normalized logarithmic limit of a sequence. This is an interchange of infima justified by monotonicity at fixed $n$. For example, when $p>1/2$, the threshold proved in question 2 and uniqueness give an infinite percolation cluster with root probability $\theta (p)>0$. The Harris-FKG inequality makes the probability that both endpoints belong to it at least $\theta (p{)}^2$, so $q(n)\ge \theta (p{)}^2$. The whole-plane rate, and hence ${\lim }_k{f}_k(p)$, is then zero, despite the strict positivity of every finite-strip rate.

Put $q_k(n)={\mathbb{P}}_p((0,0)\leftrightarrow (n,0)\text{ inside }{T}_k)$, with $q_k(0)=1$. The connection event is increasing in the open edges. Apply the Harris-FKG inequality to connection from $(0,0)$ to $(n,0)$ and connection from $(n,0)$ to $(n+m,0)$, both constrained to the strip. Their intersection implies connection between the outer endpoints, while horizontal translation invariance makes the second event's probability $q_k(m)$. HenceThese events may depend on infinitely many strip edges. To justify the inequality directly, first restrict each connecting graph path to a finite box, apply Harris-FKG inequality in the finite independent edge family, and then let the box grow. Every connection has a finite witness graph path, so continuity from below gives the stated inequality.

The direct horizontal graph path supplies $q_k(n)\ge p^n>0$, and probabilities are at most one. Thus $a_k(n)=-\log q_k(n)$ is a finite nonnegative subadditive sequence with $a_k(n)\le -n\log p$. The Fekete lemma now gives a finite connection decay rate in a percolation strip:Each individual ratio is at least its infimum, so $-\log q_k(n)\ge n{f}_k(p)$ and thereforeAt $n=0$ this reads $1\le 1$. Only nonnegative separations are intended by this notation; horizontal reflection supplies the analogous bound with $|n|$ for negative separations.

We use planar duality for rectangle crossings, Harris-FKG inequality and the permitted exponential decay of subcritical percolation. A critical-point detail matters: uniqueness stated only for $p>p_c$ cannot by itself be applied at $p=p_c$. We prove the Burton-Keane theorem by the boundary counting proof of percolation uniqueness, establishing at most one infinite percolation cluster at every parameter and avoiding that gap.

For $0<p<1$, the number $N$ of infinite percolation clusters is almost surely constant by translation ergodicity of Bernoulli percolation. It cannot be a finite constant greater than one. A sufficiently large box meets two different infinite percolation clusters with positive probability; opening its finitely many interior edges joins them without creating a new infinite percolation cluster, decreasing $N$. Finite modification of Bernoulli percolation gives positive probability to this modification, contradicting constancy.

Nor can $N=\infty$. A finite box then meets three different infinite percolation clusters with positive probability. Select one infinite exterior branch from each. In the box and a finite collar retain just an embedded three-armed tree joining those branches and close all other incident edges there. Its branching graph vertex separates three infinite components when removed, so is a trifurcation vertex in percolation. All selections concern finitely many possible entrances and edge patterns; finite-energy property of Bernoulli percolation gives positive probability for at least one such pattern. Translation invariance then gives a common positive trifurcation probability $q$.

Here is the counting contradiction in detail. For a finite box $W$, contract each open component outside $W$ that touches $W$ to a boundary terminal. In each cluster, the resulting incidence graph is finite and connected. Every trifurcation in $W$ separates at least three sets of terminals, since each infinite branch must leave $W$. A minimal subtree joining all terminals must therefore contain that graph vertex with degree of a vertex at least three. Its leaves are terminals. The tree identity ${\sum }_v(\mathrm{deg}(v)-2)=-2$ bounds the number of those branching graph vertices by the terminal count. Different terminals are represented by different exterior graph neighbours of $W$. Thus the trifurcation boundary-counting lemma givesFor $W=[-r,r{]}^2\cap {\mathbb{Z}}^2$, the two sizes are $(2r+1{)}^2$ and $8r+4$. Letting $r\to \infty$ contradicts $q>0$. Therefore $N\in \{0,1\}$ almost surely, with the same conclusion for the translated planar dual graph.

Now prove absence of percolation at $p=1/2$ by Zhang's argument. Suppose instead that $\theta (1/2)>0$. Almost surely an infinite primal cluster exists, and self-duality gives an infinite dual cluster as well. An infinite connected subgraph of this locally finite lattice contains a graph ray, by the König infinity lemma. Expanding square boxes meet the infinite percolation clusters with probability tending to one.

For $B_r=[-r,r{]}^2\cap {\mathbb{Z}}^2$, let $A_N,A_E,A_S,A_W$ mean that an open graph ray takes an outward edge through the indicated side and subsequently uses graph vertices outside $B_r$. The union occurs whenever the box meets an infinite percolation cluster: take the last exit of an infinite graph ray from the finite box. Conversely an exterior arm supplies an infinite percolation cluster meeting its boundary. Quarter-turn symmetry makes the four probabilities equal. Their complements are decreasing events, so the square-root trick for positively associated events givesFor the planar dual graph use the box with boundary coordinates $\pm (r+1/2)$ and define its exterior side-arm events in the same way. This box also has quarter-turn symmetry and meets the dual infinite percolation cluster with probability tending to one. Thus each dual side has an infinite exterior arm with probability tending to one. Primal outward edges cross this dual-box contour on the corresponding sides; their remaining graph vertices stay outside it. A union bound makes the simultaneous event of primal north/south arms and dual east/west arms have positive probability for a sufficiently large box.

The four arms alternate around the contour. Open all edges with both endpoints in $B_r$, leaving all outward and exterior edges intact. This joins the two primal entrance points and preserves all four exterior arms. Finite-energy property of Bernoulli percolation keeps the event's probability positive. The primal north and south arms are now connected through the box. The two dual arms must lie in different infinite dual clusters. To see the separation, a hypothetical finite dual graph path joining the east and west arms cannot enter the dual-box interior: its boundary-crossing and interior edges cross primal edges that were just opened. Erase its loops and cut it at successive hits of the contour. A resulting exterior dual crosscut, together with the corresponding contour arc, encloses one of the two intervening primal entrance points. The infinite primal graph ray from that point would have to cross a dual-open edge, which is impossible. This is the planar separation used in the alternating arms argument at the self-dual percolation parameter. It contradicts uniqueness of the infinite dual cluster. HenceFinally suppose $p_c(2)>1/2$. At $p=1/2$ the permitted exponential decay of subcritical percolation would supply $C,c>0$ with ${\mathbb{P}}_{1/2}(v\leftrightarrow \partial B_n(v))\le C{e}^{-cn}$. Use the rectangle with graph vertices $\{0,\dots ,n\}\times \{0,\dots ,n-1\}$, and let $H_n$ be its left-to-right open crossing. Planar duality for rectangle crossings identifies its complement with a dual top-to-bottom crossing of a rectangle of width $n-1$ and height $n$. Rotation and translation give the same crossing law; side edges at the entrance and exit boundaries are irrelevant. Thus the exact self-dual rectangle crossing probability isBut a crossing has some starting graph vertex on its $n$-vertex left side connected to distance $n$. The union bound and exponential decay of subcritical percolation imply ${\mathbb{P}}_{1/2}(H_n)\le nC{e}^{-cn}\to 0$, a contradiction. ThereforeNo Russo-Seymour-Welsh theorem or continuity assertion for $\theta$ is being used, and critical uniqueness was proved rather than inferred from the supercritical hypothesis.

For bond percolation, the Harris-FKG inequality states that any two increasing events $A,B$ satisfyThe same inequality holds for two decreasing events, either by reversing the coordinate order or by taking complements in the two-event identity. Intersections of decreasing events are decreasing events, so induction givesWrite $a={\mathbb{P}}_p(A_1)=\cdots ={\mathbb{P}}_p(A_n)$ and $u={\mathbb{P}}_p({\bigcup }_j{A}_j)$. Then $1-u\ge (1-a{)}^n$. Taking the nonnegative $n$-th root and rearranging yields the square-root trick for positively associated events:No independence among the events is needed.

For $m\ge 1$, let $S_m$ be the set of vertices of $B(m)$ connected to $\partial B(m)$ by an open path in a graph using only edges of $B(m)$. Every infinite-cluster vertex of $B(m)$ belongs to $S_m$, by stopping an infinite open path in a graph at its first boundary visit. Hence the increasing eventhas probability at least $\theta (p)/2$ by part (a). Importantly, $E_m$ depends only on the edges with both endpoints in $B(m)$.

Open all $8m$ perimeter edges joining successive vertices of $\partial B(m)$. They form a cycle in a graph, and opening them joins every vertex of $S_m$ into one percolation cluster, containing the specified perimeter vertex $v_m=(m,0)$. The Harris-FKG inequality bounds the probability of this event together with $E_m$ below by $p^{8m}\theta (p)/2$.

Close all $8m+4$ edges in the edge boundary of $B(m)$. These edges are distinct from the internal edges already considered, so their states are independent of those events. The resulting percolation cluster of $v_m$ is finite, contained in $B(m)$, and has at least $|S_m|$ vertices. Therefore, by translating $v_m$ to the origin,For $n\ge 1$, choose $m=\lceil \sqrt{n/(2\theta (p))}\rceil$. Then the threshold is at least $n$, and $m\le (1+(2\theta (p){)}^{-1/2})\sqrt{n}$. Put $A=-\log p$ and $B=-\log (1-p)$; both are positive. The preceding lower bound is at least $e^{-c_1\sqrt{n}}$ with, for example,The construction modifies only order-$m$ boundary edges while trapping order-$m^2$ vertices, which explains the stretched exponential scale. Here $\mathbb{N}$ is interpreted as the positive integers; the proposed lower bound at $n=0$ would be false because $\theta (p)>0$.

The Harris-FKG inequality givesHere is an induction proof of the product-measure version, also called Harris' inequality. The result is immediate for one coordinate. For $n$ coordinates, defineThe sections of an increasing event are increasing events in the first $n-1$ coordinates, so the induction hypothesis givesMonotonicity gives $a_1\ge a_0$ and $b_1\ge b_0$. The difference between the right side andisThis closes the induction and proves positive association.

Set $N=3^j$. At $p=1/2$, first use the Russo-Seymour-Welsh theorem to produce, with probability bounded below independently of $N$, two separated open crossings from left to right and a closed crossing between them. Explore the interface separating the lower open cluster from the adjacent closed cluster until it reaches the opposite macroscopic boundary. The explored interface supplies two alternating arms, while fresh RSW crossings in the unexplored regions supply the other two.

Repeat this construction in the geometrically separated scale bandsThe domain Markov property of a percolation exploration leaves unrevealed sites with their original independent critical law. RSW and the Harris-FKG inequality therefore give a uniform conditional probability $c>0$ that the required open and closed connections occur in each band. On that event the exploration identifies a site having four alternating arms to the four sides of $W_N$, hence a pivotal site for $E_N$.

Choose the candidates in disjoint scale bands, so successful bands give distinct pivotal sites. If $Y_k$ indicates success at scale $k$, thenThusfor $j\ge 2$, after adjusting the positive constant. This is the required logarithmic-in-$N$ lower bound.

Both $\{\#C_N\ge \theta \#H_N/2\}$ and $\{0\in C_N\}$ are increasing events. The first has probability at least $\theta /2$ by part (a2), while the second has probability at least $\theta$ because an infinite origin cluster reaches $h_N$. The Harris-FKG inequality gives

The conditional open probability in part (b) is increasing in the states of all other edges, because adding edges can only connect the two endpoints. A common-uniform update therefore preserves the coordinatewise order. Start two copies from all closed and use the same updates, with the second copy additionally conditioned through an increasing event $B$ by the corresponding monotone censored heat-bath chain. The monotone grand coupling and convergence to stationarity show that conditioning on $B$ stochastically increases the configuration. Hence for increasing $A$,which is the Harris-FKG inequality $\varphi (A\cap B)\ge \varphi (A)\varphi (B)$. Applying this to the complement of a decreasing event $B^{\prime }$ gives

WriteThe Russo-Seymour-Welsh theorem and the Harris-FKG inequality give the standard one-arm extension estimate: there is $c>0$ such thatuniformly in $r$. Indeed, on the one-arm event to scale $r$, a fixed finite collection of open rectangle crossings in the annulus ${\Lambda }_{2r}\setminus {\Lambda }_r$, each having probability bounded below by RSW, joins that arm to $\partial {\Lambda }_{2r}$; FKG multiplies the lower bounds. Iteration shows that $\pi (ar)$ and $\pi (r)$ are comparable for every fixed $a>0$. This proves the estimate suggested in the hint.

Fix $x\in \partial {\Lambda }_n$ and choose $r=\lfloor n/4\rfloor$. If $0⟷x$, there is an open arm from $0$ to distance $r$ and another from $x$ to distance $r$. These are independent events because they use disjoint edge sets, so the extension estimate gives

For the reverse inequality, take one-arm events from $0$ and $x$ at scale comparable with $n$, in disjoint boxes. A fixed collection of open crossings of rectangles of bounded aspect ratio joins the two arms. The Russo-Seymour-Welsh theorem bounds the probability of every added crossing below uniformly in $n$ and in the position of $x$ along the four sides; the Harris-FKG inequality and the arm-extension estimate therefore giveThis is the usual RSW gluing lemma for two one-arm events. Enlarging the constants handles the finitely many small $n$, proving the claim with positive constants $c_1,c_2$.

For $j=0,\dots ,k-1$, let $E_j$ be the increasing event that $ju⟷(j+1)u$. Their intersection implies $0⟷ku$. By translation invariance and the Harris-FKG inequality,

The two-point percolation two-point connection probability ${\tau }_p(x,y)={\mathbb{P}}_p(x\leftrightarrow y)$ is the probability that two fixed graph vertices belong to the same percolation cluster. On a translation-invariant lattice it depends on their displacement. Restricting the permitted connecting graph paths to a subgraph gives the corresponding confined percolation two-point connection probability. The Harris-FKG inequality implies ${\tau }_p(x,z)\ge {\tau }_p(x,y){\tau }_p(y,z)$.

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.

For bond percolation on the square lattice, write $h_n={\mathbb{P}}_p(0\leftrightarrow (n,0))$ and $g_k={\mathbb{P}}_p(0\leftrightarrow \partial [-k,k{]}^2)$. A maximal-probability boundary graph vertex can be placed at $(k,j)$ by symmetry. Reflection about $x=k$ identifies its connection probability from $0$ with that from $(2k,0)$. The Harris-FKG inequality yields $(g_k/(8k){)}^2\le h_{2k}\le g_{2k}$ for $k\ge 1$.

For independent bond percolation on the square lattice at $p=1/2$, square crossing probabilities bounded below by self-duality extend to a uniform positive bound for rectangles of aspect ratio $3/2$. Expose an extremal vertical square crossing, reflect its geometry, and attach it to an exterior side through an unexposed independent region. Reflection symmetry bounds the attachment probability below. Two attachments and a horizontal square crossing join by the Harris-FKG inequality. Overlapping rectangles and transverse overlap-square crossings then give all fixed aspect ratios. Reflecting a geometrical graph path does not require its reflected edges to be open.

If increasing events $A_1,\dots ,A_k$ have a common probability, applying the Harris-FKG inequality to their decreasing complements gives $\mathbb{P}(A_1)\ge 1-(1-\mathbb{P}({\bigcup }_i{A}_i){)}^{1/k}$.

For independent bond percolation on the square lattice, a positive uniform lower bound for square crossings implies a positive uniform lower bound for rectangle crossings of every fixed aspect ratio $\rho >1$. The RSW reflection extension lemma first extends a square to aspect ratio $3/2$. Overlapping rectangles and transverse overlap-square crossings then glue using the Harris-FKG inequality. The required constant depends on $\delta$ and $\rho$, not on scale. At a self-dual parameter, dual crossings also give a uniform upper bound below one.