= Solution
In <bond percolation> on $\mathbb Z^d$, <edges> are independently open with <probability> $p$ and closed otherwise. An open <percolation cluster> is a connected component using only open <edges>. In <site percolation> it is the <graph vertices> that are independently declared open. The <percolation probability> $\theta(p)=P_p(0\leftrightarrow\infty)$ measures the chance that the origin lies in an <infinite percolation cluster>, and the <percolation critical probability> is $p_c=\inf\{p:\theta(p)>0\}$. <Monotone coupling of Bernoulli percolation> by independent <uniform distribution> makes $\theta$ nondecreasing.
The <sharpness of the percolation transition> says that below $p_c$ connection <probabilities> on $\mathbb Z^d$ decay exponentially with distance, while above $p_c$ the percolation <probability> is positive. There is almost surely a unique <infinite percolation cluster> in the supercritical regime. On the <square lattice>, the <Harris-Kesten theorem> gives bond threshold $1/2$, and there is no <infinite percolation cluster> at <percolation critical probability>. The emergence of connections on arbitrarily large scales is the geometric meaning of the <phase transition>.
Several observables describe this singular behavior. The <percolation susceptibility> is $\chi(p)=E_p|C(0)|$ for $p<p_c$. A directional <correlation length> can be defined through exponential two-point decay, $\xi(p)^{-1}=-\lim_n n^{-1}\log P_p(0\leftrightarrow ne_1)$. <Percolation critical exponents> describe leading powers, conveniently defined logarithmically rather than requiring exact asymptotic amplitudes:
$$
\theta(p)=(p-p_c)^{\beta+o(1)},\quad
\chi(p)=(p_c-p)^{-\gamma+o(1)},\quad
\xi(p)=(p_c-p)^{-\nu+o(1)}.
$$
The first <limit of a sequence> approaches from above, the other two from below. At <percolation critical probability>, <percolation two-point connection probability> is expected to have leading power $|x|^{-(d-2+\eta)}$, and the number per site of <percolation clusters> of size $s$ has leading power $s^{-\tau}$. This cluster-count <probability distribution> differs from the size-biased <probability distribution> seen by choosing a site. <Scaling relations for critical exponents> such as $\gamma=(2-\eta)\nu$ and $2\beta+\gamma=d\nu$ connect the exponents in their applicable scaling regime; <hyperscaling relation> is not valid above the <upper critical dimension of percolation> in ordinary mean-field regimes.
The <percolation universality hypothesis> predicts that dimension and large-scale symmetry, rather than microscopic <graph> details, determine these exponents and scaling functions. Thus square-lattice bond and triangular-lattice site models are expected to share the <planar graph> <universality class> despite different microscopic definitions. Universality does not mean identical critical <probabilities>, amplitudes or finite-lattice distributions. Exact values and existence of exponents require model-specific theorems, not merely this prediction.
Two dimensions have an additional structure: <conformal maps> preserve angles, and critical crossing <probabilities> are expected to be unchanged under conformal transport of the domain and its marked boundary arcs. A crossing <probability> then depends only on the conformal shape of that marked quadrilateral. The <Cardy boundary crossing formula> gives this dependence in terms of its boundary cross ratio. For critical triangular-lattice site percolation, conformal invariance and the resulting exponents are established; examples are $\beta=5/36$, $\gamma=43/18$, $\nu=4/3$, with the rigorous statements in leading-power form. These values are summarized in https://www.unige.ch/~smirnov/papers/smw-j.pdf[Smirnov and Werner's original paper]. Their universality extension to arbitrary planar lattices should not be presented as the same theorem.
An exploration path separating the two colors, with opposite boundary colors on two arcs, has a <domain Markov property of a percolation exploration>: conditional on the explored portion, the unexplored colors retain their independent laws with the newly exposed boundary colors. Conformal invariance, this domain Markov property and suitable regularity lead to <Schramm–Loewner evolution>. In the upper half-plane its chordal equation is
$$
\partial_tg_t(z)=\frac2{g_t(z)-U_t},\qquad U_t=\sqrt\kappa\,B_t,
$$
where $B_t$ is standard <Brownian motion> and time is normalized by half-plane capacity. Critical percolation corresponds to $\kappa=6$. Thus the complicated discrete interface becomes a conformally natural random curve driven by one real <Brownian motion>. Its crossing and arm <probabilities> supply critical exponents and, through <scaling relations for critical exponents>, near-critical singularities. This makes conformal methods predictive tools for the transition, rather than only descriptions of an interface's appearance.
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