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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 cubic lattice in dimension $d$ has graph vertex set ${\mathbb{Z}}^d$ and an edge between $x,y$ exactly when ${\sum }_i|x_i-y_i|=1$. For $d=2$ it is the square lattice. Every graph vertex has $2d$ graph neighbours, so it is a locally finite graph.

For $q\ge 1$, finite random-cluster measures with all boundary graph vertices identified have a weak infinite-volume limit of a sequence. On the cubic lattice it is invariant under translations and cubic lattice reflections and has positive association of the random-cluster model. Its conditional single-edge open probability lies between $p/(p+q(1-p))$ and $p$. Connection events can be approximated by increasing finite-volume path events, so their positive-association inequalities pass to the limit of a sequence.

Suppose the one-arm probability on the cubic lattice satisfies ${\beta }_r=\exp (-\lambda r+o(r))$ with $\lambda >0$. Define $M_n$ as the largest maximum-norm radius of any cluster rooted in $[-n,n{]}^d$. Then $M_n/\log n\to d/\lambda$ convergence in probability. An upper bound is a union bound over $O(n^d)$ roots. For a lower bound, pack $\Theta ((n/\log n{)}^d)$ disjoint radius-$O(\log n)$ boxes and use the independent first-hit connection events inside them. Clusters are finite almost surely under the positive-rate hypothesis.

The mean-field approximation replaces each site's fluctuating surroundings by a self-consistent average. On a cubic lattice each spin has coordination number of a lattice $z=2D$, even though summing only over positive basis directions counts each bond once. Write ${\sigma }_n=M+\delta {\sigma }_n$ and drop products of deviations. Each Ising bond becomes ${\sigma }_n{\sigma }_m\simeq M{\sigma }_n+M{\sigma }_m-M^2$, so the approximate statistical Hamiltonian isThe constant corrects counting each bond twice. The resulting independent-spin partition function is $Z_{\mathrm{MF}}=e^{-{\beta }_{T}NzJ{M}^2/2}[2\cosh ({\beta }_{T}zJM){]}^N$, where ${\beta }_T=1/(k_{B}T)$. The one-spin probabilities giveEquating this average with the assumed $M$ givesHere $J>0$ is assumed; a ferromagnetic order parameter is not the appropriate choice for an antiferromagnetic coupling. Put $A=T_c/T$. For $A\le 1$, $\tanh (AM)<AM\le M$ at any $M>0$, so only $M=0$ is possible. For $A>1$, the function $\tanh (AM)-M$ has positive slope at zero, is strictly concave on $M>0$ and is negative at $M=1$. It therefore has exactly one positive zero and a spin-reversed negative zero. Expanding at the bifurcation givesThe transition is continuous, with order-parameter critical exponent $\beta =1/2$. Stability can also be checked from the Bragg-Williams free energy of the Ising model per Ising spin:Its derivative is $-zJM+k_{B}T\mathrm{arctanh}M$, giving the same self-consistency equation. Near zero,The quartic coefficient is positive at $T_c$, the stable minima approach zero continuously, and the minimized free energy has no first-derivative jump. Thus mean-field approximation predicts a second-order transition at $T_c=2DJ/k_B$. This is the mean-field Ising critical bifurcation.

For short-range interactions, large dimension gives many neighbours and suppresses the relative fluctuations of their average. More quantitatively, Gaussian field theory fluctuations averaged over a correlation volume have variance of order ${\xi }^{2-D}$, while the squared ordinary mean-field order parameter is of order $|t|\sim {\xi }^{-2}$. Their ratio is proportional to ${\xi }^{4-D}$. The Ginzburg criterion therefore makes neglect of critical fluctuations asymptotically consistent above four dimensions, inconsistent below four, and marginal at four, where logarithms occur. Large dimension is a route to validity; sufficiently long-range or infinite-range interactions can also produce mean-field behaviour. In low-dimensional short-range systems even the existence and location of a transition can differ from this approximation.

Order configurations coordinatewise. The FKG lattice condition on the masses of a finite Boolean lattice isThe FKG inequality states that under this condition, increasing real-valued functions $f,g$ satisfy $\mathbb{E}[fg]\ge \mathbb{E}[f]\mathbb{E}[g]$. In particular increasing events satisfy $P(A\cap B)\ge P(A)P(B)$. For a product measure with coordinate weights $w_e(0),w_e(1)$, the pair $(\min ({\omega }_e,{\eta }_e),\max ({\omega }_e,{\eta }_e))$ has the same two entries as $({\omega }_e,{\eta }_e)$. Multiplying over coordinates gives equality in the lattice condition, including degenerate Bernoulli parameters without division by zero.

For independent bond percolation on the nearest-neighbor cubic lattice, write ${\theta }_d(p)={\mathbb{P}}_p(|C(0)|=\infty )$. Define the percolation critical probability and the connective constant bywhere $c_n(d)$ counts rooted $n$-step self-avoiding walks. Translation invariance makes the root irrelevant, and part (i) proves existence of the latter limit.

First work on the square lattice, whose dual is another translated square lattice. Put $q=1-p$ and assume $q\mu (2)<1$. If the open cluster at the origin is finite, its exterior edge boundary contains a closed dual simple circuit surrounding the origin. To see this planar fact, surround its finitely many vertices by their unit square cells and follow the exterior boundary: its crossing primal edges are closed. Resolving repeated boundary vertices into simple circuits leaves a circuit separating the origin from infinity.

Let $a_n$ count such dual circuits of length $n$. Every one crosses the positive horizontal ray at distance at most $n$, because it surrounds the origin and its horizontal span is at most its length. Choosing a ray-crossing edge, an orientation and all but the last edge encodes it by one of at most $Cn$ rooted length-$n-1$ self-avoiding walks. Thus $a_n\le Cn{c}_{n-1}(2)$. Choose $b>\mu (2)$ with $qb<1$. The root-count limit gives $c_{n-1}(2)\le C_b{b}^n$, soThis summable tail alone need not make the probability of every enclosing circuit less than one. Choose a large $R$ and let $A_R$ be the absence of enclosing closed dual circuits of length at least $R$, with $P(A_R)>1/2$. Let $B_R$ require every primal edge inside $[-R,R{]}^2$ to be open. It has positive probability. Both events are increasing. The finite-measure Harris-FKG inequality extends to $A_R$ by decreasing limits over finitely many circuit exclusions, so $P(A_R\cap B_R)\ge P(A_R)P(B_R)>0$.

On $B_R$ the origin is connected to every graph vertex of that box. Any closed dual circuit enclosing the origin must then enclose the whole box, hence have length at least $R$. On $A_R\cap B_R$ no such circuit exists, so the origin cluster is infinite. This proves the connective-constant Peierls bound, $p_c(2)\le 1-1/\mu (2)$. Finally the lattice in dimension $d\ge 2$ contains a coordinate copy of the square lattice, whose edge law is unchanged. Percolation in that subgraph implies percolation in the full graph, and hence

Independent bond percolation turns a deterministic graph into a random network: each edge is open with probability $p$, independently, and a percolation cluster is a connected component of its open subgraph. Site percolation instead randomizes vertices. On the cubic lattice the percolation probability $\theta (p)=P_p(0\leftrightarrow \infty )$ is the order parameter and $p_c=\inf \{p:\theta (p)>0\}$ is the critical probability. A monotone coupling, using independent uniform edge labels and opening labels below $p$, makes the growth of connectivity with $p$ transparent.

For $d\ge 2$, $0<p_c<1$: self-avoiding walk counting gives $p_c\ge 1/\mu (d)$, and the planar bound proved above gives an upper bound below one. Below $p_c$, clusters are finite and connection probabilities decay exponentially, a percolation sharpness theorem rather than a consequence of the threshold definition alone. Above $p_c$ there is almost surely a unique infinite percolation cluster. At criticality, the presence of an infinite percolation cluster must be addressed for the particular model. In two dimensions there is none. The transition replaces a finite characteristic length by scale-free geometry, followed by a macroscopic connected component.

Percolation critical exponents quantify different aspects of this change. Use power-law notation for leading exponents; it may suppress amplitudes and logarithmic corrections. From the subcritical side, define the percolation susceptibility $\chi (p)={\mathbb{E}}_p|C(0)|$ and an exponential correlation length $\xi (p)=1/\lambda (p)$, using a fixed norm for the one-arm rate. Above criticality the unrestricted expectation is infinite, so a finite-cluster percolation susceptibility must exclude the infinite percolation cluster. The principal observables are:

The last distribution is not the distribution seen from a uniformly chosen site: the latter is weighted by cluster size. In the scaling description this accounts for $1/\delta =\tau -2$. Under the usual below-upper-critical-dimension scaling hypotheses, further scaling relations for critical exponents includeThey organize the exponents into a small number of independent quantities; they are not automatic identities following only from their definitions. A percolation cluster fractal dimension $D_f$ describes the mass of a large critical cluster of radius $r$ as roughly $r^{D_f}$, with the scaling relation for critical exponents $D_f=d-\beta /\nu$ in this regime.

The percolation universality hypothesis is that ordinary short-range independent models in the same spatial dimension share these exponents and their continuum behavior despite differing microscopic lattices or using bonds rather than sites. The threshold itself is not universal. Long-range connections, correlations or changes of geometry can change the universality class, so the hypothesis does not include every random graph called percolation.

Two dimensions have a particularly strong geometric structure. Closed planar dual graph circuits obstruct primal open paths. The Harris-Kesten theorem fixes the square lattice bond percolation threshold at $p_c=1/2$, using planar duality and crossing estimates. Self-duality alone is not a proof; uniform rectangle-crossing control is a crucial extra ingredient. The Russo-Seymour-Welsh theorem keeps critical crossing probabilities of rectangles of fixed aspect ratio bounded away from zero and one at all scales. Closed dual circuits on disjoint annular scales then rule out an infinite critical cluster and reveal why arbitrarily large finite structures persist.

Critical site percolation on the triangular lattice has rigorously established conformal invariance of planar percolation for crossing limits and an SLE description of interfaces. These results are specific to this model. The exact planar valuesare rigorously established there; its one-arm probability has exponent $5/48$. In the corresponding scaling description these values give $\eta =5/24$, $\delta =91/5$, $\tau =187/91$ and $D_f=91/48$. The percolation universality hypothesis predicts the same planar exponents for square lattice bond percolation, but the triangular-lattice theorem by itself does not prove that transfer. This distinction separates exact model-specific mathematics from a broader physical prediction.

Dimension changes the importance of loops and correlations between growing branches. The mean-field percolation exponents come from approximating critical clusters by branching processes. Their values are $\beta =1$, $\gamma =1$, $\delta =2$, $\nu =1/2$, $\eta =0$ and $\tau =5/2$. The predicted upper critical dimension of percolation is six. One way to see its role is the triangle diagram appearing in the percolation triangle condition: if the long-distance Fourier connectivity behaves like $|k{|}^{-2}$, its infrared contribution is proportional to ${\int }_0^{\epsilon }{r}^{d-7}dr$. It is finite above six and logarithmically divergent at six. This motivates mean-field behavior above six and logarithmic corrections at six, without turning that heuristic into a proof for every lattice.

Lace expansion makes the mean-field picture rigorous in suitable high-dimensional regimes. For nearest-neighbor percolation, sufficiently high dimensions are covered; sufficiently spread-out models have mean-field results for every $d>6$. Between two and six the exponents depend nontrivially on dimension, and exact three-dimensional values are not supplied by the planar theory. Hyperscaling relation illustrates the change: mean-field values give $2\beta +\gamma =3$, while $d\nu =d/2$, so the naive equality fails above six. In $d=1$, $p_c=1$ and no infinite percolation cluster exists for $p<1$; this is a degenerate endpoint case rather than the ordinary interior transition.

Give each nearest-neighbor edge of the cubic lattice an independent Bernoulli distribution state, open with probability $p$. The resulting product measure is denoted ${\mathbb{P}}_p$. In this bond percolation model, $C(0)$ is the percolation cluster of the origin, andThe uniform-label monotone coupling of Bernoulli percolation shows that ${\theta }_d$ is increasing.

Let $c_n(d)$ count $n$-step self-avoiding walks starting at the origin, with $c_0(d)=1$. Splitting a walk after $n$ steps, and discarding the avoidance constraint between the two pieces, gives $c_{n+m}(d)\le c_n(d)c_m(d)$. The Fekete lemma therefore gives the connective constantThe locally finite graph structure means that an infinite percolation cluster at the origin supplies an open self-avoiding walk of every length. Each specified walk has $n$ distinct edges and is open with probability $p^n$. The union bound givesFor $p\mu (d)<1$ the right side tends to zero. Hence the connective-constant lower bound for percolation is $p_c(d)\ge \mu (d{)}^{-1}$.

For the upper bound first work on the square lattice. A finite open percolation cluster has an outer boundary containing a simple closed graph cycle of dual edges, all crossing closed primal edges. Such a dual bond percolation circuit separates that cluster from infinity. Write $N_{\ell }$ for the number of simple dual circuits of length $\ell$ surrounding the origin. Each circuit meets the positive horizontal ray at distance at most $\ell$: its bounding box contains the origin and its diameter is bounded by its length. Choose such an intersection as an anchor and orient the circuit. Removing its last edge leaves a rooted self-avoiding walk of length $\ell -1$ in the translated square lattice. Consequently, for an absolute constant $K$,The exact constant and this possible overcount do not matter. If $(1-p)\mu (2)<1$, the root test givesA summable circuit count alone need not give a total sum below one. To use its tail correctly, let $B_R=[-R,R{]}^2\cap {\mathbb{Z}}^2$ and condition every edge internal to $B_R$ to be open. This finite event has positive probability. A closed dual circuit surrounding all of $B_R$ crosses no internal edge of $B_R$, and so its closed-edge probability remains $(1-p{)}^{\ell }$ under this conditioning. Its length tends to infinity with $R$. Choose $R$ so large that the union bound for all such circuits is below one. With positive conditional probability, none occurs.

On that event, $C(0)$ contains $B_R$ and cannot be finite: a finite cluster containing $B_R$ would have an enclosing closed dual circuit. Thus ${\theta }_2(p)>0$ whenever $p>1-\mu (2{)}^{-1}$. This is the connective-constant Peierls bound, proved by excluding short circuits through the open-box conditioning. An embedded coordinate plane in the cubic lattice has exactly the same bond percolation law as the square lattice, so $p_c(d)\le p_c(2)$. Together,There are $2d$ choices for the first step of a self-avoiding walk and at most $2d-1$ thereafter, because immediate reversal is forbidden. Hence $c_n(d)\le 2d(2d-1{)}^{n-1}$ and $\mu (d)\le 2d-1$. In particular $\mu (2)\le 3$. Substituting with the correct directions of the inequalities gives

The purpose of the RSW lemma is to turn local crossing information into control at every fixed shape and scale. Work with independent bond percolation on the square lattice. Let $H(a,b)$ be the event of an open left-to-right graph path in $[0,a]\times [0,b]\cap {\mathbb{Z}}^2$, and write $h_p(a,b)={\mathbb{P}}_p(H(a,b))$.

A useful precise uniform version of the RSW lemma is the following uniform RSW crossing estimate. If for a fixed parameter $p$ and some $\delta >0$,then for every fixed aspect ratio $\rho >1$ there is $c(\delta ,\rho )>0$, independent of $n$, such thatIn particular this bounds length $2n$ in terms of the uniform square bound at length $n$. The constants need not be sharp. The hypotheses used in the proof are planarity, translation and reflection symmetries, the Harris-FKG inequality and independence of unexplored edges. At a self-dual parameter the same argument for dual crossings gives an upper bound strictly below one as well. Neither an exact value of $p_c$ nor the existence of an infinite cluster is an assumption of this estimate.

Here is the gluing picture behind the RSW lemma. For increasing crossing or attachment events, the Harris-FKG inequality provides a lower bound on their joint probability. If an event of probability at least $\delta$ is a union of two reflection-related increasing alternatives $A_1,A_2$, the square-root trick for positively associated events givesIndeed their decreasing complements are positively associated too, so ${\mathbb{P}}_p(A_1^c\cap A_2^c)\ge {\mathbb{P}}_p(A_1^c{)}^2$. This prevents a square crossing from concentrating all its useful attachment locations on just one side.

The nontrivial first gluing step enlarges a square to aspect ratio $3/2$. Explore an extremal square crossing, revealing the edges on the explored side but leaving its other side unexamined. In that unexplored region the conditional law is still independent bond percolation. Compare possible attachments to the crossing with their reflected alternatives. Reflection symmetry and the preceding square-root estimate give a positive bound, depending only on $\delta$, for the required attachment after averaging over the explored crossing. Carry out the reflected construction at the other end and use the Harris-FKG inequality to combine the increasing attachment events with the original crossing. Planarity ensures that the relevant transverse paths actually meet. This RSW reflection extension lemma yields a lower bound $u(\delta )>0$ for the $3n/2$-by-$n$ crossing. The exploration is important: a reflection compares the laws of fresh configurations; reflecting the picture of an open path does not make the reflected edges open.

From that first extension, longer rectangles are obtained by a genuinely transverse gluing. Two $3n/2$-by-$n$ rectangles shifted by $n/2$ overlap in an $n$-by-$n$ square. Require a horizontal crossing in each long rectangle and a vertical crossing of the overlap square. Each horizontal crossing crosses that overlap from left to right, and therefore meets its vertical crossing. Their union crosses the $2n$-by-$n$ rectangle. The Harris-FKG inequality gives a lower bound $u(\delta {)}^2\delta$. Repeat a bounded number of times for any fixed $\rho$. Integer rounding uses neighboring lattice rectangles and bounded additional gluing steps; finitely many smallest scales can be absorbed into the constant. The constants deteriorate with $\rho$ but not with $n$. Merely requiring horizontal crossings in adjacent squares would not suffice, since their endpoints need not coincide; the overlap crossing solves that problem.

I apply the RSW lemma to the exact threshold for bond percolation on the square lattice. The planar duality for rectangle crossings says that an open horizontal crossing and a closed dual vertical crossing are complementary. At $p=1/2$, the dual edge states have the same law as the primal ones. For the balanced lattice rectangle with side lengths $n+1,n$, the rotated dual crossing rectangle has those same side lengths: its transverse lengths before rotation are $n,n+1$. Thus the exact self-dual rectangle crossing probability is $1/2$. Restricting a crossing of this rectangle to its first visit to the shorter vertical side givesThe one-unit balance avoids assuming an exact one-half probability for every finite vertex-square convention. The RSW lemma now gives scale-independent positive bounds for all fixed-aspect-ratio primal and closed dual rectangle crossings.

Arrange four appropriately overlapping long rectangles around a square ring. Closed dual crossings along the four sides, with transverse overlap crossings if needed, join to a closed dual graph cycle surrounding the inner square. The Harris-FKG inequality, applied to the closed dual states, and the RSW lemma give a constant $b>0$ for this circuit event, uniformly over ring size. Choose disjoint rings with radii increasing, for example, by a factor of four. Their circuit events depend on disjoint edge sets, so they are independent. An open graph path from the origin to infinity would have to avoid every one of these dual barriers. Its probability is at most $(1-b{)}^j$ after $j$ rings and therefore zero. HenceThis is the independent annular barriers for percolation argument. It is worth separating it from the other inequality: the absence of an infinite cluster at one parameter alone does not prove that every larger parameter percolates.

For the reverse inequality use sharpness of the percolation transition: below $p_c$, independent bond percolation on the cubic lattice has exponentially decaying connection probabilities. One can see why this is the relevant general ingredient through the finite-set criterion for percolation sharpness. For a finite set $S$ containing the origin putIf ${\phi }_p(S)<1$, split a long open self-avoiding walk at its first exit from $S$. Its internal connection, exit edge and subsequent connection have disjoint witnesses. The BK inequality gives a contraction by ${\phi }_p(S)$ each time distance decreases by the diameter of $S$ plus a fixed step. Iteration proves exponential decay of subcritical percolation at such $p$.

To identify this finite-set threshold with $p_c$, the Margulis–Russo formula expresses the derivative of $g_n(p)$ as the sum of pivotal-edge probabilities. Explore the cluster attached to the box boundary, and let $S$ be its complement. On failure of the origin-to-boundary event, $0\in S$; all edges from $S$ to the boundary cluster are closed, while the internal edges of $S$ remain fresh. A boundary edge is pivotal precisely when its endpoint in $S$ is connected to the origin inside $S$. Removing the factor $1-p$ for a closed pivotal edge yields the general differential inequalityFor $p>{\tilde{p}}_c$ every finite-set quantity on the right is at least one. Integrating from any $p_0\in ({\tilde{p}}_c,p)$ gives a positive lower bound for $g_n(p)$ independent of $n$, and taking $n\to \infty$ gives $\theta (p)>0$. Combined with the contraction below ${\tilde{p}}_c$, this proves $p_c={\tilde{p}}_c$ and the stated sharpness conclusion. This outline supplies the extra threshold argument rather than assuming the desired critical value.

If $p_c(2)>1/2$, sharpness would give $g_n(1/2)\le C{e}^{-cn}$. But a square crossing starting somewhere on the left side entails an open connection from that starting graph vertex to distance $n$. A union bound over the $n+1$ possible starting graph vertices givescontradicting exponential decay. Therefore $p_c(2)\le 1/2$, and the two directions establishThis is the Harris-Kesten theorem. The overall mechanism is that self-duality supplies a square crossing, the RSW lemma transports it between shapes and creates barriers at every scale, and sharpness converts the finite-scale crossing information into the exact threshold.

The cubic lattice has graph vertices ${\mathbb{Z}}^d$ and an edge between $x,y$ when ${\sum }_i|x_i-y_i|=1$. In independent site percolation, every graph vertex is open with probability $p$ and closed with probability $1-p$, independently. The open subgraph contains only open graph vertices and the edges between them. Let $C_p(0)$ be the open connected component of a graph containing $0$, with $C_p(0)=\varnothing$ when $0$ is closed. The percolation probability isSince the cubic lattice is a locally finite graph, the König infinity lemma identifies an infinite open connected component of a graph with the existence of an infinite open graph ray from the origin. In particular, the origin itself must be open. The percolation critical probability is $p_c=\inf \{p:\theta (p)>0\}$.

Write $a=V_c^{1/3}$ for the primitive cubic lattice spacing. With wavevector convention $e^{i\mathbf{k}\cdot \mathbf{r}}$, reciprocal-lattice primitive vectors have length $2\pi /a$. The first Brillouin zone is the cube $-\pi /a\le k_j\le \pi /a$, henceIt has the same volume as any primitive reciprocal unit cell. A reciprocal coordinate defined in cycles per metre would instead give $1/V_c$; that is a different convention from this $k$-space convention.

The negative sign of the exchange term and $J>0$ make this a Heisenberg ferromagnet. Each classical bond minimizes its energy when its two fixed-length spins are parallel. On the connected cubic lattice all ground states are therefore uniform:using length $S$ for the classical large-spin convention. Any two orientations are related by a common spatial rotation of every spin. The global spin-rotation symmetry is $SO(3)$, and a chosen direction retains its $SO(2)$ subgroup of rotations about that axis.

Spontaneous symmetry breaking means selecting one member of this continuous set in the thermodynamic limit, for example by taking volume to infinity before sending an aligning field to zero. The zero-field Hamiltonian operator remains rotationally invariant even though the selected state is not. There are gapless transverse spin waves. Although two rotation generators are broken, an isotropic ferromagnet has one quadratic magnon branch, whose two transverse spin components form a conjugate dynamical pair. It is not necessary to invent two independent linear-dispersion branches from the number of broken generators. The long-wavelength dispersion is derived in d.

Quantize along the selected field direction by the right-handed relabeling ${\tilde{S}}^z=S^x$, ${\tilde{S}}^x=S^y$, ${\tilde{S}}^y=S^z$. Then the exact Holstein–Primakoff transformation in c gives $S_j^x=S-n_j$. Keep exchange terms of order $S^2$ and $S$, with $J$ and the printed $B$ held fixed. For a nearest-neighbor bond,On a periodic cubic lattice with $N$ sites and spacing $a$, there are $3N$ undirected bonds. Consequently,Use the unitary lattice Fourier transform $a_j=N^{-1/2}{\sum }_{\mathbf{k}}{e}^{i\mathbf{k}\cdot {\mathbf{r}}_j}{a}_{\mathbf{k}}$. Each positive lattice direction contributes $2(1-\cos k_{\alpha }a)$, so the Hamiltonian operator is already diagonal in momentum space:There are no anomalous $aa$ or $a^{†}{a}^{†}$ terms, so no Bogoliubov transformation or extra zero-point correction is needed. At long wavelengths,At $B=0$ this is the quadratic ferromagnetic Goldstone mode; for $B>0$ it is gapped by $BS$. Energies are written with $\hslash =1$; an excitation frequency is ${\omega }_{\mathbf{k}}={\epsilon }_{\mathbf{k}}/\hslash$. In this isotropic ferromagnet the fully aligned vacuum and its one-magnon dispersion are also exact; higher-magnon interactions are among the terms discarded from the linear spin-wave Hamiltonian operator.

Take $q\ge 1$ and $0<p\le 1$ for the infinite-volume wired random-cluster measure on the cubic lattice. Positive association of random variables and translation invariance give $C_{m+n}\ge C_m{C}_n$ for axis connection probabilities. The Fekete lemma applied to $-\log C_n$ proves the displayed limit of a sequence and the bound $C_n\le e^{-n\alpha }$. The lower single-edge conditional bound gives $C_n\ge [p/(p+q(1-p)){]}^n$, so the decay rate is finite and nonnegative. It may be zero.

At $p=0$, every distinct-vertex connection probability is zero, so the inverse correlation length is $+\infty$ under the convention $-\log 0=+\infty$. At $p=1$, every connection probability is one and the rate is zero. Thus finiteness of the displayed rate requires the positive-$p$ hypothesis.

Choose a graph vertex $x=(n,x_2,\dots ,x_d)$ on the boundary of a cubic lattice box. Translation and reflection invariance identify the probabilities of $0\leftrightarrow x$ and $x\leftrightarrow 2n{e}_1$. Their intersection implies $0\leftrightarrow 2n{e}_1$, so positive association of random variables gives $C_{2n}\ge \varphi (0\leftrightarrow x{)}^2$. A union bound and cubic lattice symmetries choose such an $x$ with probability at least the box-boundary connection probability divided by the number of boundary graph vertices. This polynomial factor disappears in the exponential rate, identifying boundary and axial inverse correlation lengths.

Independent bond percolation on the cubic lattice has exponential connection decay at every $p<p_c$ and positive percolation probability at every $p>p_c$. The finite-set criterion for percolation sharpness gives decay below its auxiliary threshold. The Margulis–Russo formula and exploration from a finite box boundary give $g_n^{\prime }(p)\ge (1-g_n(p)){\inf }_{S\ni 0}{\phi }_p(S)/(p(1-p))$, with the infimum over interior finite sets. Above the auxiliary threshold the infimum is at least one; integrating gives positive percolation probability and identifies the two thresholds. This theorem does not identify the value of $p_c$ for a particular lattice.

Independent site percolation and bond percolation on the cubic lattice have continuous percolation probability on $(p_c,1]$. Fix $p>p_c$ and an intermediate $q\in (p_c,p)$. By uniqueness of the infinite percolation cluster, every origin in the infinite $p$-cluster has a finite $p$-open graph path to the infinite $q$-cluster. In the uniform-label monotone coupling of Bernoulli percolation, the finitely many labels on this graph path are strictly below fixed $p$ almost surely, so it persists at some parameter below $p$. The coupled onset parameter for origin percolation has no attained atom there, giving left continuity, including at $1$. Combine this with right continuity of percolation probability.

For independent site percolation or bond percolation on the cubic lattice ${\mathbb{Z}}^d$, there is almost surely at most one infinite percolation cluster at every fixed parameter. Above the percolation critical probability, there is exactly one almost surely. This theorem is called the Burton-Keane theorem. Existence uses translation ergodicity together with positive percolation probability; uniqueness is a separate conclusion. At $p=1$ uniqueness is deterministic.

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 $|\partial {\Lambda }_m|g_m$.