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Take the connected correlation functionFor a massive phase its large-distance form is an algebraic prefactor times $e^{-|r|/\xi }$, defining the exponential correlation length $\xi$. To specify the normalization of magnetic susceptibility, add $-h{\sum }_r{\sigma }_r$ to the energy and differentiate the mean spin. Translation invariance gives the correlation-function susceptibility sum ruleUsing the dimensionless source ${\beta }_{\mathrm{th}}h$ removes the prefactor ${\beta }_{\mathrm{th}}$. Below the transition, use a selected pure phase rather than a mixture of the two magnetizations, which would have a macroscopic disconnected fluctuation.

Choose a nonnegative blocking kernel $K_b({\sigma }^{\prime },\sigma )$ normalized by ${\sum }_{{\sigma }^{\prime }}{K}_b({\sigma }^{\prime },\sigma )=1$. It may enforce a block average, a majority rule or a chosen rescaled coarse spin. Define the transformed weight bySumming this identity over coarse spins shows exact preservation of the partition function. The geometrical changes are $a^{\prime }=ba$ and $N^{\prime }=N/b^D$. A coordinate rescaling subsequently restores the reference cutoff if desired. The transformation is exact only if all generated operators are retained; a finite-coupling truncation is an approximation.

Define the coupling-independent term generated when $C=0$ byRestoring the original $C$ gives the identity-operator contribution to renormalization-group free energy:The parameter $C_p$ is an accumulated additive energy per coarse site. It multiplies the identity operator and leaves normalized spin probabilities unchanged, but it must be kept when calculating the absolute free energy. It records the eliminated-mode entropy and the normalization of their statistical measure.

Use an energy-per-site convention: $F(u,C)=-\log Z(u,C,N)/({\beta }_{\mathrm{th}}N)$ and $f(u)=F(u,0)$, so $F=f+C$. Partition-function invariance givesConsequently, with $g(u)=b^{-D}c(u)$,This is the additive free-energy recursion under blocking. In a dimensionless free-energy convention multiply $f,C,c,g$ by ${\beta }_{\mathrm{th}}$; the same recursion holds. The inhomogeneous term comes from the generated identity operator, not from a change in the spin-dependent couplings alone.

The affine equation naturally applies to the unsubtracted free energy. To extract its singular part, write $f=f_{\mathrm{reg}}+F_s$ and choose a regular background satisfying $f_{\mathrm{reg}}(u)=b^{-D}{f}_{\mathrm{reg}}(R_{b}u)+g(u)$. Subtraction then gives $F_s(u)=b^{-D}{F}_s(R_{b}u)$. This analytic subtraction of an inhomogeneous renormalization recursion is justified when the eliminated finite-shell contribution is analytic near the fixed point and admits a regular subtraction. It can fail in a resonant direction, producing additive logarithms. The simple homogeneous formulas also assume the remaining irrelevant operators are not dangerously irrelevant couplings, and that marginal operators do not produce unaccounted logarithmic corrections. Calling the quantity in the affine equation “singular” does not make $g$ vanish automatically.

Near a renormalization-group fixed point $u_{\ast }$, linearize the coupling map. In scaling coordinates $v_i$, write $v_i^{\prime }=b^{{\lambda }_i}{v}_i$ to leading order. Here $b^{{\lambda }_i}$ is a discrete multiplier, whereas ${\lambda }_i$ is its logarithmic scaling exponent. A relevant operator has ${\lambda }_i>0$ and grows under coarse-graining; an irrelevant operator has ${\lambda }_i<0$ and decays. A marginal operator needs nonlinear analysis. Tuning the relevant fields to zero places the system on the critical surface. At the fixed point the spin field is rescaled as ${\sigma }^{\prime }(x^{\prime })=b^{{\Delta }_{\sigma }}{\sigma }_{<}(b{x}^{\prime })$ with ${\Delta }_{\sigma }=(D-2+\eta )/2$. Its conjugate source therefore has ${\lambda }_h=D-{\Delta }_{\sigma }=(D+2-\eta )/2$. This accounts for both coupling and spin rescaling; the field normalization is part of the transformation, not a change in the physical microscopic spin length.

Assume the thermal field $t=(T-T_c)/T_c$ and magnetic field $h$ are the only remaining relevant directions. The homogeneous scaling hypothesis for critical phenomena givesChoose the stopping scale with $b^{p{\lambda }_t}|t|=1$. Thenwhere $f_{+}$ is the branch for $t>0$ and $f_{-}$ the branch for $t<0$. Nonuniversal metric factors can be absorbed into the definitions of $t,h$ and these functions. Correlation lengths transform as $\xi (t)=b^p\xi (b^{p{\lambda }_t}t)$, so $\nu =1/{\lambda }_t$. Differentiating the singular free energy twice with respect to temperature gives $C_s\propto |t{|}^{D/{\lambda }_t-2}$, and hence the hyperscaling relationTwo source derivatives similarly give $\gamma =(2{\lambda }_h-D)/{\lambda }_t$. A single source derivative gives the ordered-branch exponent $\beta =(D-{\lambda }_h)/{\lambda }_t$, and setting $t=0$ and scaling $h$ to one gives $\delta ={\lambda }_h/(D-{\lambda }_h)$ when the corresponding magnetization branch exists. These show explicitly how the critical exponents follow from the fixed-point scaling eigenvalues.

The independent correlation derivation uses $G(r)=A{r}^{-(D-2+\eta )}{f}_G(r/\xi )$ with a single crossover scaling function. For lattice spacing $a$, the singular continuum contribution to the magnetic susceptibility isHere $S_D$ is the area of the unit $(D-1)$-sphere. The final integral is finite and nonzero when $\eta <2$, $f_G(0)$ is finite and nonzero, and the tail decays exponentially. Microscopic distances contribute a regular background. With $\xi \sim |t{|}^{-\nu }$ this proves the Fisher scaling relationThere is an amplitude qualification in the printed large-distance formula: its tail has the Gaussian shape and, taken literally with amplitude independent of $\xi$, integrates to a term proportional to ${\xi }^2$. For a nonzero anomalous exponent it must instead be matched to the critical short-distance normalization. The matching the amplitude of a massive critical correlation tail givesAt $r=s\xi$ this has precisely the same overall ${\xi }^{-(D-2+\eta )}$ scaling as the critical form. Thus the requested identity uses the full scaling hypothesis, or equivalently this implicit amplitude factor. The printed tail without it is consistent as written in the Gaussian case $\eta =0$.

Split the Fourier modes into $|p|<\Lambda /b$ and the shell $\Lambda /b<|p|<\Lambda$. The source is uniform, so it couples only to the zero mode, which belongs to the retained part. The Gaussian field theory has no couplings between the shell and retained modes. Integrating out the shell therefore contributes only a source-independent determinant to the free energy, without changing the quadratic coefficients of the retained Hamiltonian.

Restore the cutoff by $x=b{x}^{\prime }$ and choose the field normalization ${\varphi }_{<}(b{x}^{\prime })=b^{-(D-2)/2}{\varphi }^{\prime }(x^{\prime })$. Counting the measure, derivative and field factors explicitly givesThus in the convention that preserves the kinetic normalization, the Gaussian momentum-shell scaling isThe field has engineering dimension $(D-2)/2$ and zero anomalous dimension. The measure Jacobian and shell determinant supply the additive constant described in Q2. A different choice of field rescaling can redistribute factors into $\alpha$, but must give the same physical transformed correlation length ${\xi }^{\prime }=\xi /b$.

At $m^2=0$ and $h=0$ the model is at a Gaussian fixed point. Its connected correlation function has the power-law critical form $r^{-(D-2)}$ rather than a finite exponential decay length. Approaching from $m^2>0$, the uniform susceptibility ${\tilde{G}}_0(0)=1/m^2$ and the correlation length both diverge. If the thermal scaling field is linear in $m^2$ with $\alpha$ finite and positive, thenThe same result follows from the mass multiplier $b^2$: its logarithmic thermal eigenvalue is two, so $\nu =1/2$.

The determinant also shows a thermodynamic singularity. Two derivatives of the free-energy density with respect to $m^2$ give, up to sign and a nonzero normalization,as $m^2\downarrow 0$. Rescaling $p=\sqrt{\alpha m^2}q$ establishes the power, and the dimensionless integral is ultraviolet-convergent for $D<4$. Integrating back gives a nonanalytic free-energy term proportional to $(m^2{)}^{D/2}$ after regular backgrounds are removed. This is critical behavior of the quadratic model.

There is a stability qualification: a pure Gaussian potential with $m^2<0$ is unbounded below in the uniform mode, so it does not by itself provide a stable ordered phase on the other side. The ordered-phase obstruction in a Gaussian scalar model is removed by adding a positive quartic interaction. In $2<D<4$ that interaction is relevant, so the resulting generic scalar critical point need not retain the Gaussian exponents. Thus the calculation proves the Gaussian critical singularity and $\nu =1/2$, without claiming that the unmodified quadratic integral defines both phases of a physical ordered transition.

Use the connected correlation function $G(r)=⟨{\sigma }_0{\sigma }_r⟩-⟨{\sigma }_0⟩⟨{\sigma }_r⟩$. Away from criticality its large-distance envelope decays exponentially on the correlation length $\xi$, often with an algebraic prefactor. At criticality $\xi$ diverges and a short-range scalar model instead has the power law $G(r)\sim r^{-(D-2+\eta )}$, with anomalous dimension $\eta$.

A blocking kernel $K_b({\sigma }^{\prime },\sigma )$ assigns coarse spins to each block of side $b$ lattice spacings, with ${\sum }_{{\sigma }^{\prime }}{K}_b({\sigma }^{\prime },\sigma )=1$. Define the coarse Hamiltonian byThe operator basis must be sufficiently complete to include interactions generated by blocking. Summing over coarse spins preserves the partition function, while the physical lattice spacing and site count change to $a^{\prime }=ba$, $N^{\prime }=b^{-D}N$. After $p$ steps they are $a_p=b^{p}a$ and $N_p=b^{-pD}N$. Exact blocking preserves the long-distance predictions; truncating the generated interaction space introduces an approximation.

Define the dimensionless free energy per site $F(u,C)=-N^{-1}\log Z(u,C,N)=f(u)+\beta C$. Preservation of $Z$ gives $F(u_0,C_0)=b^{-pD}F(u_p,C_p)$. For one step, removing the explicitly additive constant givesThe function $g$ records the additive free-energy density generated by eliminated degrees of freedom; it is usually regular near a finite blocking step's critical point. Iteration provesThis is the additive free-energy recursion under blocking. After removing regular backgrounds, its singular part obeys homogeneous scaling.

A renormalization-group fixed point satisfies $R_b(u^{\ast })=u^{\ast }$. Linearizing the map gives scaling coordinates $v_i^{\prime }=b^{{\lambda }_i}{v}_i$. A relevant operator has ${\lambda }_i>0$, an irrelevant operator has ${\lambda }_i<0$, and a marginal operator has ${\lambda }_i=0$ and needs nonlinear analysis. The critical surface consists of points flowing into the critical fixed point after all relevant directions are tuned. An outgoing or repulsive trajectory lies in the unstable manifold and describes perturbations that grow under coarse-graining.

With relevant thermal and magnetic coordinates $t,h$, the singular free-energy density obeys $F_s(t,h)=b^{-D}{F}_s(b^{{\lambda }_t}t,b^{{\lambda }_h}h)$. Choose $b=|t{|}^{-1/{\lambda }_t}$ to obtainHere ${\lambda }_t,{\lambda }_h$ are the positive scaling eigenvalues, and $f_{\pm }$ refer to the signs of $t$. Since $\xi (t,h)=b\xi (b^{{\lambda }_t}t,b^{{\lambda }_h}h)$, the correlation-length exponent is $\nu =1/{\lambda }_t$. Two temperature derivatives of $F_s(t,0)$ give $\alpha =2-D/{\lambda }_t$, establishing the hyperscaling relationThe magnetization exponent similarly is $\beta =(D-{\lambda }_h)/{\lambda }_t$. These expressions assume no additional dangerous scaling variables.

For the Gaussian field theory, decompose the field into low and high Fourier modes, integrate out $\Lambda /b<|p|<\Lambda$, and restore the cutoff using $x=b{x}^{\prime }$ and $\varphi (x)=b^{-(D-2)/2}{\varphi }^{\prime }(x^{\prime })$. The gradient coefficient remains fixed, the mass coefficient transforms as $m^{\prime 2}=b^2{m}^2$, and the uniform source as $h^{\prime }=b^{(D+2)/2}h$. Hence ${\lambda }_t=2$, ${\lambda }_h=(D+2)/2$, and the Gaussian momentum-shell scaling givesThe sign convention $+h\varphi$ reverses the magnetization's sign but not these exponents. For $2<D<4$, these are Gaussian scaling predictions, not the exponents of a generic interacting scalar critical point: the quartic interaction is relevant there. The free quadratic Hamiltonian also has no stable ordered phase at negative $m^2$, so the quoted $\beta$ is its formal order-parameter scaling index, not a spontaneously magnetized state of the unstabilized Gaussian integral. The relevance distinction is also explained in Tong's renormalization-group lecture notes.

The ultraviolet cutoff specifies which fluctuations remain explicit. Changing it integrates out some modes; to preserve long-distance predictions, their effect must be absorbed into the mass, interactions, and other couplings. Thus bare coefficients depend on $\Lambda$. The long-distance inverse connected correlation function has the form $\tilde{\Gamma }(p)\simeq Z^{-1}(p^2+{\xi }^{-2})$. After fixing the gradient normalization, the infrared mass is an inverse correlation length. With the question's unrescaled gradient coefficient, ${\xi }^{-2}=\alpha m^2$ at Gaussian level, so proportionality rather than literal equality is appropriate until that normalization is chosen.

As a concrete momentum-shell renormalization group step, split $\varphi ={\varphi }_{<}+{\varphi }_{>}$ into modes below $\Lambda /b$ and in the shell $\Lambda /b<|p|<\Lambda$, and defineThen rescale coordinates and the field to restore the cutoff and gradient coefficient. A local derivative expansion of $H_{\mathrm{eff}}$ gives new quadratic, quartic, and higher even couplings. If the system has short-range interactions, the field varies slowly, the effective potential is analytic, and remaining fluctuations are small, retaining the leading gradient and lowest stabilizing powers and minimizing the resulting functional gives Landau-Ginzburg theory. Suppressing the omitted fluctuations is the substantive Landau approximation; it is justified sufficiently far from the narrow fluctuation region or above the appropriate upper critical dimension.

The truncated two-point function in this question is the one-particle-irreducible two-point vertex, $\tilde{\Gamma }(p)=\tilde{G}(p{)}^{-1}$. It is the Hessian of the Legendre effective action, not merely the connected two-point function. In perturbation theory,Here ${\tilde{G}}_0$ is the Gaussian propagator about the chosen reference mass, $\delta m^2$ is the mass counterterm or bare-to-reference mass shift, and $\Sigma$ is the interaction self-energy, computed from proper diagrams. Fix the gradient normalization so ${\tilde{G}}_0(p)=(p^2+M^2{)}^{-1}$, and choose the infrared reference mass $M^2=m^2(0,T)$. Then $\delta m^2=m^2(\Lambda ,T)-M^2$ at this order. The mass renormalization condition is $\tilde{\Gamma }(0)=M^2$.

At one loop the only two-point diagram is the quartic tadpole diagram. There are $4\cdot 3$ choices for attaching its two external lines, leaving the other pair contracted; dividing by $4!$ gives its factor $1/2$. It is independent of external momentum, soThe renormalization condition therefore gives the self-consistent one-loop equationUsing $M$ inside the loop is a self-consistent resummation; replacing it by the bare mass gives the same formal first-order expansion away from infrared problems.

Subtract the equation at $T_c$, where $M=0$, and absorb regular temperature dependence of $g$ into the thermal coefficient. Let $\tau =m^2(\Lambda ,T)-m^2(\Lambda ,T_c)\propto T-T_c$ and $I(M)={\int }_{|q|<\Lambda }{d}^{D}q/(2\pi {)}^D(q^2+M^2{)}^{-1}$. ThenFor $D>4$, the integral divided by $M^2$ tends to a finite cutoff-dependent constant, since its small-momentum radial integrand is $q^{D-5}$. Thus $\tau$ is proportional to $M^2$, consistent with the Landau prediction. For $2<D<4$, set $q=Mz$: the correction instead scales as $M^{D-2}$ and dominates $M^2$. At $D=4$ it is proportional to $M^2\log (\Lambda /M)$. Hence a pure linear Landau mass law without logarithmic modification is consistent only aboveFor $D\le 2$, the massless integral itself is infrared divergent, so the subtraction already needs additional care. The sub-four-dimensional self-consistent approximation locates the breakdown of mean-field behavior; it is not an exact calculation of interacting critical exponents.

At a tricritical point the quartic coupling is tuned away and the leading stabilizing interaction is $g_6{\varphi }^6$. At the Gaussian fixed point, the field has engineering dimension $(D-2)/2$, so the sextic coefficient rescales as $g_6^{\prime }=b^{D-3(D-2)}{g}_6=b^{6-2D}{g}_6$. It changes from irrelevant to relevant atThis is the sextic case of the upper critical dimension of an even scalar interaction; marginality at three dimensions allows logarithmic corrections.

A first-order phase transition exchanges two distinct global minima of the Landau free energy. The equilibrium free energy remains continuous, but a first derivative such as the magnetization or entropy jumps. A temperature-driven transition generically has latent heat; crossing $h=0$ between symmetry-related ordered phases instead has a magnetization jump without requiring latent heat. A continuous phase transition has no such order-parameter jump and usually has a divergent correlation length and nonanalytic higher responses.

For $u>0$, $h=0$ and $r$ crossing zero, minimizing $V=r{M}^2/2+u{M}^4/4$ gives $M=0$ for $r>0$ and $M=\pm \sqrt{-r/u}$ for $r<0$. The minima merge continuously at zero. The curvature vanishes there, so the long-distance response becomes singular.

For $u<0<v$, put $z=M^2$. A nonzero stationary minimum satisfies $r+uz+v{z}^2=0$. Equality with the $M=0$ free energy givesThe finite separation demonstrates the first-order phase transition. Metastability occurs where a nonglobal local minimum persists; its disappearance at a spinodal point is not the equilibrium coexistence criterion.

Use a selected translationally invariant pure thermodynamic phase and define the connected lattice correlation functionAway from criticality its long-distance tail has an exponential factor $e^{-r/\xi }$, possibly multiplied by a power. Equivalently ${\xi }^{-1}=-{\lim }_{r\to \infty }{r}^{-1}\log |G(r)|$ when this limit exists; different directions can have different lengths on an anisotropic lattice. At a thermodynamic critical point the correlation length diverges and the tail becomes algebraic. Differentiate $M=N^{-1}⟨{\sum }_n{\sigma }_n⟩$ with respect to the physical field appearing in the energy Statistical Hamiltonian. With ${\beta }_{\mathrm{th}}=1/(k_{B}T)$, the correlation-function susceptibility sum rule isIn physical continuum coordinates the sum becomes $a^{-D}\int d^{D}rG(r)$. Differentiating with respect to the dimensionless source ${\beta }_{\mathrm{th}}h$ instead removes the explicit inverse-temperature factor. Below the transition, take a one-sided pure-phase limit rather than a macroscopic symmetric mixture.

A normalized blocking kernel $B_b({\sigma }^{\prime },\sigma )$ assigns a probability or a delta constraint to a coarse configuration for each microscopic one, with ${\sum }_{{\sigma }^{\prime }}{B}_b({\sigma }^{\prime },\sigma )=1$. Define the blocked Boltzmann weight bySumming over ${\sigma }^{\prime }$ proves exact equality of the partition functions. Repeated real-space renormalization group steps compose the kernels and integrate out shorter-scale variables. To retain equality one must keep all generated interactions and the field-independent constant; a finite truncation is an approximation. Long-distance observables are mapped through the corresponding coarse observables, not identified blindly with microscopic spins. In physical units,A subsequent coordinate rescaling restores the reference lattice spacing while expressing couplings in cutoff units.

Let $F(u,C)=-({\beta }_{\mathrm{th}}N{)}^{-1}\log Z=C+f(u)$ be free energy per microscopic site. Invariance of $Z$ givesThe constant transforms as $C_1=b^D{C}_0+c(u_0)$. Therefore, with $g(u)=b^{-D}c(u)$, the identity-operator contribution to renormalization-group free energy givesThe second equality follows by substitution and induction. The inhomogeneous term records eliminated-mode entropy, determinants, and normalization factors, all multiplying the identity operator; it is not an interaction with the retained spins. Strictly, this $f$ can contain regular background terms. After an analytic subtraction of an inhomogeneous renormalization recursion, the genuinely singular part satisfies the homogeneous scaling equation, except for possible logarithmic resonances.

A renormalization-group fixed point satisfies $R_b(u_{\ast })=u_{\ast }$. Linearize in scaling coordinates $s_i$: $s_i^{\prime }=b^{{\lambda }_i}{s}_i$. A relevant operator has ${\lambda }_i>0$, an irrelevant operator has ${\lambda }_i<0$, and a marginal operator has ${\lambda }_i=0$ and needs nonlinear analysis. These ${\lambda }_i$ are logarithmic scaling eigenvalues; the discrete Jacobian eigenvalues are $b^{{\lambda }_i}$, not ${\lambda }_i$. The critical surface is the stable manifold obtained by tuning all relevant coordinates to zero. A repulsive renormalization-group trajectory leaves the fixed point along a relevant direction as the coarse-graining length grows. Stable and unstable manifolds become curved away from linear order.

The left section has $h=0$, one irrelevant coordinate $w$ and a relevant thermal coordinate $t$. Its critical surface is $t=0$ in that section. With two relevant variables, the full critical surface has codimension two: the second section shows repulsion in the $(t,h)$ plane at $w=0$.

Assume an isolated fixed point with two relevant scaling fields, analytic changes from physical controls to $(t,h)$, short-range isotropic scaling, no dangerously irrelevant variable in the thermodynamic scaling function, and no marginal or resonant logarithm. Let ${\lambda }_t,{\lambda }_h>0$ denote their scaling eigenvalues. The homogeneous singular free-energy density obeysChoose a stopping scale $L=b^p\simeq |t{|}^{-1/{\lambda }_t}$ so the renormalized thermal field is order one. Irrelevant fields vanish at this scale. With $s=D/{\lambda }_t$ and $\Delta ={\lambda }_h/{\lambda }_t$,Continuous scale interpolation absorbs the immaterial integer stopping-scale convention. Nonuniversal thermal, field and free-energy metric factors can be included explicitly; the normalized functions depend only on the universality class. The $+$ and $-$ labels refer to $t>0$ and $t<0$, which flow to different sides of the critical surface.

For clarity about the additive terms, let $a(t,h)$ solve $a(t,h)=b^{-D}a(b^{{\lambda }_t}t,b^{{\lambda }_h}h)+g(t,h)$. If $g=\sum g_{mn}{t}^m{h}^n$, its analytic coefficients areprovided the denominators do not vanish. Subtracting $a$ makes the recursion homogeneous. A vanishing denominator is a resonance and generically produces logarithms, which the simple power-law hypothesis excludes. In a convention where the regular field-dependent background has been subtracted, write ${\Phi }_{\pm }=f_{\pm }+C_{\pm }$ and $I_{\pm }(t)=C_0+a(t,0)$. This yields the requested formThe $C_{\pm }$ contain branch-dependent matching contributions; only $f_{\pm }+C_{\pm }$ is physically determined. The $I_{\pm }$ are nonsingular functions analytic through the thermodynamic critical point and represent the same regular Taylor background on the two sides, though either branch may be used to describe them. For the full free energy at nonzero field, the regular background is generally $I(t,h)$, not just $I(t)$. Adding an analytic identity-operator contribution proportional to $h^2$ provides a direct counterexample to an exact unrestricted $I(t)$ formula; it changes no singular exponents. The printed expression is justified for the leading singular scaling form with that analytic field background removed. Confluent corrections from irrelevant fields are also omitted in this leading form.

Differentiate the singular scaling form, with one-sided derivatives in the ordered phase. It givesThe length transforms as $\xi (t,h)=L\xi (L^{{\lambda }_t}t,L^{{\lambda }_h}h)$, so $\nu =1/{\lambda }_t$. Thus the Rushbrooke scaling relation and hyperscaling relation follow by direct substitution:The same reasoning also gives $\gamma =\beta (\delta -1)$. Hyperscaling relation relies on the assumptions above and can fail above an upper critical dimension because of a stabilizing dangerously irrelevant coupling.

For the universal specific-heat amplitude ratio, write $F_s=A_f|a_{t}t{|}^s{\Phi }_{\pm }(0)$ at $h=0$. Two temperature derivatives multiply both branches by the same dimensional and thermal metric factors. Away from logarithmic resonances and exceptional zero amplitudes, $A_{+}/A_{-}={\Phi }_{+}(0)/{\Phi }_{-}(0)$ with the same heat-capacity sign convention on both sides. HenceThe functions and their matched branch constants are tied to the fixed-point theory; the constants are not independent arbitrary experimental parameters. If $\alpha <0$, this means the amplitudes of the subleading singular contribution after subtracting the analytic specific-heat background. A finite background ratio itself need not be universal.

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.

Along an isolated two-phase coexistence line, two distinct stable minima have equal free energy. If this line ends at an ordinary interior endpoint without encountering another phase, the distinction between the two phases must disappear: their minima and the intervening barrier merge. For separated nondegenerate minima, their stationary values are smooth functions of the controls. The difference $\Delta V=V(m_1)-V(m_2)$ has ${\partial }_h\Delta V=-m_1+m_2\ne 0$. The implicit function theorem therefore continues the equation $\Delta V=0$ as a local coexistence curve. It cannot end while those distinct stable minima persist.

At the merger the potential satisfiesThe third-derivative condition is essential: a stationary zero-curvature point with nonzero third derivative is an inflection, not a stable equilibrium. Shifting to $\psi =m-m_c$ gives an ordinary quartic Landau free energy with two independent control directions, temperature-like and field-like. The curvature and the order-parameter discontinuity vanish, the magnetic susceptibility diverges, and the correlation length diverges. The endpoint is therefore associated with a continuous transition. This is the critical endpoint of a quartic Landau free energy mechanism.

In the symmetric quartic example, the coexistence line is $h=0,r<0$. Its jump $2\sqrt{-r/u}$ vanishes at $(r,h)=(0,0)$, where $V=u{m}^4/4$. This supplies a concrete realization. The word “must” requires the stated ordinary-endpoint assumptions: a first-order line can instead meet a third phase, a multicritical point, a boundary of parameter space, or continue without a finite endpoint. A spinodal point alone is not an equilibrium critical endpoint.

A first-order phase transition has a discontinuity in a first derivative of the equilibrium free energy, such as the entropy or order parameter. It can have latent heat when the entropy jumps. A continuous phase transition has a continuously vanishing order parameter and no latent heat, with singular higher derivatives and a diverging correlation length. The LG theory compares global minima, not merely the points where a local minimum loses stability.

For the uniform quartic Landau free energythe equation of state is $rm+u{m}^3=h$. At zero field the stable minimum is $m=0$ for $r>0$, and $m=\pm \sqrt{-r/u}$ for $r<0$. Thus tuning $r$ through zero gives a continuous transition. For a fixed $r<0$, varying $h$ through zero instead switches between the two ordered minima and makes $m$ jump, giving a first-order phase transition in the conjugate field.

A temperature-like first-order transition can occur at zero field when $u<0$ and a positive sextic term $v{m}^6/6$ stabilizes the potential. Write $q=m^2$. Stationarity of a nonzero phase gives $r+uq+v{q}^2=0$, while equality with $V(0)=0$ gives $rq/2+u{q}^2/4+v{q}^3/6=0$. Solving these two conditions yieldsThe order parameter jumps from zero to $\pm \sqrt{q_{\mathrm{coex}}}$. At this pointso the competing stationary points are genuinely global minima. This phase coexistence condition differs from the spinodal points $r=0$ and $r=u^2/(4v)$, which mark loss or creation of local stability, not equilibrium coexistence.

If the symmetry permits a cubic term $w{m}^3/3$, a positive quartic coefficient does not preclude a first-order transition. For $V=r{m}^2/2+w{m}^3/3+u{m}^4/4$ with $w\ne 0,u>0$, stationarity and coexistence give $m=-2w/(3u)$ and $r=2w^2/(9u)$. This is the first-order transition in a cubic-quartic Landau potential; the symmetry restriction on the expansion is therefore part of the prediction.

Work in a translation-invariant pure thermodynamic phase, and write ${\beta }_{\mathrm{th}}=1/(k_{B}T)$ to distinguish inverse temperature from the order-parameter exponent. The connected correlation function isAway from criticality in the massive scalar order channel, it decays exponentially, possibly multiplied by an algebraic factor. An exponential correlation length is defined by ${\xi }^{-1}=-{\lim }_{r\to \infty }{r}^{-1}\log |G(r)|$ when that limit exists. At a continuous transition $\xi$ diverges and the critical decay is algebraic. In an ordered symmetric mixture, a nondecaying contribution can remain; selecting a pure branch prevents confusing it with connected critical fluctuations.

Let $M={\sum }_{\mathbf{n}}{\sigma }_{\mathbf{n}}$ and $m=⟨M⟩/N$. Differentiating the partition function with the energy term $-hM$ gives the correlation-function susceptibility sum ruleIf the source is instead the dimensionless ${\beta }_{\mathrm{th}}h$, the explicit inverse-temperature factor is absent. In continuum physical coordinates the lattice sum is approximately $a^{-D}\int d^{D}rG(\mathbf{r})$.

Choose a normalized blocking kernel $B_b({\sigma }^{\prime },\sigma )\ge 0$ such that ${\sum }_{{\sigma }^{\prime }}{B}_b({\sigma }^{\prime },\sigma )=1$, replacing the sum by an integral for continuous variables. A deterministic example is a product of delta functions imposing that each coarse variable equals the average of spins in a block of $b^D$ sites, with its field normalization included consistently. Define the effective Boltzmann weight byKernel normalization makes the blocked partition function exactly equal to the original. Its coarse-grained variables retain the long-distance observables through their defining relation to the original fields. An exact RG step generally generates all symmetry-allowed operators; keeping only a small coupling set is an approximation, not an exact closure assumption. The geometrical coarse lattice hasso its physical volume is unchanged. A subsequent coordinate rescaling may restore the numerical lattice spacing to its initial value, which is the equivalent rescaled-coordinate convention.

Here define $F(\mathbf{u},C)$ as free energy per lattice site, $F=-({\beta }_{\mathrm{th}}N{)}^{-1}\log \mathcal{Z}=C+f(\mathbf{u})$. Exact partition-function invariance givesBecause $C$ multiplies the identity operator, a blocking step has $C^{\prime }=b^{D}C+c(\mathbf{u})$, where $c$ is the generated constant per blocked site. With $g(\mathbf{u})=b^{-D}c(\mathbf{u})$ this impliesThe identity-operator contribution to renormalization-group free energy includes the eliminated modes' entropy, connected vacuum terms and field-measure normalization. It cannot be discarded when calculating an absolute free energy, even though it cancels out of normalized correlation functions.