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For dimensionless free energy per site $F(u,C)=f(u)+\beta C$, an exact blocking kernel preserves the partition function and gives $F(u,C)=b^{-D}F(u^{\prime },C^{\prime })$. Thus $f(u)=b^{-D}f(u^{\prime })+g(u)$, where $g(u)=\beta [b^{-D}{C}^{\prime }-C]$ is the additive contribution from the eliminated degrees of freedom. Repeating this equality gives the displayed sum. After subtracting analytic backgrounds, the singular free-energy density obeys the homogeneous scaling hypothesis for critical phenomena, subject to marginal or dangerous-variable qualifications.

Write ${\beta }_T=(k_{B}T{)}^{-1}$ for inverse temperature to distinguish it from the order-parameter critical exponent. A normalized blocking kernel $B_b({\sigma }^{\prime },\sigma )$ is a nonnegative conditional distribution of coarse spins given the fine spins, normalized byFor deterministic blocking it can impose a majority rule or block average, with a Dirac delta replacing the sum constraint for continuous coarse variables. Each block occupies $b^D$ original cells, so the new lattice spacing and number of sites areThe physical volume $N{a}^D$ is unchanged. After $p$ blockings, $a_p=b^{p}a$ and $N_p=b^{-pD}N$.

Define the coarse statistical Hamiltonian by the exact Boltzmann-weight identityThe operator family must include all generated interactions for this renormalization-group transformation to be exact; restricting it to a few couplings is an approximation. The scalar $g(u)$ has energy units and is the contribution per original site that is independent of ${\sigma }^{\prime }$. Restoring the original constant term $NC$ giveshenceThis convention measures $g$ per original site; defining a constant per new site would move a factor $b^D$ into its definition.

The identity-operator contribution to renormalization-group free energy $C_p$ records the configuration-independent normalization generated by the eliminated spins. It includes their contribution to free energy, including entropy and measure factors, and need not be independent of temperature. It has no effect on normalized probabilities of the retained spins, but is indispensable for the total partition function and its temperature derivatives. Iterating its recursion givesThus $N_p{C}_p$ is the original constant $N{C}_0$ plus the accumulated free-energy contributions of the eliminated degrees of freedom.

To obtain the additive free-energy recursion under blocking, use energy-valued free energy per current lattice site,The blocking kernel normalization impliesSubstituting the recursion for $C^{\prime }$ cancels $C$, givingA second step gives $f_0(u_0)=b^{-2D}{f}_0(u_2)+g(u_0)+b^{-D}g(u_1)$, and induction provesFor dimensionless free energy $-\log Z/N$, multiply $F,f_0,g,C$ throughout by ${\beta }_T$.

The singular free-energy density must be distinguished from the exact per-site representative $f_0$. If $a(u)$ is a regular background and $f_s=f_0-a$, thenThus the same displayed inhomogeneous recursion holds with $f_s,g_s$ after subtraction. When a suitable analytic background solves $a(u)=g(u)+b^{-D}a(Ru)$, it leaves a homogeneous singular recursion. Analytic subtraction of an inhomogeneous renormalization recursion explains this separation. A renormalization-group free-energy resonance can obstruct analytic subtraction and produce a logarithm. The original function $g$ originates in the short-distance spins that were summed out, rather than in an extra external interaction.

Near a renormalization-group fixed point $u_{\ast }$, linearize the transformation:An eigenvector of $\mathsf{A}$ with eigenvalue $b^{y_i}$ defines a scaling coordinate $t_i$ with $t_i^{\prime }=b^{y_i}{t}_i$ to leading order. A relevant operator has $y_i>0$: its coupling grows towards large distances, so it must be tuned to reach the fixed point. An irrelevant operator has $y_i<0$ and dies away; a marginal operator has $y_i=0$ and requires higher-order flow analysis. An operator of scaling dimension $x_i$ has $y_i=D-x_i$.

The coarse spins also require a multiplicative normalization. In continuum notation, writewhere ${\varphi }_{<}$ is the retained field and $\eta$ is its anomalous dimension. This restores the normalization of the critical long-distance connected correlation function. Since a uniform field conjugate to an order parameter enters through $\int d^{D}xH\varphi$, it has eigenvalue exponent $y_h=D-x_{\varphi }=(D+2-\eta )/2$.

Let $t$ be the thermal relevant coordinate with exponent $y_t>0$. After regular backgrounds have been removed, the leading singular free-energy density obeysfor a large total scale factor $B=b^p$. Choosing $B=|t{|}^{-1/y_t}$ gives the scaling hypothesis for critical phenomenaTwo temperature derivatives give the heat-capacity critical exponent; one conjugate-field derivative gives the order-parameter critical exponent; two such derivatives give the magnetic-susceptibility critical exponent. Scaling the correlation length gives $\xi (t)=B\xi (B^{y_t}t)$. Consequently,At $t=0$, instead choose $B=|H{|}^{-1/y_h}$; differentiating gives $\delta =y_h/(D-y_h)$. These scaling relations for critical exponents assume that irrelevant variables can be set to their fixed-point values without singular consequences. A dangerously irrelevant coupling or logarithmic correction requires keeping the corresponding dependence.

For the Gaussian field theory, integrate out the Fourier modes in $\Lambda /b<|q|<\Lambda$ and retain ${\varphi }_{<}$. The modes are independent at zero external field, so this Gaussian thinning transformation changes the normalization by a Gaussian determinant and leaves a quadratic statistical Hamiltonian for the retained modes. Restore the cutoff withIn the integrated statistical Hamiltonian, the volume element contributes $b^D$, each gradient contributes $b^{-1}$, and each field contributes $b^{-(D-2)/2}$. Thus the coefficient ${\kappa }^{-1}$ of the gradient energy is unchanged, whileIn particular $y_t=2$, $y_h=(D+2)/2$ and $\eta =0$. The Gaussian critical exponents and the Gaussian order-parameter scaling index therefore give the requested indicesThe sign convention $+h\varphi$ corresponds to the physical conjugate field $H=-h$ and does not affect these powers.

There are physical qualifications to these Gaussian indices. The strictly quadratic Gaussian field theory is stable for positive $m^2$ and has no stable ordered phase for negative $m^2$; its purported spontaneous $\beta$ is therefore the formal Gaussian order-parameter scaling index. A stabilizing quartic term gives an actual ordered phase. Below four dimensions that term is relevant and changes the exponents; above four dimensions it is a dangerously irrelevant coupling and gives the actual mean-field $\beta =1/2$, rather than the Gaussian hyperscaling value.

The heat capacity power can also be checked directly. Apart from constants, the zero-field Gaussian free-energy density isIf $m^2\propto t$, its singular second temperature derivative is controlled by $\int d^{D}q(q^2+\kappa m^2{)}^{-2}$. For $0<D<4$, rescaling $q=\sqrt{\kappa m^2}k$ gives a power $t^{D/2-2}$, as above. At $D=4$ there is a logarithm, and for $D>4$ an analytic background dominates the heat capacity. The negative formal Gaussian $\alpha$ then describes the subleading singular piece, with logarithmic resonances at appropriate even dimensions. This is the Gaussian specific-heat infrared threshold, and it prevents confusing singular indices with the entire observable.

Partition the lattice into blocks of side $b$ sites, with $b>1$, and retain one coarse spin ${\sigma }_R^{\prime }$ per block. A blocking kernel is a nonnegative conditional weight $K_b({\sigma }^{\prime },\sigma )$ normalized by ${\sum }_{{\sigma }^{\prime }}{K}_b({\sigma }^{\prime },\sigma )=1$ for every microscopic configuration. A deterministic majority rule, with fair resolution of ties, is one example.

Define the blocked statistical Hamiltonian and its field-independent constant bySumming over ${\sigma }^{\prime }$ proves exact equality of the old and new partition functions. The lattice spacing becomes $ba$; expressing lengths again in the original units completes the real-space renormalization group step. Blocking generally generates many-body and longer-range interactions, so exactness requires retaining the full operator space and the additive constant. A finite-coupling truncation is a calculational approximation.

A blocking kernel is a conditional probability $K({\sigma }^{\prime }\mid \sigma )$ for replacing the original spins by block spins. It must satisfyfor each original configuration. For example, divide the lattice into blocks of side $b$ and assign each block the sign of its total spin; a tie can be resolved with equal probabilities for the two signs. The product of these block rules is a normalized blocking kernel.

The real-space renormalization group defines the effective Boltzmann weight by summing the old weights against this kernel. Its normalization preserves the full partition function. The blocked lattice has spacing $ba$ and $N^{\prime }=N/b^D$ sites. Rescaling lengths restores the original lattice-spacing convention. An exact transformation generally produces every symmetry-allowed interaction, not just the nearest-neighbor term; retaining only a few couplings is a further approximation.

Use ${\beta }_T=1/(k_{B}T)$ for inverse temperature, to distinguish it from the order-parameter critical exponent $\beta$. In a translation-invariant pure thermodynamic phase, the connected two-point correlation function isFor finite correlation length, at $|r|\gg \xi$ it decays exponentially, possibly with an algebraic prefactor; ${\xi }^{-1}$ is the asymptotic exponential decay rate. Below an ordered transition the pure phase is selected by taking the thermodynamic limit before $h\to 0^{+}$ or $h\to 0^{-}$. A zero-field symmetric mixture of the two ordered phases has a nondecaying connected term and does not describe the usual pure-phase magnetic susceptibility.

Let $\mathcal{F}=-k_{B}T\log Z$ denote extensive physical free energy and $S={\sum }_r{\sigma }_r$. Since the field contributes $-hS$ to the statistical Hamiltonian, differentiation givesExpanding the variance and using translation invariance proves the correlation-function susceptibility sum rule:The sum includes $r=0$. If the source is instead defined as the dimensionless quantity ${\beta }_{T}h$, the corresponding derivative omits the displayed inverse-temperature factor.

A real-space renormalization group step of length factor $b>1$ uses a local blocking kernel $B_b({\sigma }^{\prime },\sigma )\ge 0$ satisfying ${\sum }_{{\sigma }^{\prime }}{B}_b({\sigma }^{\prime },\sigma )=1$. For example, a deterministic majority-spin rule assigns one coarse Ising spin to each block; tied blocks can be resolved with equal probabilities. Define the blocked statistical Hamiltonian byThe spacing becomes $a^{\prime }=ba$. All generated interactions, including multispin terms and the field-independent constant, must be retained for an exact transformation. Summing over ${\sigma }^{\prime }$ uses normalization to prove $Z(u^{\prime },C^{\prime },N^{\prime })=Z(u,C,N)$. Iteration gives $N_p=b^{-pD}N$ and $Z(u_p,C_p,N_p)=Z(u_0,C_0,N)$.

The coarse probability law is the exact marginal law of these block variables. Indeed, for any coarse observable $A$, its expected value is the microscopic expected value of ${\sum }_{{\sigma }^{\prime }}{B}_b({\sigma }^{\prime },\sigma )A({\sigma }^{\prime })$. A blocking rule that retains the long-wavelength order-parameter mode, with source and observable transformations tracked, therefore retains long-distance response and correlation functions. Their field normalization may change, while their physical length scales and critical singularities do not. Partition-function equality alone would not justify this assertion for an arbitrary kernel that discards all slow variables. Truncating the generated operator family is likewise an approximation.

For the free-energy recursion use dimensionless free energy per site,in the thermodynamic limit. Exact partition-function invariance gives $F(u_0,C_0)=b^{-pD}F(u_p,C_p)$. If one step sends $C$ to $C^{\prime }=b^{D}C+c(u)$, the coefficient $c(u)$ records the identity-operator contribution from eliminated fluctuations and normalization. ConsequentlyRepeated substitution derives the additive free-energy recursion under blocking:In physical energy units, divide this convention by ${\beta }_T$. The additive term is essential for absolute free energies even when it does not affect the leading critical powers.

A renormalization-group fixed point satisfies $R_b(u_{\ast })=u_{\ast }$. Linearize in scaling coordinates $v_i$ so that $v_i^{\prime }=b^{y_i}{v}_i+O(v^2)$. A relevant operator has $y_i>0$, an irrelevant operator has $y_i<0$, and a marginal operator has $y_i=0$ with behaviour decided by nonlinear terms. Thus the eigenvalues of the discrete linear map are $b^{y_i}$; the sign criterion applies to the generator exponents $y_i$, not to the discrete multipliers. The critical surface is the stable manifold on which every relevant scaling field has been tuned away. Irrelevant coordinates flow to the renormalization-group fixed point along it. A repulsive renormalization-group trajectory leaves the renormalization-group fixed point along an unstable relevant direction, ultimately entering a noncritical phase or another scaling regime.

To separate the singular free-energy density from its regular background, suppose the eliminated-mode contribution $g$ is analytic near the renormalization-group fixed point. Look for an analytic background $a(u)$ satisfying $a(u)=b^{-D}a(R_{b}u)+g(u)$. Then $F_s=f-a$ satisfies $F_s(u)=b^{-D}{F}_s(R_{b}u)$. In linear thermal and field coordinates, an analytic monomial $g_{mn}{t}^m{h}^n$ has background coefficientprovided the denominator is nonzero. This explicitly explains when analytic subtraction of an inhomogeneous renormalization recursion works. The printed inhomogeneous equation may describe the free-energy function before this subtraction, or a singular part defined only up to analytic terms. The inhomogeneous sum need not be small compared with the full free energy: an analytic constant can dominate a vanishing singular contribution. It can be omitted in deriving leading singular powers after regular subtraction, provided no renormalization-group free-energy resonance, marginal logarithm or dangerously irrelevant coupling obstructs the homogeneous critical asymptotics. At a renormalization-group free-energy resonance $m{\lambda }_t+n{\lambda }_h=D$, additive logarithms can remain, so simply dropping the term would miss them.

Suppose there are exactly two relevant scaling fields, thermal $t$ and magnetic $h$, with positive exponents ${\lambda }_t,{\lambda }_h$, and the remaining fields approach a regular limit. Physical reduced temperature and magnetic field agree with these coordinates to leading order, up to nonuniversal metric factors. The homogeneous scaling hypothesis for critical phenomena isas an asymptotic scaling equation. Choose $\ell =|t{|}^{-1/{\lambda }_t}$ to make the thermal argument $+1$ or $-1$. Defining $f_{\pm }(z)=F_s(\pm 1,z)$ givesThe two functions describe respectively $t>0$ and $t<0$: approaching the transition from disordered and ordered sides need not give the same function. In particular the ordered zero-field function has one-sided derivatives associated with spontaneous magnetization. Metric factors multiply this formula in different microscopic normalizations.

The correlation length in Bravais lattice units obeys $\xi (t,h)=\ell \xi ({\ell }^{{\lambda }_t}t,{\ell }^{{\lambda }_h}h)$. At $h=0$, the same choice of $\ell$ gives $\nu =1/{\lambda }_t$. Two thermal derivatives of $F_s(t,0)$ give $2-\alpha =D/{\lambda }_t$. One and two field derivatives give, respectively, $\beta =(D-{\lambda }_h)/{\lambda }_t$ and $\gamma =(2{\lambda }_h-D)/{\lambda }_t$, assuming the relevant branch amplitudes are nonzero. At $t=0$, choose $\ell =|h{|}^{-1/{\lambda }_h}$; then $M\propto \mathrm{sgn}(h)|h{|}^{(D-{\lambda }_h)/{\lambda }_h}$, so $\delta ={\lambda }_h/(D-{\lambda }_h)$. HenceSimple algebra now gives the Rushbrooke scaling relation $\alpha +2\beta +\gamma =2$ and the hyperscaling relation $\alpha =2-D\nu$. Also $\beta \delta ={\lambda }_h/{\lambda }_t=\beta +\gamma$, giving the Widom scaling relation. These statements refer to leading singular contributions with the homogeneous two-field scaling assumptions. Above the upper critical dimension, the quartic coupling is dangerously irrelevant because it is still needed to stabilize the ordered phase; ordinary mean-field critical exponents then satisfy Rushbrooke and Widom but generally not hyperscaling. Marginal logarithms also require qualifications at the upper critical dimension.

For the connected-correlation scaling form $G(r)=|r{|}^{-(D-2+\eta )}{f}_G(|r|/\xi )$, the function $f_G(x)$ tends to a finite nonzero value as $x\to 0$ and decays exponentially at large $x$, up to powers of $x$, in a massive short-range phase. The scaling ansatz alone does not fix that prefactor. If the usual massive Gaussian field correlation tail $G(r)\propto r^{-(D-1)/2}{e}^{-r/\xi }$ applies, matching the powers gives $f_G(x)\sim B{x}^{(D-3)/2+\eta }{e}^{-x}$, with a correlation-length-dependent amplitude in $G$ fixed by matching the critical scaling form.

In physical coordinates the lattice sum has site-density factor $a^{-D}$. Its long-distance singular part therefore giveswhere ${\Omega }_{D-1}$ is the area of the unit sphere in $D$ dimensions. For $\eta <2$, finite $f_G(0)$, an integrable exponential tail and a nonzero integral, the last factor tends to a finite nonzero constant as $\xi \to \infty$. Microscopic distances contribute a regular background. Thus ${\chi }_s\propto {\xi }^{2-\eta }\propto |t{|}^{-(2-\eta )\nu }$, proving the Fisher scaling relationEquivalently the field has scaling dimension $(D-2+\eta )/2$, so its conjugate field has ${\lambda }_h=(D+2-\eta )/2$, which gives the same identity from the exponent formulas above.

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 $\eta =0$.

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.