The Renormalization Group Idea: Coarse-Graining and Rescaling

Right at a continuous phase transition a system develops fluctuations of every size, from the atomic spacing all the way up to the correlation length $\xi$ , which itself diverges. There is no dominant scale and no small parameter to expand in: a bubble of ordered spins the size of your thumb is statistically as important as one the size of an atom. Mean-field theory, which averages over fluctuations, and ordinary perturbation theory, which needs a small coupling, both break down. For decades critical phenomena were a wall physics could measure but not compute. The renormalization group walks through the wall by refusing to treat all scales at once. Instead it removes them one thin layer at a time. Coarse-grain: average over (integrate out) the shortest-wavelength degrees of freedom — the spins inside a block, or the Fourier modes in a thin shell near the cutoff — leaving an effective description of the coarser variables. Rescale lengths by the coarse-graining factor $b$ so the new theory has the same cutoff as the old and the two are comparable. Renormalize the field so fluctuation amplitudes stay normalized. The composition of these three moves is one renormalization-group transformation $\mathcal{R}_b$ , and it maps a set of coupling constants $\{K\}$ to a new set $\{K'\}=\mathcal{R}_b\{K\}$ . Iterating $\mathcal{R}_b$ turns the impossible sum-over-all-scales into a flow through the space of all possible Hamiltonians. Long-distance physics is governed not by any single step but by where the flow ends up: its fixed points $\{K^*\}=\mathcal{R}_b\{K^*\}$ and the way trajectories approach them. Because the correlation length contracts by $b$ at every step, a fixed point can only have $\xi^*=0$ or $\xi^*=\infty$ ; the infinite value picks out critical points. Systems with utterly different microscopic Hamiltonians that flow to the same fixed point share the same critical exponents — this is the precise origin of universality. This article builds that idea structurally; the concrete machinery (block spin, decimation, momentum shells, the Wilson–Fisher fixed point) lives in the neighbouring articles it organizes.

Key takeaways

The renormalization group is not one transformation but a repeatable procedure: coarse-grain (integrate out short-wavelength degrees of freedom), rescale lengths by the factor $b$ to restore the cutoff, and renormalize the field — producing an effective theory for the surviving long-wavelength fluctuations.
Each step maps the coupling set $\{K\}$ to $\{K'\}=\mathcal{R}_b\{K\}$ , defining a flow in the (infinite-dimensional) space of Hamiltonians; iterating generates an RG trajectory.
The correlation length transforms as $\xi'=\xi/b$ ; a fixed point therefore has $\xi^*=0$ (trivial) or $\xi^*=\infty$ (critical), so critical points sit at fixed points.
Coarse-graining is irreversible, so $\mathcal{R}_b$ forms a semigroup, not a group: $\mathcal{R}_{b_1}\mathcal{R}_{b_2}=\mathcal{R}_{b_1 b_2}$ with no inverse.
Universality is the basin-of-attraction structure of fixed points: microscopic differences are irrelevant directions washed out by the flow, so distinct systems share critical behaviour.
Key idea: repeatedly remove short-distance degrees of freedom and rescale — the resulting flow in coupling space has fixed points where $\xi=\infty$ , and long-distance critical physics is controlled entirely by those fixed points and the flow around them.

The core insight

Repeatedly remove short-distance degrees of freedom and rescale: the resulting flow in coupling space has fixed points whose neighbourhoods control critical behaviour, and universality is the basin-of-attraction structure of these fixed points.

Where this lives today

The renormalization group is now one of the load-bearing ideas of theoretical physics, reaching far beyond the critical magnets it was invented for. In condensed matter it explains why the 3D Ising model, the liquid–gas critical point of a simple fluid, and a uniaxial ferromagnet all share the same critical exponents: they sit in the same basin of attraction. It underlies finite-size scaling, the workhorse method for extracting exponents from finite-lattice Monte Carlo simulations, precisely because the only diverging length at criticality is $\xi$ , cut off by the system size $L$ . The same coarse-graining logic became the foundation of effective field theory: keep only the relevant and marginal operators consistent with symmetry, and the infinitely many irrelevant ones merely shift nonuniversal constants. That is why low-energy particle physics, the physics of ordinary materials, and even aspects of turbulence and machine-learning theory can be described by a handful of couplings — the microscopic details flow to zero. Wilson's picture of "remove one layer of scales and repeat" is now taught as a way of thinking, not just a calculational trick.

The physical picture

Photograph a fluid at its critical point, then blur the photograph and shrink it back to the original size. Because fluctuations exist at every scale, the blurred, shrunk image looks statistically like the original — the system is scale-invariant. The RG makes this precise: coarse-graining is the blurring, rescaling is the shrinking, and at criticality the statistical description is unchanged. That "unchanged under blurring" is exactly a fixed point. Away from criticality there is a special length, the correlation length $\xi$ . Blur past $\xi$ and the system looks featurely uniform — a trivial fixed point ( $\xi^*=0$ in rescaled units). The number of blurring steps needed to wash the system out measures $\ln\xi$ . As $T\to T_c$ , $\xi\to\infty$ , so it takes infinitely many steps — you never reach the trivial sink, and the flow is instead captured by a nontrivial fixed point. Think of coupling space as a landscape and $\mathcal{R}_b$ as the rule for one step. Most starting points drain into trivial sinks — the high-temperature disordered fixed point or the low-temperature ordered one. On the ridge separating those basins lies the critical surface; flow within it is drawn to a special nontrivial fixed point. Two materials whose microscopic Hamiltonians are wildly different but which both sit on the same critical surface flow to the same fixed point, and therefore share critical exponents. Universality is the geometry of this flow. The genius is replacing one impossible problem (sum over all scales at once) with infinitely many easy ones (peel off one thin layer of scales, repeat). Each peel is a controlled — often perturbative — calculation because only a narrow band of modes is integrated out. The nonperturbative physics is reconstructed by composing the steps. And all three operations are load-bearing: coarse-graining alone shrinks the system and changes the cutoff; rescaling lengths restores comparability; rescaling the field keeps fluctuation amplitudes finite. Only the full coarse-grain–rescale transformation possesses nontrivial fixed points.

Figure

One RG step: coarse-grain, rescale, and watch ξ shrink by b

Figure

The flow: two microscopically different systems reach one fixed point

The second figure is the payoff: two systems that look nothing alike at the atomic scale, but which both lie on the critical surface, flow into the same fixed point. Everything that distinguished them was an irrelevant direction, washed out by the flow. Only the one relevant direction — how far off criticality you sit — survives, and it is the same for both. That collapse onto one fixed point is universality.

The core equations

The whole construction is one map on the space of couplings, applied over and over. We write it abstractly first, because its power is that the form of the Hamiltonian need not be preserved — new couplings simply appear.

\mathcal{R}_b:\quad \{K'\} = \mathcal{R}_b\{K\} \tag{1}

where: $\mathcal{R}_b$ — the RG transformation that coarse-grains by linear factor $b>1$ then rescales; $\{K\}$ — the full set of coupling constants of the (reduced, i.e. divided by $k_BT$ ) Hamiltonian; $b$ — the linear rescaling factor. Read it as one step of a discrete dynamical system on coupling space. The first move, coarse-graining, is a partial trace: sum over the short-wavelength degrees of freedom only, keeping the long-wavelength field, to define an effective Hamiltonian.

e^{-\mathcal{H}'[\phi']} = \int_{\text{short}} \mathcal{D}\phi_<\; e^{-\mathcal{H}[\phi_<,\,\phi']} \tag{2}

where: $\mathcal{H}$ — the reduced Hamiltonian of the original system; $\mathcal{H}'$ — the effective Hamiltonian for the surviving long-wavelength field $\phi'$ ; $\phi_<$ — the short-wavelength modes being integrated out. By construction the partition function is preserved, $Z=\int\mathcal{D}\phi'\,e^{-\mathcal{H}'}$ , so the free energy is inherited exactly and every observable built only from $\phi'$ keeps its expectation value. The second and third moves restore comparability: contract lengths by $b$ so the cutoff matches the original, and rescale the field by $\zeta$ so fluctuation amplitudes stay normalized.

x \to x' = x/b,\qquad \phi'(x') = \zeta^{-1}\,\phi_{\text{coarse}}(x),\qquad \zeta = b^{(d-2+\eta)/2} \tag{3}

where: $x'$ — the rescaled coordinate restoring the original spacing; $\zeta$ — the field-rescaling factor, fixed at a fixed point by the dimension $d$ and the anomalous dimension $\eta$ ; at the Gaussian fixed point $\eta=0$ so $\zeta=b^{(d-2)/2}$ . Only after all three moves is one theory comparable to the next. The single most important consequence is how the correlation length transforms. Every physical length is measured in the new, contracted units, so $\xi$ shrinks by $b$ .

\xi' = \xi/b \tag{4}

where: $\xi$ — the correlation length of the original system; $\xi'$ — the same length measured in rescaled units. This relation expresses how the correlation length transforms under one RG step. Fixed points satisfy $\xi^{\ast}=\xi^{\ast}/b$ , whose only solutions are $\xi^{\ast}=0$ (stable phase) or $\xi^{\ast}=\infty$ (critical point). A fixed point is a set of couplings the transformation leaves alone — a self-similar, scale-invariant theory.

\{K^*\} = \mathcal{R}_b\{K^*\} \tag{5}

where: $\{K^*\}$ — a scale-invariant coupling set. Near the fixed point, the nonlinear RG map linearizes to a matrix whose eigenvalues determine the local scaling behavior.

\delta K'_a = \sum_b T_{ab}\,\delta K_b,\qquad T_{ab}=\left.\frac{\partial K'_a}{\partial K_b}\right|_{K^*},\qquad T\,\mathbf{e}_i=\lambda_i\,\mathbf{e}_i,\quad \lambda_i=b^{\,y_i} \tag{6}

where: $\delta K_a$ — the deviation of coupling $a$ from $K^*$ ; $T_{ab}$ — the linearized RG matrix (Jacobian at the fixed point); $\mathbf{e}_i$ — an eigenvector defining a scaling field $u_i$ ; $y_i$ — the RG eigenvalue ( $y_i>0$ relevant, $<0$ irrelevant, $=0$ marginal). Each scaling field scales by a pure power, $u_i'=b^{y_i}u_i$ . The full construction of $T$ , its eigenvalues, and the classification of the $y_i$ is developed by name in Linearization About a Fixed Point and Eigenvalue Spectrum and Relevant, Irrelevant, and Marginal Operators; here we only need that the exponents come from these eigenvalues. Because the singular free-energy density is extensive, one RG step divides it by $b^d$ while the relevant fields scale up — a homogeneity relation that, remarkably, shows that the homogeneity law follows from the linearized RG near the fixed point, rather than being introduced as an independent hypothesis.

f(u_t,u_h,\dots) = b^{-d}\,f\!\big(b^{y_t}u_t,\;b^{y_h}u_h,\dots\big) \tag{7}

where: $f$ — the singular part of the free-energy density; $u_t$ — the thermal scaling field $\sim(T-T_c)/T_c$ with eigenvalue $y_t$ ; $u_h$ — the magnetic scaling field $\sim H$ with eigenvalue $y_h$ . The scaling relations of Scaling Hypothesis and Scaling Relations follow from this equation as theorems of the RG. The single cleanest consequence — that the relevant thermal eigenvalue is the reciprocal of the correlation-length exponent — is derived below.

Derivation AWhy the correlation-length exponent is ν = 1 ⁄ y_t

The point of the whole machine is that an eigenvalue of a linearized map turns out to be a measurable critical exponent. This derivation is the cleanest demonstration: it takes the single relevant thermal field $u_t$ and the transformation $\xi'=\xi/b$ , and shows in four lines that $\xi$ must diverge as $|t|^{-1/y_t}$ . The strategy is to use the freedom in $b$ — the coarse-graining factor is ours to choose — and pick it so the renormalized thermal field becomes order one, where $\xi$ is order one and no longer critical. Watch how the arbitrary $b$ drops out and leaves a pure power law with a universal exponent.

Step 1. Combine the two transformation laws. Expressed in original units, $\xi$ is $b$ times the rescaled correlation length, and the rescaled system carries thermal field $b^{y_t}u_t$ :

\xi(u_t) = b\,\xi(u_t') = b\,\xi\!\big(b^{y_t}u_t\big).

Step 2. Choose $b$ to absorb the thermal field. Set the renormalized field to magnitude one:

b^{y_t}u_t = 1\ \Rightarrow\ b = u_t^{-1/y_t}.

This drives the coarse-grained system away from criticality, where $\xi(1)$ is just a finite constant.

Step 3. Substitute back:

\xi(u_t) = u_t^{-1/y_t}\,\xi(1).

Since $\xi(1)$ is a nonuniversal number, the $u_t$ -dependence is a pure power.

Step 4. Identify the exponent. With $u_t\propto t=(T-T_c)/T_c$ and the definition $\xi\sim|t|^{-\nu}$ ,

\boxed{\ \nu = 1/y_t.\ }

Why this is the whole point: $\nu$ is a property of the fixed point (through $y_t$ ), not of the microscopic Hamiltonian — so every system flowing to the same fixed point has the same $\nu$ . The arbitrary blocking factor $b$ cancelled, exactly as it must for a physical answer. The Gaussian fixed-point cross-check $y_t=2\Rightarrow\nu=1/2$ confirms the mean-field value.

Go deeper · Derivation B · 1 minNew couplings appear: why coupling space must be infinite-dimensional

Derivation BNew couplings appear: why coupling space must be infinite-dimensional

A subtle but essential fact is that coarse-graining does not preserve the form of the Hamiltonian: integrating out spins generates interactions that were absent microscopically. This derivation exhibits it on the one case that is exactly solvable — decimating the 1D Ising chain — and then explains why in higher dimensions the generated couplings proliferate, forcing the RG to act on an infinite-dimensional space and allowing fixed points of a form different from the bare theory. The strategy is to sum over every other spin on the chain and re-express the result as an effective coupling between the survivors plus a spin-independent constant.

Step 1. Start from the pure nearest-neighbour chain, $\mathcal{H}=K\sum_i s_i s_{i+1}$ (with $K=\beta J$ ).

Step 2. Trace over the central spin of a triplet $s_1,s_2,s_3$ :

\sum_{s_2=\pm1} e^{K(s_1 s_2 + s_2 s_3)} = 2\cosh\!\big(K(s_1+s_3)\big).

Summing out $s_2$ has coupled its two neighbours.

Step 3. Demand this equal an effective form $A\,e^{K's_1 s_3}$ and match the two cases $s_1 s_3=\pm1$ :

s_1 s_3=+1:\ 2\cosh 2K = A e^{K'};\qquad s_1 s_3=-1:\ 2 = A e^{-K'}.

Dividing gives $e^{2K'}=\cosh 2K$ , i.e.

K' = \tfrac12\ln\cosh 2K,\qquad A = 2\sqrt{\cosh 2K}.

The constant $A$ feeds the free energy; the recursion in $K$ is exact (developed further by name in Decimation and the One-Dimensional Ising RG).

Step 4. Go to $d>1$ . There the analogous partial trace cannot be written with only the original nearest-neighbour coupling; it generates further-neighbour terms $K_2 s_i s_{i+2}$ , four-spin plaquette terms $K_\square s_i s_j s_k s_l$ , and so on:

\mathcal{H}'\ni K_1' s_i s_{i+1} + K_2' s_i s_{i+2} + K_\square' s_i s_j s_k s_l + \cdots

Why this is not a defect but the whole design: because $\mathcal{R}_b$ maps this infinite-dimensional coupling space into itself (it is closed), the flow is well-defined, and fixed points need not resemble the bare Hamiltonian — the Wilson–Fisher fixed point, reached in momentum space, is nothing like a nearest-neighbour Ising model. As a consistency check: the recursion $K'=\tfrac12\ln\cosh2K$ and prefactor $A=2\sqrt{\cosh2K}$ are exact, and iterating from $K=1$ flows to $K^*=0$ , confirming there is no finite-temperature transition in 1D.

Limiting cases

The single idea specializes to every standard RG construction by a choice of how to coarse-grain; each is developed in a neighbouring article:

Case	Result	Meaning
Real-space (block spin)	$s'_I=\mathrm{sgn}\big(\sum_{i\in I}s_i\big)$	Spins grouped into blocks of linear size $b$ ; a majority or decimation rule defines the partial trace. Intuitive, but generates infinitely many couplings and is usually approximate.
1D Ising decimation (exact)	$K'=\tfrac12\ln\cosh 2K$	Summing over alternate spins is exact; the only fixed points are $K^=0$ (disorder) and $K^=\infty$ (order), so $T_c=0$ in 1D — no finite- $T$ transition.
Momentum shell	$\Lambda/b<\vert\mathbf{k}\vert<\Lambda$ integrated out	Fourier modes in a thin shell near the cutoff are traced over; with $k'=bk$ and field rescaling, this reaches the Wilson–Fisher fixed point and the $\epsilon$ -expansion.
Continuous (Wilson) $b\to1$	$b=e^{\ell},\ \dfrac{dK}{d\ell}=\beta(K)$	Infinitesimal steps convert the map into a flow with beta functions; fixed points are the zeros of $\beta$ .

Worked examples

Example 1 — Iterate the exact 1D decimation and confirm there is no transition.

Step 1. Use $K'=\tfrac12\ln\cosh 2K$ . Start at $K_0=1.0$ (a fairly strong coupling). Then $K_1=\tfrac12\ln\cosh 2 = \tfrac12\ln(3.7622) = \mathbf{0.6625}$ .

Step 2. Iterate: $K_2=\tfrac12\ln\cosh(1.3250)=\mathbf{0.4085}$ , then $K_3=\mathbf{0.2263}$ , $K_4=\mathbf{0.1141}$ , $K_5=\mathbf{0.0554}$ . The coupling is halving-and-more each step.

Step 3. The sequence marches monotonically toward $K^*=0$ : after ~20 steps it is below $10^{-4}$ , and the flow limit is the disordered (paramagnetic) fixed point. The only fixed points are $K^{\ast}=0$ and $K^{\ast}=\infty$ ; there is no nontrivial fixed point at finite $K$ .

Step 4. Interpret. A finite-temperature transition would require a nontrivial fixed point at some $0<K^*<\infty$ separating an ordered basin from a disordered one. The 1D flow has none, so the 1D Ising model orders only at $T=0$ ( $K^*=\infty$ ) — the RG reproduces the exact transfer-matrix result (developed by name in One-Dimensional Ising Model and Transfer Matrix Method).

Example 2 — Check $\xi'=\xi/b$ against the exact 1D correlation length.

Step 1. The exact 1D Ising correlation length is $\xi=-1/\ln\tanh K$ (lattice units). Take $K=1.0$ : $\tanh 1=0.76159$ , so $\xi=-1/\ln(0.76159)=\mathbf{3.6719}$ spacings.

Step 2. One decimation step ( $b=2$ ) sends $K\to K'=0.6625$ ; there is an exact identity for $b=2$ decimation, $\tanh K'=\tanh^2 K$ . Check: $\tanh 0.6625=0.58003$ and $\tanh^2 1=0.76159^2=0.58003$ . ✓

Step 3. Compute the new correlation length: $\xi'=-1/\ln(0.58003)=\mathbf{1.8359}$ spacings (now in the doubled lattice spacing).

Step 4. Verify the RG law. Measured in the original spacing, the coarse-grained length is $\xi/b=3.6719/2=\mathbf{1.8359}$ — exactly $\xi'$ . So $\xi'=\xi/b$ holds identically, confirming the transformation law that forces critical points to sit at the $\xi^*=\infty$ fixed point.

Origins · 1 minrenormalization group and how this was discovered

renormalization group

the framework built by Kadanoff, Wilson and Fisher, 1966–1972 · The renormalization group is not a person but the central framework of critical phenomena: coarse-grain a system by a factor

b

, rescale, and follow how the couplings flow. Leo Kadanoff's 1966 block-spin picture supplied the intuition; Kenneth Wilson (1936–2013) made it an explicit flow of couplings in two 1971 Physical Review B papers; and Wilson with Michael Fisher introduced the ε-expansion in 1972, computing critical exponents for the first time from a controlled scheme — work honoured with Wilson's 1982 Nobel Prize in Physics. The "group" is really a semigroup, because coarse-graining throws information away and cannot be undone.

History

1954

— Gell-Mann and Low introduce a renormalization-group equation in quantum electrodynamics

1965

— Widom proposes the scaling form of the equation of state near a critical point

1966

— Kadanoff's block-spin picture suggests coarse-graining as the route to scaling

1971

— Wilson formulates the renormalization group for critical phenomena (two Phys. Rev. B papers)

1972

— Wilson and Fisher introduce the ε-expansion, computing critical exponents systematically

1982

— Wilson receives the Nobel Prize in Physics for the theory of critical phenomena

Going further

References. Kardar, Statistical Physics of Fields, Ch. 4–5 · Pathria & Beale, Statistical Mechanics 3rd ed., Ch. 14 · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Ch. 12 · Wilson & Kogut, Physics Reports 12 (1974) 75 · Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Ch. 9 · Cardy, Scaling and Renormalization in Statistical Physics, Ch. 3. Key results to keep in view: the correlation-length transformation $\xi'=\xi/b$ forces the fixed-point dichotomy $\xi^*=0,\infty$ ; the 1D Ising decimation recursion $K'=\tfrac12\ln\cosh 2K$ has fixed points $K^*=0,\infty$ ; the free-energy homogeneity relation $f=b^{-d}f(b^{y}u)$ yields $\nu=1/y_t$ , $2-\alpha=d/y_t$ , and the gap exponent $\Delta=y_h/y_t$ ; the field-rescaling exponent is $\zeta=b^{(d-2+\eta)/2}$ ; and the Gaussian fixed-point cross-check gives $y_t=2\Rightarrow\nu=1/2$ (the mean-field value).

Builds on: Correlation length and critical phenomena · Partition function and effective Hamiltonians · Mean-field theory and Landau expansion · Fourier analysis and the cutoff. Leads to: RG Flow in Coupling Space · Fixed Points of the RG · Linearization About a Fixed Point and Eigenvalue Spectrum · Block-Spin (Real-Space) Renormalization Group · Momentum-Shell Renormalization Group · Critical Exponents from the RG.

Why can't I just do ordinary perturbation theory at a critical point?

Because there is no small parameter. Perturbation theory expands in a coupling and assumes fluctuations are a small correction to some reference state. At criticality, fluctuations exist at every length scale from the lattice spacing up to $\xi\to\infty$ , and they all contribute comparably — the "corrections" are the same size as the leading term. Mean-field theory fails for the mirror-image reason: it averages fluctuations away, but near $T_c$ they dominate. The RG works precisely because it never handles all scales at once; each step integrates out only a thin band of modes, which is a controlled calculation.

Is the renormalization group the same thing in statistical mechanics and in particle physics?

It is the same core idea — systematically changing the scale of description and tracking how couplings run — but the two traditions emphasize different things. The particle-physics RG (Gell-Mann–Low, 1954) grew from removing ultraviolet divergences and gives running coupling constants. Wilson's statistical-mechanics RG (1971) is a real-space/momentum-space coarse-graining that computes critical exponents from fixed points. Wilson himself unified them: renormalizability in field theory is the statement that a theory sits near a fixed point with only a few relevant couplings. So "the RG" is one concept with two dialects.

What does it actually mean for a coupling to be "generated" by coarse-graining?

When you integrate out the short-wavelength spins, the effective Hamiltonian for the survivors generally contains interactions that were not in the original. Decimating a 1D chain of nearest-neighbour bonds is special — it stays nearest-neighbour — but in $d>1$ tracing out spins couples further neighbours and produces multi-spin (e.g. four-spin plaquette) terms. So even if you start with a "simple" model, after one step you are somewhere in an infinite-dimensional space of couplings. This is why the RG must be formulated on that whole space, and why fixed points can look nothing like the model you started from.

If the choice of block size b is arbitrary, how can the answer be physical?

Because the universal quantities do not depend on $b$ . You can see it in the $\nu=1/y_t$ derivation: the factor $b$ was introduced and then chosen ( $b=u_t^{-1/y_t}$ ) to absorb the thermal field, and it cancelled out of the final power law. The eigenvalues $y_i$ themselves are $b$ -independent (that is a consistency requirement, since $\mathcal{R}_{b_1 b_2}=\mathcal{R}_{b_1}\mathcal{R}_{b_2}$ forces $\lambda_i(b_1 b_2)=\lambda_i(b_1)\lambda_i(b_2)$ , i.e. $\lambda_i=b^{y_i}$ ). Nonuniversal numbers like $T_c$ and amplitudes do depend on the scheme, but exponents and scaling functions do not.

Why does the correlation length shrink under the RG instead of grow?

Because you are measuring it in the new, contracted units. Coarse-graining does not physically change any correlation — the far-apart spins are still correlated. But after you rescale lengths $x\to x/b$ to restore the original cutoff, every length, including $\xi$ , is reported as $b$ times smaller: $\xi'=\xi/b$ . Intuitively, each blurring step "uses up" a factor $b$ of the correlation length, so a system with large $\xi$ takes many steps to look uniform. Only $\xi=\infty$ survives infinitely many steps unchanged — which is exactly why critical points are fixed points.

How do the eigenvalues of the linearized flow become critical exponents in general?

Near a fixed point the map linearizes to a matrix $T$ ; its eigenvalues are written $\lambda_i=b^{y_i}$ , and the eigen-combinations of couplings (the scaling fields $u_i$ ) transform multiplicatively, $u_i'=b^{y_i}u_i$ . Feeding these into the free-energy homogeneity relation $f=b^{-d}f(b^{y_t}u_t,b^{y_h}u_h)$ and choosing $b$ to absorb one field turns $f$ into a scaling form whose powers are ratios of the $y_i$ — giving $\nu=1/y_t$ , $2-\alpha=d/y_t$ , the gap exponent $\Delta=y_h/y_t$ , and so on. The full extraction of all six exponents is carried out by name in Critical Exponents from the RG.