Linearization About a Fixed Point and Eigenvalue Spectrum

The renormalization group turns a statistical model into a point $\mu=(K_1,K_2,\dots)$ in the space of coupling constants and advances it one coarse-graining step at a time by a map $\mu'=\mathcal{R}_b[\mu]$ . That map is hopelessly nonlinear — each step generates new couplings from old ones through sums over eliminated degrees of freedom. Yet everything measurable about a critical point comes from the map's behaviour in a tiny neighbourhood of a fixed point, and there the nonlinear tangle collapses into a single matrix. Linearization is the move that makes the renormalization group calculable. Write $\mu=\mu^*+\delta\mu$ and expand to first order. The constant matrix $T_{ij}=\partial\mu_i'/\partial\mu_j$ , evaluated at the fixed point, governs how deviations evolve: $\delta\mu'=T\,\delta\mu$ . Its eigenvalues $\lambda_i$ depend on the arbitrary length-rescaling factor $b$ , but the combination $y_i=\ln\lambda_i/\ln b$ does not — the semigroup law forces $\lambda_i=b^{y_i}$ , so $y_i$ is an intrinsic property of the fixed point. These $y_i$ are the scaling dimensions of the theory. The eigenvectors define special linear combinations of couplings, the scaling fields $u_i$ , which flow independently and multiplicatively: $u_i'=b^{y_i}u_i$ . The sign of each $y_i$ tells the whole story of the flow's local geometry. Relevant fields ( $y_i>0$ ) grow under coarse-graining and span the unstable manifold; irrelevant fields ( $y_i<0$ ) decay and span the critical surface; marginal fields ( $y_i=0$ ) are decided by nonlinear terms. Because power laws are the fingerprint of multiplicative dynamics, these eigenvalues are exactly what turn into the power-law singularities of criticality. This article derives the linearization, proves the $b$ -independence of $y_i$ , and computes the spectrum explicitly for the exactly solvable one-dimensional Ising decimation — including a correction to a value that is commonly misquoted. How the two relevant eigenvalues become the full set of critical exponents is the subject of the next article; here we build the eigenvalue spectrum they depend on.

Key takeaways

Near a fixed point the nonlinear RG recursion $\mu'=\mathcal{R}_b[\mu]$ linearizes to $\delta\mu'=T\,\delta\mu$ with $T_{ij}=\partial\mu_i'/\partial\mu_j\big|_{\mu^*}$ .
The eigenvalues obey $\lambda_i=b^{y_i}$ ; the RG eigenvalue (scaling dimension) $y_i=\ln\lambda_i/\ln b$ is intrinsic — independent of the rescaling factor $b$ .
The eigenvectors define scaling fields $u_i$ that decouple the flow: each transforms multiplicatively, $u_i'=b^{y_i}u_i$ , converting a messy nonlinear map into independent rescalings.
The sign of $y_i$ classifies directions — relevant ( $y_i>0$ , unstable), irrelevant ( $y_i<0$ , stable), marginal ( $y_i=0$ ).
For the 1D Ising decimation the spectrum has one relevant eigenvalue at the $T=0$ fixed point: $\lambda=2$ , $y_t=1$ (a value often misquoted as $\lambda=4$ , $y_t=2$ ).
Key idea: diagonalizing the linearized RG map replaces a nonlinear flow by independent multiplicative rescalings $u_i'=b^{y_i}u_i$ ; the $b$ -independent eigenvalue exponents $y_i$ are the scaling dimensions that control the flow's geometry and every critical exponent.

The core insight

Diagonalizing the linearized RG map at a fixed point replaces a nonlinear flow by independent multiplicative rescalings $u_i'=b^{y_i}u_i$ ; the eigenvalue exponents $y_i=\ln\lambda_i/\ln b$ are $b$ -independent scaling dimensions that control the flow's geometry and, through the two relevant ones, every critical exponent.

Where this lives today

The linearized eigenvalue spectrum is the backbone of how critical exponents are actually measured, not just imagined. In large-scale Monte Carlo simulations the relevant eigenvalues are extracted directly from finite-size scaling: the susceptibility peak grows with system size as $L^{2y_h-d}$ and the slope of the Binder cumulant as $L^{y_t}$ , so fitting these size dependences reads $y_t$ and $y_h$ off the data. The conformal-bootstrap program achieves the same eigenvalues to many digits for the 3D Ising class by demanding self-consistency of the fixed-point theory. And in the study of quantum matter, the scaling dimensions of a quantum critical point — eigenvalues of a zero-temperature RG — set the temperature and frequency power laws seen in transport and neutron-scattering experiments on materials tuned to quantum phase transitions.

The physical picture

Stand exactly on a mountain pass — the fixed point. Release a marble in each principal direction. In some directions it rolls away downhill; in others it rolls back toward you along the ridge. Linearization is nothing more than measuring the local slopes in each principal direction, and those slopes are the eigenvalues of $T$ . A slope greater than one (in the sense that a displacement is amplified) means the marble runs away: a relevant direction. A slope less than one means it returns: an irrelevant direction. A slope of exactly one is marginal — the terrain is momentarily flat, and only the next-order curvature decides which way the marble eventually drifts. A single RG step multiplies a displacement along eigendirection $i$ by $\lambda_i=b^{y_i}$ . After $n$ steps the total factor is $b^{ny_i}=L^{y_i}$ , where $L=b^n$ is the accumulated rescaling. This is why relevant fields grow as a power of the length scale, and why the physical observables they control — the correlation length, the order parameter — acquire power-law singularities. Power laws are the signature of multiplicative (eigenvalue) dynamics; exponential or logarithmic behaviour would signal something else, and indeed marginal directions produce logarithms rather than powers.

Figure

Linearizing at the saddle: eigen-directions and their multiplicative rates

Universality becomes transparent in this language. Change the microscopic Hamiltonian and you change where you start in coupling space, but if the new starting point lies on the same critical surface — the stable manifold, spanned by irrelevant eigen-directions — it flows to the same fixed point. Only the handful of relevant eigenvalues survive the journey, so only they appear in the exponents. Two systems as different as a uniaxial magnet and a boiling liquid share their exponents precisely because their differences lived entirely in irrelevant directions that the flow eroded away. The concrete payoff is a definite number. For the one exactly solvable case — the decimation of the 1D Ising chain — the linearization is a one-line calculation whose single relevant eigenvalue sits at the zero-temperature fixed point, and it is a value that is easy to misquote.

Figure

The 1D Ising decimation cobweb: one relevant eigenvalue at the T ＝ 0 fixed point

The core equations

The starting point is the fixed-point condition and the deviation about it. Because $\mathcal{R}_b$ returns $\mu^*$ unchanged, the zeroth-order term of a Taylor expansion cancels and the leading behaviour is linear.

\mu'=\mathcal{R}_b[\mu],\qquad \mu^*=\mathcal{R}_b[\mu^*] \qquad (1)

where: $\mu=(K_1,K_2,\dots)$ — point in coupling space; $\mathcal{R}_b$ — RG transformation with length-rescaling factor $b>1$ ; $\mu^*$ — fixed point. Reading it physically: $\mu^*$ is the scale-invariant theory whose neighbourhood we are about to resolve. Expanding $\mu=\mu^*+\delta\mu$ and keeping first order defines the linearized map — a constant matrix because it is evaluated at the fixed point.

\delta K_i' = \sum_j T_{ij}\,\delta K_j,\qquad T_{ij}=\left.\frac{\partial K_i'}{\partial K_j}\right|_{\mu^*} \qquad (2)

where: $\delta K_i=K_i-K_i^*$ — deviation of coupling $i$ ; $T_{ij}$ — Jacobian of $\mathcal{R}_b$ at $\mu^*$ . Reading it physically: the whole local flow is captured by this one matrix; the nonlinearity of $\mathcal{R}_b$ only matters far from the fixed point. Diagonalizing $T$ produces eigenvalues and eigenvectors. The eigenvectors define scaling fields; the eigenvalues, remarkably, must be pure powers of $b$ .

T\,\phi^{(i)}=\lambda_i\,\phi^{(i)},\qquad u_i=\phi^{(i)}\!\cdot\delta\mu,\qquad \lambda_i=b^{\,y_i},\qquad y_i=\frac{\ln\lambda_i}{\ln b} \qquad (3)

where: $\lambda_i$ — $i$ -th eigenvalue of $T$ (depends on $b$ ); $\phi^{(i)}$ — left eigenvector; $u_i$ — scaling field, a linear combination of couplings that flows independently; $y_i$ — RG eigenvalue / scaling dimension, $b$ -independent. Reading it physically: the scaling fields are the natural coordinates in which the flow decouples, and $y_i$ is the intrinsic label of each. Under one RG step each scaling field simply rescales by its own eigenvalue — the defining property that makes them useful.

u_i' = \lambda_i\,u_i = b^{\,y_i}\,u_i \qquad (4)

where: $b^{y_i}$ — multiplicative factor per step. Reading it physically: after $n$ steps $u_i\to b^{ny_i}u_i$ , so relevant fields ( $y_i>0$ ) blow up and irrelevant ones ( $y_i<0$ ) vanish, exactly as the mountain-pass picture demands. The sign of $y_i$ classifies each eigen-operator, and this classification is the payoff of the whole construction.

y_i>0\ (\text{relevant}),\qquad y_i<0\ (\text{irrelevant}),\qquad y_i=0\ (\text{marginal}) \qquad (5)

where: relevant — grows, must be tuned away to reach the fixed point; irrelevant — decays, spans the critical surface; marginal — $\lambda_i=1$ , decided by nonlinear terms. The number of relevant directions is the codimension of the critical surface — the count of parameters you must tune to be critical.

Derivation AWhy the RG eigenvalue is independent of the rescaling factor

The eigenvalues $\lambda_i$ obviously depend on how much you coarse-grain — a bigger $b$ is a bigger step, so a bigger multiplication. The claim is that the exponent $y_i=\ln\lambda_i/\ln b$ does not depend on $b$ , so it is a genuine property of the fixed point rather than an artefact of the arbitrary step size. The strategy is to use the semigroup structure of the RG: coarse-graining by $b_1$ and then by $b_2$ must equal a single coarse-graining by $b_1b_2$ . Applying that consistency to the eigenvalue along a fixed eigendirection forces the eigenvalue to be a pure power of $b$ , and the power is $b$ -independent by construction.

Step 1. Two successive RG steps compose into one with the product rescaling factor,

\mathcal{R}_{b_1}\circ\mathcal{R}_{b_2}=\mathcal{R}_{b_1 b_2}.

This is the semigroup law: throwing away short-distance information twice, by $b_1$ then $b_2$ , is the same as throwing it away once by $b_1b_2$ .

Step 2. Linearize all three maps at the same fixed point $\mu^*$ and follow one eigendirection $\phi^{(i)}$ . The Jacobians compose by multiplication, so their eigenvalues along that shared direction do too,

\lambda_i(b_1)\,\lambda_i(b_2)=\lambda_i(b_1 b_2).

Why the eigendirection is shared. At the fixed point the eigen-directions of $\mathcal{R}_b$ are the same for every $b$ (they are set by the fixed point, not the step size); only the eigenvalues change with $b$ . So the composition acts diagonally on each $\phi^{(i)}$ .

Step 3. The functional equation $f(b_1)f(b_2)=f(b_1b_2)$ has, for a continuous positive function, only the solution

\lambda_i(b)=b^{\,y_i},

with a constant $y_i$ . Taking logarithms, $\ln\lambda_i=y_i\ln b$ , so $y_i=\ln\lambda_i/\ln b$ is manifestly independent of $b$ . Why this matters. The intrinsic quantity is $y_i$ , never $\lambda_i$ . Two calculations using $b=2$ and $b=3$ will report different $\lambda_i$ but the same $y_i$ — the scaling dimension is unambiguous, which is what lets different RG schemes agree on critical exponents. Verified: verify_rg-linearization-eigenvalues.py confirms $\lambda(b)=b^y$ solves the composition law symbolically, and demonstrates numerically that a two-coupling map gives the same $y_i$ whether read at $b=2$ or at $b=4$ (its square).

The abstract construction becomes concrete on a map we can write down in full: the decimation of the one-dimensional Ising chain. Here linearization is a one-line calculation, and it exposes a value that is easy to get wrong.

Go deeper · Derivation B · 1 minThe eigenvalue spectrum of the 1D Ising decimation

Derivation BThe eigenvalue spectrum of the 1D Ising decimation

We compute the linearized RG spectrum for the one exactly solvable case — decimating every second spin of the Ising chain — to see the general machinery produce a definite number, and to correct a value that is frequently misquoted. The recursion itself (summing over alternate spins to get an effective coupling between survivors) is derived elsewhere; we take it as the map $\tanh K'=\tanh^2K$ and linearize it about each of its two fixed points, $K^*=0$ and $K^*=\infty$ . The delicate one is $K^*=\infty$ , which we handle in the variable $x=e^{-2K}$ so that the fixed point sits at $x=0$ .

Step 1. At the disordered fixed point $K^*=0$ , differentiate the map in the form $K'=\tfrac12\ln\cosh2K$ :

\left.\frac{\mathrm{d}K'}{\mathrm{d}K}\right|_{K=0}=\tanh(2K)\big|_{0}=0.

The eigenvalue is $\lambda=0$ ( $y=-\infty$ ): the strongest possible irrelevance. Any small coupling collapses to zero in a single step — the high-temperature sink is superstable.

Step 2. At the ordered fixed point $K^*=\infty$ , switch to $x=e^{-2K}\to0$ . With $\tanh K=(1-x)/(1+x)$ and $\tanh K'=\tanh^2K$ , solve $\tanh K'=(1-x')/(1+x')$ for $x'$ and expand:

x' = 2x + O(x^2).

The eigenvalue along the single coupling is $\lambda=2$ . Why the leading factor is $2$ , not $4$ . A common slip keeps the wrong power in the small- $x$ expansion and reports $x'\approx4x$ , hence $\lambda=4$ and $y_t=2$ . The correct expansion of $\tanh K'=\tanh^2K$ gives $x'=2x$ exactly at leading order, so $\lambda=2$ .

Step 3. Convert to the scaling dimension with $b=2$ (decimation halves the lattice spacing),

y_t=\frac{\ln\lambda}{\ln b}=\frac{\ln 2}{\ln 2}=1.

There is one relevant eigenvalue, at the zero-temperature fixed point.

Step 4. Read the physics. A single relevant direction at $T=0$ means one dimension has no finite-temperature transition; criticality is pushed to $T=0$ . Consistently, the exact correlation length $\xi=1/x=e^{2K}$ diverges exponentially as $x\to0$ , not as a power law — the hallmark of a zero-temperature critical point, and exactly what $y_t=1$ with an exponential (rather than power-law) $\xi(T)$ encodes. Verified: verify_rg-linearization-eigenvalues.py expands $\tanh K'=\tanh^2K$ in $x=e^{-2K}$ and finds $x'\to2x$ (so $\lambda=2$ , $y_t=1$ ), explicitly flagging the $x'=4x$ base error; it also checks the disordered fixed point has $\lambda=0$ .

Once the spectrum is known, the two relevant eigenvalues $y_t$ and $y_h$ determine every critical exponent through the homogeneity of the singular free energy, $f_s(u_t,u_h)=b^{-d}f_s(b^{y_t}u_t,b^{y_h}u_h)$ . That homogeneity relation follows because the free energy is extensive and rescales by the volume factor $b^{-d}$ ; extracting $\nu,\alpha,\beta,\gamma,\delta,\eta$ from it is the dedicated task of the exponents article. As a consistency check we may invert the relations: given measured exponents, $y_t=1/\nu$ and $y_h=(d+2-\eta)/2$ . For the 2D Ising fixed point ( $\nu=1$ , $\eta=1/4$ ) this returns $y_t=1$ and $y_h=15/8$ , the exact Onsager values recalled from the two-dimensional solution.

Limiting cases

The spectrum specializes cleanly in the standard fixed points and regimes.

Case	Result	Meaning
Gaussian fixed point, $d>4$	$y_t=2,\ y_h=\tfrac{d+2}{2}$	Eigenvalues from engineering dimensions; quartic $y_u=4-d<0$ irrelevant, mean-field exponents.
2D Ising fixed point	$y_t=1,\ y_h=\tfrac{15}{8}$	Exact Onsager values; give $\nu=1$ , $\delta=15$ , $\eta=\tfrac14$ .
1D Ising, $T=0$	$\lambda=2,\ y_t=1$	One relevant direction at $T=0$ ; no finite- $T$ transition, $\xi\sim e^{2K}$ .
Marginal direction	$y_i=0,\ \lambda_i=1$	Linear analysis silent; the leading nonlinear term decides marginal-relevant vs marginal-irrelevant (logs).

Worked examples

Example 1 — Reading a spectrum off a Wilson–Fisher-like Jacobian. At a critical fixed point the linearized map (in the thermal/quartic basis, with $b=2$ ) is the upper-triangular matrix

T=\begin{pmatrix}1.984 & 0.55\\ 0 & 0.42\end{pmatrix}.

Find the two scaling dimensions and classify the directions.

Step 1. For an upper-triangular matrix the eigenvalues are the diagonal entries: $\lambda_t=1.984$ and $\lambda_u=0.42$ .

Step 2. Convert with $b=2$ . The thermal direction: $y_t=\ln(1.984)/\ln 2=0.6853/0.6931=\mathbf{0.988}$ , positive — relevant. The quartic direction: $y_u=\ln(0.42)/\ln 2=-0.8675/0.6931=\mathbf{-1.252}$ , negative — irrelevant.

Step 3. Interpret. There is one relevant thermal direction (so one tuning, temperature, on the symmetric critical surface) and the quartic coupling is irrelevant, which is exactly the Wilson–Fisher fixed-point signature: the interaction that destabilized the Gaussian point has itself become irrelevant at the new fixed point, stabilizing it.

Example 2 — Checking $b$ -independence numerically. Take the same matrix $T$ and compose it with itself, $T^2$ , which represents a single RG step of $b'=b^2=4$ . Verify that the thermal scaling dimension is unchanged.

Step 1. The relevant eigenvalue of $T^2$ is $\lambda_t^2=1.984^2=3.936$ .

Step 2. Convert with $b'=4$ : $y_t=\ln(3.936)/\ln 4=1.3705/1.3863=\mathbf{0.988}$ .

Step 3. This is the same $0.988$ found in Example 1 with $b=2$ . The scaling dimension is invariant under the choice of rescaling factor, as the semigroup argument guarantees — a numerical confirmation that $y_i$ , not $\lambda_i$ , is the physical quantity.

Recall the tools we lean on

The decimation recursion $\tanh K'=\tanh^2K$ linearized above is derived in Decimation and the One-Dimensional Ising RG, and the exact 1D correlation length $\xi=-1/\ln\tanh K$ comes from One-Dimensional Ising Model and Transfer Matrix Method. The fixed points whose neighbourhoods we linearize are established in Fixed Points of the RG. Turning the two relevant eigenvalues $y_t,y_h$ into the numbers $\nu,\alpha,\beta,\gamma,\delta,\eta$ is the job of Critical Exponents from the RG; the one-loop momentum-shell calculation of these eigenvalues for the Wilson–Fisher fixed point is developed in the $\epsilon$ -expansion and Wilson–Fisher articles. This one supplies the spectrum they all consume.

Origins · 1 minrunning coupling and how this was discovered

running coupling

the RG eigenvalue that fixes critical scaling · A running coupling is a parameter whose value depends on the scale at which you look — precisely what the linearized RG produces, since each scaling field

u_i

grows or shrinks as

u_i' = b^{y_i} u_i

under coarse-graining. The exponents

y_i

became the central quantities of critical phenomena through the renormalization-group work of the early 1970s: Kadanoff's 1966 block-spin scaling already assigned dimensions to the thermal and magnetic fields; Wilson's 1971 formulation identified them as eigenvalues of the linearized flow; and Wegner (1972) systematized the scaling fields. A running coupling is not a person but the eigen-direction whose flow turns coarse-graining into measurable power laws.

History

1966

— Kadanoff assigns thermal and magnetic scaling dimensions in the block-spin picture

1971

— Wilson identifies scaling dimensions as eigenvalues of the linearized RG flow

1972

— Wegner formalizes scaling fields and the classification of operators by their RG eigenvalues

1972

— Wilson and Fisher compute the leading scaling dimensions of the nontrivial fixed point in the ε-expansion

1974

— Wilson and Kogut's review sets out the eigenvalue-spectrum framework in full

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 · Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Ch. 9 · Wilson & Kogut, Phys. Rep. 12, 75 (1974).

Provenance. The linearization $T_{ij}=\partial K_i'/\partial K_j$ , the semigroup argument $\lambda_i(b)=b^{y_i}$ , and the scaling-field construction follow Kardar (Ch. 4) and Goldenfeld (Ch. 9). The 1D Ising decimation spectrum is computed here from the recursion $\tanh K'=\tanh^2K$ (Pathria Ch. 13, Reif). The homogeneity relation and exponent inversions $y_t=1/\nu$ , $y_h=(d+2-\eta)/2$ are standard; the exponent formulas themselves are derived in the dedicated exponents article. The 2D Ising eigenvalues $y_t=1$ , $y_h=15/8$ are exact (Onsager). A base-error correction is recorded: the 1D decimation eigenvalue in $x=e^{-2K}$ is $\lambda=2$ , $y_t=1$ (from $x'=2x$ ), not $\lambda=4$ , $y_t=2$ .

Builds on: The Renormalization Group Idea: Coarse-Graining and Rescaling · RG Flow in Coupling Space · Fixed Points of the RG. Leads to: Relevant, Irrelevant, and Marginal Operators · Scaling Fields and Scaling Variables · Critical Exponents from the RG · The Epsilon-Expansion.

What exactly is a "scaling field," and how is it different from a coupling constant?

A coupling constant like $K_1$ or $K_2$ is a bare parameter of the Hamiltonian; under the RG the different couplings mix, so no single one flows cleanly. A scaling field $u_i$ is a special linear combination of the deviations $\delta K_j$ — an eigenvector of the linearized map — chosen precisely so that it does not mix with the others. Each scaling field flows on its own, $u_i'=b^{y_i}u_i$ . Near a fixed point the scaling fields are the natural coordinates: they diagonalize the flow, turning a coupled nonlinear tangle into independent rescalings.

Why does the eigenvalue have to be a power of

b

rather than some other function?

Because RG steps compose: coarse-graining by $b_1$ then $b_2$ equals one step by $b_1b_2$ . Along a fixed eigendirection this forces the eigenvalue to satisfy $\lambda(b_1)\lambda(b_2)=\lambda(b_1b_2)$ . The only continuous positive function obeying that multiplicative law is a power, $\lambda(b)=b^y$ . So the power-law form is not an assumption — it is dictated by the semigroup structure of the renormalization group, and the exponent $y$ is what carries the physics.

If a direction is marginal (

y=0

), what determines its fate?

Linear analysis is silent, because $\lambda=1$ means the deviation neither grows nor shrinks at first order. The answer comes from the leading nonlinear term, say $u'=u+au^2$ . If $a$ has the sign that makes $u$ grow, the direction is marginally relevant and flows slowly to large coupling; if it makes $u$ shrink, it is marginally irrelevant and produces logarithmic corrections to scaling. The upper critical dimension $d=4$ is the classic marginal case — hence the logarithmic corrections to mean-field behaviour there.

Why is the 1D Ising eigenvalue so often quoted as

\lambda=4

It comes from a sign-innocent but wrong expansion. In the variable $x=e^{-2K}$ , one must expand $\tanh K'=\tanh^2K$ carefully; using $\tanh K=(1-x)/(1+x)$ gives $\tanh^2K=1-4x+\dots$ , but the conversion back to $x'$ through $\tanh K'=(1-x')/(1+x')$ halves that, yielding $x'=2x$ , not $4x$ . Dropping the second step leaves the spurious factor of $4$ . The correct eigenvalue is $\lambda=2$ and $y_t=1$ , consistent with the exact exponential divergence $\xi\sim e^{2K}$ of the 1D chain. This article carries the corrected value.

Why don't the irrelevant eigenvalues appear in the critical exponents?

Because they decay under coarse-graining. A field with $y_i<0$ shrinks by $b^{y_i}<1$ each step, so by the time the flow reaches the fixed point it has vanished — its initial value cannot influence the asymptotic long-distance physics. Only the relevant eigenvalues ( $y_t,y_h$ ) survive to the fixed point, so only they enter the exponents. The irrelevant ones are not entirely invisible, though: the slowest of them sets the leading correction to scaling, which matters when you extract exponents from finite data.

How are

y_t

and

y_h

measured in a simulation?

Through finite-size scaling. At criticality, thermodynamic quantities scale with the system size $L$ as powers set by the relevant eigenvalues: the susceptibility peak grows as $L^{2y_h-d}$ , and the temperature-derivative of the Binder cumulant grows as $L^{y_t}$ . Running the model at several sizes and fitting these growths gives $y_h$ and $y_t$ directly, without ever writing down the RG map explicitly. Every other exponent then follows from the two eigenvalues via the scaling relations.