F002 · What survives a change of scale?

Coarse Graining and Renormalization Flow

Why can a coarser field keep the same long-distance predictions? Integrating out short-scale variables changes the parameters of the remaining field. Matching a few long-wavelength quantities can preserve other low-energy predictions, with an error that grows when the coarse model omits important terms.

Keep every second field value and integrate out the sites between them. Match two long-wavelength modes, then test a third that was left out of the match.

Coarse spacing $1$

Matched mass squared $0.16$

Matched stiffness $1$

Held-out covariance error $0\%$

MATCH TWO MODES, TEST ANOTHER

How much of the spectrum survives?

Exact marginalDashed: local approximationGold point: held-out mode

Which modes determine the match?

The zero mode and the first nonzero mode determine the two parameters. The next mode tests the covariance prediction. The full exact marginal is retained in the exported trajectory.

What the twelve field samples estimate

Sample mean and one standard errorExact covarianceDashed: matched local covariance

Compare gold estimates with the exact covariance.

How the sample uncertainty is calculated

Each point averages Fourier-mode power from the same twelve retained configurations shown in the sample gallery. Whiskers use the sample variance across these independent configurations. Matching uses exact modes zero and one; the sampled points show finite-record uncertainty. That sampling uncertainty and the exact held-out truncation error above are different.

A SEPARATE INTERACTING THEORY

One direction approaches the Wilson–Fisher point

Thick cyan: tuned critical trajectoryWilson–Fisher fixed pointDrag the gold initial point

Drag gold to compare nearby flow trajectories.

Flow interpretation and critical exponents

The thick cyan curve numerically traces the stable direction of Tong’s one-loop equations (section 3.5, Figure 30), with arrows toward the purple point. Thin curves show nearby starting conditions; the small epsilon belongs to this separate model near four dimensions.

$\nu=\frac12+\frac{\epsilon}{12}+O(\epsilon^2)$ $\nu_{\mathrm{leading}}=0.508$

$\gamma=1+\frac{\epsilon}{6}+O(\epsilon^2)$ $\gamma_{\mathrm{leading}}=1.02$

Model equations, source reading and limits

The main model is a periodic one-dimensional Euclidean Gaussian field with an optional higher-derivative term. Samples are independent configurations, not a movie in physical time. A separate one-loop scalar flow explores dimensions close to four.

Lengths use a fixed reference unit; field amplitudes are scaled so that the Euclidean action is dimensionless. The 64-site periodic ring has positive mass and nonnegative bending stiffness. Exact elimination uses a Schur complement. A two-parameter nearest-neighbor action then matches the first two modes. With zero bending stiffness this one-dimensional Gaussian family closes exactly under decimation.

r=\mu^2/\Lambda^2,\quad u=9\widetilde g/(2\pi^2),\quad \frac{dr}{ds}=2r+\frac{u}{3(1+r)},\quad \frac{du}{ds}=\epsilon u-\frac{u^2}{(1+r)^2}

The separate flow keeps only the one-loop scalar mass and quartic coupling in dimensions slightly below four. The plot shows a small-coupling window. The code finds the fixed point of these truncated equations; exponent labels retain only first order in epsilon. No three-dimensional precision exponent or nonperturbative continuum limit is inferred.

Tong: chapter 3, section 3.5, printed pages 90–91 and Figure 30 (equations and figure read)

Wilson & Fisher, PRL 28, 240 (1972): the published abstract’s first-order susceptibility exponent (abstract read)

Wilson’s 1982 Nobel lecture: the physical meaning of changes of scale

Start with the essentials

Focus question: How far can two matched long-wavelength modes predict a third?
One-sentence intuition: Integrating out a variable preserves the probability distribution of the variables left behind; replacing the resulting action by a shorter formula introduces an approximation.

Core mathematical model

A positive Gaussian lattice action

S=\frac a2\sum_i\left[m^2\phi_i^2+\left(\frac{\phi_{i+1}-\phi_i}{a}\right)^2+\kappa\left(\frac{\phi_{i+1}-2\phi_i+\phi_{i-1}}{a^2}\right)^2\right]

The ring has 64 sites. Length and field amplitude use fixed reference units, and the Euclidean weight is the exponential of minus the action.

Integrating out the eliminated sites

Q_{\mathrm{eff}}=Q_{rr}-Q_{re}Q_{ee}^{-1}Q_{er},\qquad C_{rr}=Q_{\mathrm{eff}}^{-1}

The retained sites keep exactly their original covariance. The field values on the gold row are copied from the corresponding sites of the turquoise row.

A two-parameter local approximation

A_{\mathrm{local}}(q)=c_0+4c_1\sin^2(q/2),\qquad q_n=2\pi n/N_{\mathrm{coarse}}

The modes $n=0,1$ determine the two coefficients. The mode $n=2$ is left out and tests the resulting prediction.

Common difficulties

Decimation is not averaging

Typical misconceptionThe gold field is the average of each block.

Better mental modelThis experiment retains every second, fourth or eighth site. The probability of the retained values includes the integral over all eliminated values.

Two matching points do not validate a curve

Typical misconceptionThe fitted spectrum passes through two points, so the coarse model is correct.

Better mental modelThose points fixed its parameters. Check a third mode and then inspect the shorter-wavelength part of the spectrum.

Run the experiment

01
Keep the large features
Choose “Long waves survive” and drag across the field.
What to observe: The gold sites keep the same values as their turquoise counterparts. The fitted short formula stays closest at long wavelengths.
02
Find exact local closure
Choose “Exact local closure”.
What to observe: With no higher-derivative term, decimating this one-dimensional nearest-neighbor Gaussian chain gives another nearest-neighbor Gaussian chain. The held-out error falls to rounding precision.
03
Test the approximation
Choose “Too coarse for the prediction”, then compare the gold held-out point with the dashed spectrum.
What to observe: The third mode was not used for matching. Its covariance error measures a real consequence of truncating the effective action.