F003 · How big is big enough?

Finite-size Scaling

Can a beautiful data collapse give the wrong critical point? Small systems can hide a misplaced critical point because their curves change slowly over the fitted range. A larger size magnifies the same error. Keeping those larger samples aside gives the proposed scaling law a useful test.

Compare Gaussian fields on five lattice sizes. Try a convincing fit on the small lattices, then uncover the larger samples and see whether the prediction survives.

Small-size residual $1.00$

Held-out residual $1.00$

Continuum correlation length / interval $0.500$

Effective saved draws $3600$

45 recorded chains · five sizes

A prediction must work on a new size.

Small sizes used for comparisonLarger sizes held asideDashed: continuum reference

Before rescaling

\chi_L=\frac{a^2}{L}\sum_{i,j}\langle\phi_i\phi_j\rangle

After rescaling

x=(r-r_c)L^{1/\nu},\qquad y=\chi_L/L^2

Comparing recorded samples.

Adjust the proposed scaling law

Proposed critical point

r_c

0

Correlation-length exponent

\nu

0.5

The small sizes set an interpolated comparison curve. Larger sizes never enter that curve. The residual is an RMS difference in combined standard-error units; interpolation creates correlations, so it is a diagnostic rather than a significance test.

Sampling errors and raw records

Each chain stores 16,384 projections of the full field. Neighboring draws are correlated. The displayed uncertainty uses 128-draw batch means; the generator and tests also compare a doubled batch length. The model starts in its stationary Gaussian distribution.

The effective-draw readout uses the nearest saved mass for the selected size. The animated field is a separate live sample from the same model.

Saved scaled mass used for that readout: $rL^2=4$

$q=\frac{a}{\sqrt L}\sum_i\phi_i,\qquad\widehat\chi_L=\overline{q^2}$

15 sites · raw JSON 23 sites · raw JSON 31 sites · raw JSON 63 sites · raw JSON 127 sites · raw JSON

Model and scientific sources

The field has fixed zero endpoints and a quadratic Euclidean action. Successive samples follow a sampling algorithm, not physical time. This exact Gaussian benchmark has a correlation-length exponent of one half; interacting systems can have different exponents.

This is a one-dimensional free Euclidean scalar field with zero Dirichlet endpoints. The spacing is one reference length unit. The action is dimensionless. At fixed positive mass, correlation length is finite; the benchmark approaches its Gaussian critical limit from nonnegative mass squared. No interacting critical exponent is estimated here.

S=\frac12\sum_{j=0}^{N}\frac{(\phi_{j+1}-\phi_j)^2}{a}+\frac{ar}{2}\sum_{j=1}^{N}\phi_j^2,\quad L=(N+1)a

The dashed curve solves the continuum boundary-value problem. It is separate from the recorded lattice data. Finite spacing produces quadratic corrections in spacing divided by interval length.

\frac{\chi_L}{L^2}\longrightarrow\frac{1-2\tanh(\sqrt{x}/2)/\sqrt{x}}{x},\qquad x=rL^2

David Tong · Statistical Field Theory, sections 2.2–2.3

Start with the essentials

Focus question: How can small-size data collapse convincingly around a wrong critical point?
One-sentence intuition: A critical-point error grows when multiplied by the square of the interval length, so larger held-out sizes test a fit much more sharply.

Core mathematical model

Uniform-source response from fluctuations

q=\frac{a}{\sqrt L}\sum_i\phi_i,\qquad \chi_L=\langle q^2\rangle

The action is quadratic and the mean field is zero. The squared projection measures the same response that a weak uniform source would produce.

Gaussian finite-size scaling

\chi_L/L^2\longrightarrow F(rL^2),\qquad F(x)=\frac{1-2\tanh(\sqrt{x}/2)/\sqrt{x}}{x}

The continuum limit keeps $x=rL^2$ fixed as the spacing becomes small compared with the interval. At zero mass, $F(0)=1/12$ .

Common difficulties

A neat plot is one part of the test

Typical misconceptionIf the three small-size curves sit together, the chosen critical point must be right.

Better mental modelThe wrong critical point used in the preset barely shifts the small sizes. Its shift grows enough on the two larger sizes to separate their curves.

Successive samples share information

Typical misconceptionSixteen thousand stored draws give sixteen thousand independent measurements.

Better mental modelEach new field retains part of the previous field. The uncertainty uses batches longer than that memory, and the effective number of draws is smaller.

Run the experiment

01
Predict a collapse
Choose “One curve, five sizes” and compare the plots before and after rescaling.
What to observe: The unscaled response grows strongly with size. Rescaling brings all five sizes close to the continuum reference.
02
Make a tempting mistake
Choose “A tempting wrong fit”. Predict where the larger samples will lie.
What to observe: The small-size residual remains close to one even though the proposed critical point has shifted.
03
Use an independent size
Choose “Ask the larger lattices”. Compare the orange curves with your prediction.
What to observe: The held-out residual rises much more than the small-size residual. The larger interval exposes the wrong critical point.

Finite-size Scaling

A prediction must work on a new size.

Before rescaling

After rescaling

Test a scaling law on a size it has never seen

Core mathematical model

Uniform-source response from fluctuations

Gaussian finite-size scaling

Common difficulties

A neat plot is one part of the test

Successive samples share information

Run the experiment

Predict a collapse

Make a tempting mistake

Use an independent size

Related experiments and articles

From field modes to critical phenomena