FR01 · A field remembers its neighbors

Euclidean Field Correlations

How far do fluctuations of a free field remain correlated? A positive mass makes correlations fall with distance. A lighter field stays correlated farther across the lattice; the boundaries and the lattice spacing change the curve in ways that extra samples cannot remove.

Move two probes apart on a sampled scalar field. Compare the correlation estimated from many field configurations with the lattice prediction and its continuum limit.

Probe distance $1.5$

Sample correlation · one standard error $0.238\pm0.0229$

Effective samples · known sampler $1024$

Infinite-lattice correlation length $1.34$

What many samples tell us

Correlation falls with distance

Gold: samples with standard errors; cyan: exact lattice; violet: continuum.

$r/\ell_0\quad\longrightarrow\quad G(r)/\ell_0$

Samples can remember earlier samples

Gold: sample autocorrelation; cyan: sampler prediction.

$k\quad\longrightarrow\quad\rho^{2k}$

Units, uncertainty and model

The one-dimensional Euclidean model describes oscillator paths: a periodic interval gives finite-temperature paths, and fixed ends condition the endpoints. Replay advances the sample index; field height is enlarged automatically.

1024 field samples are loaded on entry.

Gold: sample mean and one-standard-error bars. Cyan: exact lattice covariance. Violet dashes: continuum covariance with the same boundary and length.

Gold: estimated autocorrelation of the selected two-point product. Cyan: the exact sampler prediction. The horizontal axis counts updates.

Lengths use an arbitrary reference length. The field has the square root of that length as its unit. The action and its sampling weight are dimensionless. In one Euclidean dimension this is also the imaginary-time path integral of a harmonic oscillator.

x_j=ja,\quad S_E=\frac{a}{2}\sum_j\left[\left(\frac{\phi_{j+1}-\phi_j}{a}\right)^2+m^2\phi_j^2\right],\qquad P(\phi)\propto e^{-S_E}

The ring has one bond per site. Fixed ends add two zero-valued boundary sites outside the sampled interior. The cyan anchor is at one quarter of the interior; the gold probe moves right from it.

\widehat G(r)=\frac1M\sum_{s=1}^M\phi_{j_0}^{(s)}\phi_{j_0+r/a}^{(s)},\quad\phi^{(s+1)}=\rho\phi^{(s)}+\sqrt{1-\rho^2}\,\eta^{(s)},\quad\eta^{(s)}\sim\mathcal N(0,A^{-1})

The initial field is already an exact Gaussian draw. For this sampler the centered two-point product has lag correlation equal to persistence squared per update. Error bars use that known correlation, including finite record length; they are not estimated by assuming independent samples.

F_M=1+2\sum_{k=1}^{M-1}\left(1-\frac{k}{M}\right)\rho^{2k},\quad M_{\mathrm{eff}}=\frac{M}{F_M},\quad\widehat{\mathrm{SE}}^2=\frac{s^2(M-1)}{M-F_M}\frac{F_M}{M}

Naive standard error $0.0229$

Variance inflation $1$

Domain length $12$

Nyquist cutoff $4\pi$

The model is free, massive and finite. Sampling noise, lattice-spacing effects and finite-volume effects are separate. The exported JSON contains every field sample, seed, boundary and model version. Real-time quantum measurements require a different description.

David Tong · Gaussian correlations, §2.2 · Ulli Wolff · Monte Carlo errors, §2

Start with the essentials

Focus question: How can we tell statistical noise from an effect of the lattice?
One-sentence intuition: Repeat the sampling to reduce noise; change the spacing and the volume separately to study their systematic effects.

Core mathematical model

A correlation estimated from saved fields

\widehat G(r)=\frac1M\sum_{s=1}^{M}\phi_{j_0}^{(s)}\phi_{j_0+r/a}^{(s)}

Each term multiplies the field at the fixed cyan point by the field at the moving gold point, in the same saved configuration.

The periodic lattice prediction

G_n=\frac1N\sum_{q=0}^{N-1}\frac{\cos(2\pi qn/N)}{am^2+(4/a)\sin^2(\pi q/N)}

The finite sum includes every mode allowed on the ring. Its highest wave number is $\pi/a$ .

The infinite-lattice decay length

\xi_a=\frac{a}{2\operatorname{arsinh}(am/2)}\longrightarrow\frac1m\quad(a\longrightarrow0)

The spacing changes the decay length. A finite ring also adds correlations that return around the other side.

Common difficulties

Zero average, nonzero product

Typical misconceptionIf both field averages are zero, their product must average to zero too.

Better mental modelThe two values are drawn together. If positive and negative excursions tend to occur together, their product has a positive average.

More samples from a persistent chain

Typical misconceptionA thousand saved fields always carry the information of a thousand independent fields.

Better mental modelNearby samples can share much of the same field. Their correlation increases the error of the mean; the effective sample count reports that loss for this sampler.

Run the experiment

01
Predict the effect of distance
Keep the default settings and drag the gold probe away from the cyan point.
What to observe: The sample correlation follows a decaying curve. The error bars grow more prominent where the correlation is weak.
02
Change the resolution at fixed length
Set the spacing to 0.5 and the sites to 24, then compare with spacing 0.25 and 48 sites.
What to observe: Both periodic rings have length 12. The change in the exact lattice curve isolates a spacing effect at fixed volume.
03
Give the sampler a memory
Choose Samples with a memory, then compare the effective sample count with the retained count in Sampling settings.
What to observe: The autocorrelation decays slowly. A longer record is needed to reach the uncertainty of independent sampling.