INTERACTIVE COMPANION · WINTER 2025 · STATISTICAL PHYSICS

The Percolation Bench

A live model accompanying “Measuring a Universal Constant on a Laptop”

One number holds this page together: 0.59274605. Fill the cells of a square grid at random and below that density you get islands. Above it you get a path from the top edge to the bottom. The constant has no closed form, so the only way anyone knows it is by measuring, and both models here measure rather than look it up.

The first bench is a lattice you can fill by hand. The second is the club's actual measurement procedure, running live in your browser, with the same arithmetic the article used.

Model 1. The lattice itself

Each cell is occupied independently with probability $p$. Occupied cells that share an edge belong to the same cluster, and the drawing colours clusters by size, palest for the smallest. If some cluster reaches from the top row to the bottom row, the lattice spans and that cluster is drawn in the darkest ink.

Drag the occupation slider slowly through the high fifties. There is no fanfare at the threshold, which is the honest lesson: at any single finite lattice size the transition is a smooth ramp, and the sharp constant only exists in the limit.

Occupation probability p 0.5927 Start where it sits, just under the accepted threshold, and nudge it up by one step at a time. Watch how often the spanning banner flickers on and off across the low 0.59s. At $L = 64$ the club measured a crossing probability of 0.5026 at exactly $p_c$, so a coin flip is the right expectation there. Lattice side L 64 Slide it from 16 up to 192 with $p$ held at 0.5925. The small lattice spans erratically; the large one makes up its mind. That narrowing is Figure 1 of the article, and it is the whole reason an extrapolation is possible. Cell shading Switch to “spanning cluster only” and then walk $p$ upward. Below threshold the panel is nearly blank, and the first time a spanning cluster appears it arrives already enormous rather than growing in from small. That abruptness is the phase transition. Redraw Same $p$, fresh randomness. Press it ten times at $p = 0.5925$ with $L = 64$ and count the spans. You are running the club's experiment by hand, at a sample size of ten.

occupied cells: – largest cluster: – as a fraction of the lattice: – spans top to bottom: –

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Model 2. The measurement run

This is the club's procedure, not a cartoon of it. Each trial is one Newman-Ziff sweep: shuffle all $L^2$ sites, add them one at a time with a union-find, and record the occupancy at which the lattice first spans. The crossing curve is then rebuilt from those records by exact binomial convolution,

$$R_L(p) \;=\; \sum_{n} \binom{N}{n} p^{n}(1-p)^{N-n}\,P(n_{\mathrm{span}} \le n),$$

and $p^*(L)$ is the value of $p$ where that curve crosses one half. The curve you see fills in as the trials accumulate, and the error bar on $p^*$ shrinks with it.

Underneath, the finite-size fit $p^*(L) = p_c + a L^{-1/\nu}$ runs on the club's published table. At the default settings it reproduces the article's headline exactly: $p_c = 0.5928171$, slope $a = -0.007456$, chi square 2.41 on 4 degrees of freedom.

Lattice size for the live run 64 Bigger lattices give a curve that stands closer to the true threshold but cost roughly $L^2$ per trial, so they accumulate trials far more slowly. That trade is the entire design problem of the study. Trials to run 2,000 Watch the curve rather than the number. The first fifty trials produce a staircase; by five hundred it is recognisably a sigmoid; after that only the error bar keeps improving, and it improves as $1/\sqrt{m}$, which is slow. Smallest lattice in the fit The one that matters. Including $L = 16$ gives 0.5930250 with a tighter error bar and a chi square of 18.04, which is the fit shouting that the model is wrong. Dropping it gives 0.5928171 and a chi square of 2.41. Exponent 1/ν used in the fit 0.750 0.75 is exact in two dimensions and is theory, not data. Drag it from 0.60 to 1.00 and the intercept moves only from 0.5928692 to 0.5927658, under one error bar, which is why the club could not measure $\nu$ but could measure $p_c$. Use the live run in the fit Leave it on “no” to see the article's exact numbers. Switch it to “yes” and your own noisy point replaces the club's at that size, so you can watch one under-sampled lattice drag the intercept around. Run control The run pauses itself when the tab goes to the background, so nothing is spent on a page nobody is looking at.

trials done: 0 p*(L) live: – fitted p_c: – vs accepted: –

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Club estimate: p_c = 0.5928171 ± 0.0001110
Accepted value: 0.59274605 (Jacobsen 2014)
Difference: +0.0000710, which is +0.64 of the club's error bar
Slope of the fit: a = −0.007456, chi square 2.41 on 4 d.o.f.
With L = 16 included: 0.5930250, a tighter bar and 2.92 sigma out
Total cost: 114,520 sweeps, 526.5 seconds, one laptop

What to try, and what it will teach you

Set the smallest lattice in the fit to $L \ge 16$. The intercept jumps to 0.5930250 and the quoted error bar gets smaller, because adding a point always tightens a least-squares intercept whether or not the point belongs there. The chi square jumps to 18.04. This is the single most useful thing on the page: a number that looks more precise and is further from the truth, with the fit itself telling you so if you bother to read it.

Then set the live lattice to 16 with 200 trials, switch the fit to use the live run, and watch what one badly sampled point does to a six-point fit. Then raise the trials to 6,000 and watch it settle back. The error bar you are watching is not decoration.

Last, drag the exponent. The intercept barely moves, and that is the honest limitation the article states in §9: this data set constrains the threshold and says almost nothing about the correlation-length exponent, because $p_c$ is an intercept and $\nu$ only reshapes the approach to it.