The big idea. Whether someone votes isn't an independent coin-flip. This model — from a 2012 physics-of-society paper by Borghesi, Raynal & Bouchaud — treats each person's decision as nudged by an invisible “cultural field”: a slowly-shifting local mood about civic participation that spreads between neighbouring towns like heat diffusing through a material. A town's turnout = a personal/idiosyncratic part + that town's own quirks + this shared regional field.
Why the “log-turnout rate” (τ). Raw turnout is awkward because it's trapped between 0 and 100%. Taking τ = ln(p/(1−p)) unfolds it into a clean, comparable, bell-shaped quantity — the natural variable for the maths.
What the distribution P(u) represents. Strip out each election's overall level and spread, and the shape that remains is remarkably the same election after election, decade after decade — a stable civic “fingerprint” of a country. Its asymmetry is diagnostic: France leans positive (a tail of unusually high-turnout towns), Italy leans negative — a real, reproducible difference in political culture, not noise.
The headline result — turnout is spatially correlated, logarithmically. Take any two towns: how alike their turnout is fades with distance not sharply, but as the logarithm of distance — astonishingly slowly. Towns 100 km apart are still visibly linked. That gentle, long-range coupling is exactly what a diffusing field produces, and it's the strongest evidence that voting habits propagate across geography rather than being purely local. The dashboard's C(r) curve and the map are two views of this same fact.
Because both the shape P(u) and the spatial coupling are so stable over time, they act as a baseline “normal.” For a new election you can: (1) anticipate the geographic pattern of turnout from the persistent field — last election's map is a strong prior for the next; (2) flag anomalies — towns or regions that depart from the stable pattern by more than history allows, which is what makes this a candidate screen for data errors or irregularities; and (3) summarise a whole election in a few structural numbers (level, spread, field strength, heterogeneity) and watch them drift election-to-election in the Time-series tab. In short: it turns “turnout” from one national percentage into a predictable, spatially-structured field.
Scope note: second-round and local elections only cover places where a runoff was held, so their maps are partial by design (noted under the map). Overseas territories are excluded, following the paper.
Each commune coloured by its size-detrended log-turnout rate. Long-range, spatially correlated patterns (paper Fig. 6).
u = (τ − ⟨τ⟩)/σ across communes vs a standard Gaussian. Non-Gaussian, positively skewed (paper Fig. 1).
Conditional mean and s.d. of τ vs commune size N (registered voters), binned (paper Figs. 2, 5).
⟨τ′(R+r)τ′(R)⟩ vs distance, from an exact all-pairs estimator (smooth at short range). Logarithmic decay (dashed fit) + diffusive-field model rescaled to the data over 5–150 km (R²≈0.98; below ~4 km, commune spacing limits the comparison).
Extrapolating the size-binned variance (minus the binomial term) to N→∞ isolates the persistent cultural-field variance (paper Fig. 3, Eq. 13); compared with the C(r→0⁺) plateau.
Rebuilt numbers vs the values reported by Borghesi–Raynal–Bouchaud (2012).