Absence of critical scaling in the Schelling segregation model
2026-08-17 • Computer Science and Game Theory
Computer Science and Game TheoryMultiagent Systems
AI summaryⓘ
The authors studied the Schelling segregation model to see if it behaves like systems that have critical points, where small changes cause big effects. They tested this on different grid sizes and neighborhood ranges but found no signs of such critical scaling. Instead, their analysis suggests that segregation happens through predictable, small cascades rather than large-scale shifts. They also showed that differences in individuals’ tolerance contribute to segregation without needing complex critical behaviors. Overall, the authors conclude that the segregation observed in this model does not involve critical phenomena.
Schelling segregation modelcritical scalingMoore neighborhoodfinite-size scalingBinder cumulantcascade mechanismtolerance distributionlong-range correlationphase transitionIsing model
Authors
Sam Rifaki
Abstract
We find no evidence of critical scaling in the Schelling segregation model, in either the Moore neighborhood or its dense-spectrum extension to Chebyshev radii up to $r_0 = 6$ ($k = 168$ neighbors). On periodic grids up to $L = 320$ with 50 trials per point (> 12,500 runs), every finite-size scaling diagnostic in the Moore baseline fails: the per-$L$ $T_c$ does not drift, Var$(S) \sim L^{-2.02 \pm 0.09}$ matches trivial averaging, $γ/ν\approx 0$, and the scaling collapse never reaches a finite optimum. The 8-site Moore neighborhood restricts satisfaction to ratios $j/k$ with $k \leq 8$, giving $S(T)$ a staircase structure with 23 rational thresholds; discreteness alone does not forbid criticality (cf. the Ising model), but the scaling evidence rules it out empirically. A branching-ratio calculation predicts subcritical cascades of mean size $1/(1-R)$ and is validated by perturbation experiments to within 15%; the multiscalar dissimilarity length stays finite across the transition. The dense-spectrum extension strengthens the negative verdict: across $r_0 \in {3,4,5,6}$ on $L \in {40,80,160}$ the Binder cumulant has no $L$-curve crossing and the per-$L$ $T_c$ drift is monotonic and unsaturated; at $r_0 = 4$, extending to $L = 320$ gives $α= -2.70$, below the critical boundary $α= -2$, dissolving an apparent $α= +0.81$ signal visible only on $L \in {40,80}$. The mechanism is the absence of long-range correlation in equilibrium plus deterministic high-$k$ dynamics, not the staircase structure. With a Beta-distributed heterogeneous tolerance, the intolerant tail drives segregation even at moderate population-average tolerance. The staircase theorem and cascade mechanism together account for the Schelling transition without invoking critical phenomena.