Absence of critical scaling in the Schelling segregation model

Mouhssine Rifaki

cond-mat.stat-mechAug 17, 2026

Absence of critical scaling in the Schelling segregation model

Authors: Mouhssine Rifaki

Organizations: Stanford University, Stanford, CA 94305, USA

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.

Figures & tables

Figure 1: Segregation index $S$ and interface density versus tolerance $T$ for $L=80$ , averaged over 50 trials with per-agent noise $\sigma=0.02$ . Shaded regions show $\pm 1$ standard deviation.

Figure 2: Equilibrium configurations on an $L=80$ grid for selected tolerance values.

Figure 3: Phase diagram in the $(T,\rho)$ plane. Color encodes the equilibrium segregation index. The mixed-segregated boundary is sharp across all densities.

$L$	$T_{c}$	Error
20	0.278	0.031
40	0.278	0.031
80	0.271	0.037
160	0.278	0.031
320	0.271	0.024

Table 1: Critical tolerance extracted independently for each system size. No systematic drift is detectable.

Figure 4: Left: Segregation index vs. tolerance for five system sizes. The curves steepen with $L$ but share the same midpoint. Right: Variance of $S$ at $T_{c}$ versus system size, with a power-law fit $\mathrm{Var}(S)\sim L^{-\alpha}$ , $\alpha=2.02\pm 0.09$ .

Figure 5: Left: Susceptibility $\chi(T)$ for five system sizes. Right: Peak susceptibility vs. $L$ . The scaling exponent $\gamma/\nu=-0.015\pm 0.086$ is indistinguishable from zero.

Figure 6: Binder cumulant $U_{4}$ vs. tolerance for five system sizes. The rapid convergence to $2/3$ and the non-universal crossing values indicate a crossover rather than a critical point.

Figure 7: Left: Best attempted collapse at $\nu=3.0$ . Right: Collapse quality (lower is better) as a function of $\nu$ . The absence of a minimum rules out a finite correlation-length exponent.

Figure 8: Power-law growth of the segregation index above $T_{c}$ : $\beta=0.306\pm 0.011$ .

Quantity	Schelling	2D Ising
$T_{c}(L)$ drift	none detected	$\sim L^{-1}$
$\mathrm{Var}(S)\sim L^{-\alpha}$	$2.02\pm 0.09$	$\gamma/\nu=1.75$
$\gamma/\nu$	$-0.015\pm 0.086$	1.75
$\beta$	$0.306\pm 0.011$	0.125
Collapse $\nu$	$>3$ (no minimum)	1.0

Table 2: Measured scaling exponents vs. 2D Ising values.

Figure 9: Relaxation dynamics for $T\in\{0.30,0.45,0.60\}$ . Left: agents moved per step. Right: mean satisfaction.

Figure 10: Dissimilarity profiles $D(r)$ at eight tolerance values. The dashed curve is the null model (random permutation of types). Higher $T$ produces profiles that decay more slowly, corresponding to larger segregated domains.

Figure 11: Statistics from the dissimilarity profiles: integrated area, decay slope, characteristic length $r^{*}$ , and local ( $r{=}1$ ) vs. global ( $r{=}12$ ) dissimilarity.

Figure 12: Interface density and segregation index vs. tolerance. The simultaneous transition in both quantities rules out a two-stage scenario.

$r_{0}$	$k$	$\|\mathcal{F}_{k}\|$	$T_{c}$
1	8	23	0.251
2	24	181	0.332
3	48	713	0.334
4	80	1 967	0.347
5	120	4 387	0.374
6	168	8 611	0.403

Table 3: Transition midpoint and satisfaction spectrum size for six neighborhood radii (

L=40

, 50 trials per tolerance value).

$L$	$\mathrm{Var}(S)$	per-octave slope
40	0.066	—
80	0.116	$+0.81$
160	0.012	$-3.27$
320	0.000 27	$-5.46$
4-point fit $\alpha(40,80,160,320)$		$\boldsymbol{-2.70}$

Table 4: Variance progression at

r_{0}=4

T=T_{c}=0.347

, with per-octave slopes and the cumulative power-law fit. The 4-point fit lies cleanly below the

\alpha=-2

critical boundary.

Figure 13: Dissolution of the apparent super-criticality at $r_{0}=4$ . Top: $\mathrm{Var}(S)$ at $T=T_{c}=0.347$ versus $L\in\{40,80,160,320\}$ , log-log. The dashed gray line is the $\alpha=-2$ critical boundary. Bottom: variance exponent $\alpha=d\log\mathrm{Var}(S)/d\log L$ as a function of $L_{\max}$ in the fit. The 2-point fit on $L\in\{40,80\}$ gives $\alpha=+0.81$ and was bootstrap-disjoint above zero, a misleading super-critical signal. Adding $L=160$ flips the sign ( $\alpha=-1.23$ ). Adding $L=320$ takes the fit deep into the sub-critical regime ( $\alpha=-2.70$ ). The trajectory rules out super-criticality at this radius.

$r_{0}$	$T_{c}(L=40)$	$T_{c}(L=80)$	$T_{c}(L=160)$	$\Delta T_{c}(80\to 160)$
4	0.362	0.348	0.341	$-0.007$
5	0.384	0.362	0.353	$-0.009$
6	0.404	0.380	0.367	$-0.013$

Table 5: Per-

L

T_{c}

at the

U_{4}=0.5

midpoint. Drift

L=80\to L=160

is unsaturated at every

r_{0}

Figure 14: Finite-size $T_{c}$ drift at the $U_{4}=0.5$ Binder midpoint across $L\in\{40,80,160\}$ for $r_{0}\in\{4,5,6\}$ . The drift is monotonic and unsaturated by $L=160$ , in direct contrast to the size-invariant fixed-point crossing that defines a second-order critical transition.

$r_{0}$	$k$	peak $\|dS/dT\|$
1	8	97.5
2	24	61.0
3	48	57.9
4	80	52.8
5	120	43.5
6	168	39.1

Table 6: Peak

|dS/dT|

L=40

versus neighborhood radius. Monotonic decrease, opposite to the sharpening expected at criticality.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 15: Segregation curves for heterogeneous tolerance with $\kappa\in\{2,5,20\}$ and the homogeneous baseline. Lower $\kappa$ (wider spread) shifts the transition leftward and broadens it.

Figure 16: Critical tolerance as a function of concentration $\kappa$ . The transition sharpens and $T_{c}$ increases toward the homogeneous limit as $\kappa\to\infty$ .

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