谢林隔离模型中不存在临界标度
Absence of critical scaling in the Schelling segregation model
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中文总结 AI 辅助
该研究通过大规模数值模拟和标度分析，证实谢林隔离模型（含摩尔邻域及密谱扩展）不存在临界标度，提出阶梯定理和级联机制解释其相变，排除临界现象的作用。
中文摘要 AI 辅助
我们在谢林隔离模型中未发现临界标度的证据，无论是在摩尔邻域还是其扩展到切比雪夫半径$r_0=6$（对应$k=168$个邻居）的密谱版本中。在周期网格$L=320$上，每个参数点进行50次试验（总运行次数超过12500次），摩尔基线的所有有限尺寸标度诊断均失效：随$L$变化的$T_c$无漂移，Var$(S) \backsim L^{-2.02 \text{±}0.09}$符合平凡平均，$\text{γ/ν} \text{≈}0$，且标度塌缩从未达到有限最优值。8位点摩尔邻域将满意度限制在$k \text{≤}8$的比例$j/k$，使$S(T)$呈现含23个有理阈值的阶梯结构；仅离散性本身并不排除临界性（对比伊辛模型），但标度证据从经验上排除了临界性。分支比计算预测亚临界级联的平均大小为$1/(1-R)$，并通过扰动实验验证，误差在15%以内；多标量相异长度在整个相变过程中保持有限。密谱扩展进一步强化了否定结论：在$r_0 \text{∈}\{3,4,5,6\}$、$L \text{∈}\{40,80,160\}$范围内，Binder累积量无$L$曲线交叉，随$L$变化的$T_c$漂移单调且未饱和；在$r_0=4$时，扩展到$L=320$得到$\text{α}=-2.70$，低于临界边界$\text{α}=-2$，消除了仅在$L \text{∈}\{40,80\}$上可见的表观$\text{α}=+0.81$信号。该机制是平衡态中缺乏长程关联加上确定性高$k$动力学，而非阶梯结构。在具有Beta分布异质容忍度的情况下，即使在中等总体平均容忍度下，不容忍的尾部也会驱动隔离。阶梯定理和级联机制共同解释了谢林相变，无需引入临界现象。
英文摘要
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.
