AI 中文总结
本文证明无序XY模型中平均关联的下界比较，结合相存在定理得到临界阈值的Lipschitz正则性，并给出有限尺寸磁化率下界及噪声对代数相温度的二次抑制。
AI 中文摘要
我们证明了无序平均铁磁XY关联的一个下界比较：独立的非负随机耦合$K_e$可被替换为$-\frac12\log\mathbb{E}e^{-2K_e}$。Ginibre不等式使得每个归一化关联在$1-e^{-2K_e}$上是凸的，因此条件Jensen不等式和稀疏化可比较伯努利键密度；两次星型Jensen处理可覆盖二部图上的位点稀释。结合Dario和Garban的相存在定理，这些比较给出了$\mathbb{Z}^2$上超临界密度区间内平均代数相阈值和磁化率发散阈值的局部Lipschitz正则性。van Engelenburg和Lis的干净有限域判据还给出了弱稀释区域中显式的有限尺寸磁化率下界。对于有界中心化键噪声，代数相温度的可能抑制至多为噪声振幅的二次方。最后，在有限正阈值处，我们证明了有界独立同分布耦合律的全变差连续性，其本质 supremum 等于公共支撑上限，当参考律在该上限处有原子时，具有局部Lipschitz控制。
英文摘要
We prove a lower comparison for disorder-averaged ferromagnetic XY correlations: independent nonnegative random couplings $K_e$ can be replaced by $-\frac12\log\mathbb{E}e^{-2K_e}$. Ginibre's inequality makes each normalized correlation convex in $1-e^{-2K_e}$, so conditional Jensen and thinning compare Bernoulli bond densities; two star-wise Jensen passes handle site dilution on bipartite graphs. Together with the phase-existence theorem of Dario and Garban, these comparisons give local Lipschitz regularity of both the averaged algebraic-phase and susceptibility-divergence thresholds throughout the supercritical density intervals on $\mathbb{Z}^2$. The clean finite-domain criterion of van Engelenburg and Lis also yields explicit finite-size susceptibility lower bounds in a weak-dilution region. For bounded centered bond noise, the possible suppression of the algebraic-phase temperature is at most quadratic in the noise amplitude. Finally, at finite positive thresholds we prove total-variation continuity for bounded iid coupling laws whose essential supremum equals the common support cap, with local Lipschitz control when the reference law has an atom at that cap.
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