无序伊辛模型中的边界自由能
Boundary Free Energies in Disordered Ising Models
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中文总结 AI 辅助
研究无序伊辛模型边界自由能，给出反例说明相关问题，证明特定条件下自由能差收敛，推导高斯边界耦合恒等式，证明缝翻转自由能差方差不等式，指出一维例子中相关问题未解决。
中文摘要 AI 辅助
无序伊辛模型自由能的边界校正取决于边界条件、归一化和有限体积序列。给出一维独立同分布反例，表明表面自由能密度不必独立于范霍夫序列且相应随机校正不必依概率收敛。对于\(d\geq2\)维立方盒上多布鲁申唯一性区域中的有界耦合，证明归一化自由到固定边界自由能差在期望、几乎必然和每个\(L^p\)（\(1\leq p<\infty\)）下收敛。对于高斯边界耦合，推导精确有限体积插值恒等式并解释其为何本身不意味着低温表面极限。对于跨方差为\(v\)的独立对称键集\(S_L\)的缝翻转自由能差\(D_L\)，证明\(\Var(D_L)\leq4v\abs{S_L}\)；一维例子表明仅对称性和有限矩不能确定刚度指数，低温高斯爱德华兹 - 安德森问题仍未解决。
英文摘要
The boundary correction to the free energy of a disordered Ising model depends on the boundary condition, the normalization, and the finite-volume sequence. We give a one-dimensional i.i.d. counterexample showing that the surface free-energy density need not be independent of the van Hove sequence and that the corresponding random correction need not converge in probability. For bounded couplings in the Dobrushin uniqueness regime on cubic boxes in dimensions $d\geq2$, we prove convergence of the normalized free-to-fixed boundary free-energy difference in expectation, almost surely, and in every $L^p$, $1\leq p<\infty$. For Gaussian boundary couplings, we derive an exact finite-volume interpolation identity and explain why it does not by itself imply a low-temperature surface limit. For a seam-flip free-energy difference $D_L$ across a set $S_L$ of independent symmetric bonds of variance $v$, we prove $\Var(D_L)\leq4v\abs{S_L}$; one-dimensional examples show that symmetry and finite moments alone do not determine a stiffness exponent, and the low-temperature Gaussian Edwards--Anderson problem remains open.
