Nishimori 点以下的垂直和再入铁磁边界
Vertical and reentrant ferromagnetic boundaries below the Nishimori point
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中文总结 AI 辅助
本文证明有偏 Ising 自旋玻璃在 Nishimori 点以下的铁磁边界可为垂直或再入，并给出 SK、p-自旋和晶格模型的严格结果。
中文摘要 AI 辅助
我们证明，在多临界（Nishimori）点以下，有偏 Ising 自旋玻璃的铁磁边界（Nishimori（1986）认为其恰好垂直）可以是垂直的，也可以是再入的，因此在固定无序下冷却会破坏有序。在具有 Curie-Weiss 偏置 $J_0$ 的 Sherrington-Kirkpatrick (SK) 模型中，该边界是垂直的；在全连接 $p$-自旋玻璃的三相点附近以及装饰平面 $\mathbb{Z}^2$ 周期晶格上（在某一无序强度下），该边界是再入的。对于每个 $T<J$，SK 磁化强度在 $J_0\le J$ 时为零，在 $J_0=J$ 处连续，在 $J_0>J$ 时为宏观的。对于每个整数 $p\ge3$，在每个略大于三相点偏置的偏置下，$p$-自旋磁化强度在 Nishimori 温度下是宏观的，并在一个明确给出的较低温度下消失；这是 Nishimori 最近以双温度形式给出的复制品预测。在晶格上（$\mathbb{Z}^2$ 的每条键替换为固定的串并联装置；最大度为四），具有独立同分布的 $\pm J$ 耦合，且 $\mathbb{P}(J_e=-1)=9/10000$，吉布斯态在高温下几乎必然唯一，在包含 Nishimori 温度的窗口内不唯一，并在低温下再次唯一。正如 Dey 和 Kang 所观察到的，SK 边界由零场磁化率以及场中自由能增益的包络界决定，后者可由规范不等式得出。我们证明该磁化率是零场 Parisi PDE 解在原点的曲率，Lopatto 通过其 Parisi 测度支撑集在原点累积的定理对每个 $\beta>1$ 进行了评估；我们给出该定理的一个简短替代证明。$p$-自旋和晶格结果以及替代证明是计算机辅助的。这些模型是 $\mathbb{Z}^2$ 上最近邻 $\pm J$ 模型的代理，对 $\mathbb{Z}^d$ 不提供任何信息。
英文摘要
We prove that below the multicritical (Nishimori) point the ferromagnetic boundary of a biased Ising spin glass, argued by Nishimori (1986) to be exactly vertical, can be vertical and can also be reentrant, so that cooling at fixed disorder destroys order. It is vertical in the Sherrington-Kirkpatrick (SK) model with a Curie-Weiss bias $J_0$, and reentrant near the triple point of fully connected $p$-spin glasses and, at one disorder strength, on a decorated planar $\mathbb{Z}^2$-periodic lattice. For every $T<J$ the SK magnetization vanishes for $J_0\le J$, continuously at $J_0=J$, and is macroscopic for $J_0>J$. For every integer $p\ge3$, at each bias slightly larger than the triple-point bias, the $p$-spin magnetization is macroscopic at the Nishimori temperature and vanishes at an explicit lower temperature; this is Nishimori's recent replica prediction in two-temperature form. On the lattice ($\mathbb{Z}^2$ with each bond replaced by a fixed series-parallel gadget; maximum degree four), with iid $\pm J$ couplings at $\mathbb{P}(J_e=-1)=9/10000$, the Gibbs state is almost surely unique at high temperature, non-unique in a window containing the Nishimori temperature, and unique again at low temperature. As Dey and Kang observed, the SK boundary is set by the zero-field susceptibility together with an envelope bound on the free-energy gain in a field, which follows from the gauge inequality. We prove that this susceptibility is the curvature at the origin of the zero-field Parisi PDE solution, which Lopatto evaluated for every $β>1$ via his theorem that the support of the Parisi measure accumulates at the origin; we give a short alternative proof of that theorem. The $p$-spin and lattice results and the alternative proof are computer-assisted. The models are proxies for the nearest-neighbour $\pm J$ model on $\mathbb{Z}^2$ and say nothing about $\mathbb{Z}^d$.
