配分函数的无零区域对外场的依赖性
On the dependence of the zero-free region of a partition function on the external field
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中文总结 AI 辅助
该研究推导了布尔立方体上1-利普希茨函数线性组合的指数期望非零的条件，将其推广到±1自旋系统，得出多自旋相互作用能量线性增长时，外场对数增长即可维持配分函数无零区域的结论。
中文摘要 AI 辅助
设{0,1}^n为布尔立方体，赋予概率乘积测度，其中P(1)=p，P(0)=q且0 < p ≤ q=1-p。设φ_i: {0,1}^n→ℂ为汉明度量下的1-利普希茨函数，每个φ_i依赖于x∈{0,1}^n中至多r个坐标，且rp≥12。对j=1,…,n，令I_j为依赖于第j个坐标的索引i的集合。我们证明，当λ_i∈ℂ满足对所有j，∑_{i∈I_j}|λ_i| ≤ 1/(10√(rp))时，E exp{∑_{i=1}^m λ_i φ_i}≠0。这一结论可推广到±1自旋系统的情形：多自旋相互作用的能量线性增长时，仅需外场对数增长即可保持配分函数无零区域，使系统远离相变。
英文摘要
Let $\{0, 1\}^n$ be the Boolean cube, endowed with the probability product measure, where ${\Bbb P}(1)=p$ and ${\Bbb P}(0)=q$ with $0 < p \leq q$ and $p+q=1$. For $i=1, \ldots, m$, let $ϕ_i: \{0, 1\}^n \longrightarrow {\Bbb C}$ be $L_i$-Lipschitz functions in the Hamming metric, such that each $ϕ_i$ depends on at most $r$ coordinates of $x \in \{0, 1\}^n$, where $rp \geq 12$. For $j=1, \ldots, n$, let $I_j $ be the set of indices $i$ such that $ϕ_i$ depends on the $j$-th coordinate. We prove that $E \exp\left\{ \sum_{i=1}^m ϕ_i \right\} \ne 0$ provided $\sum_{i \in I_j} L_i \leq {1 \over 10 \sqrt{rp}}$ for all $j$. This translates into a regime for $\pm 1$ spin systems, where a linear increase in the energy of multi-spin interactions requires only a logarithmic increase of the external field to keep the partition function zero-free and the system away from the phase transition. As a corollary, we obtain efficient deterministic algorithms to approximate the partition function in the zero-free region.
