论文标题
自由边界条件对伊辛模型在高维度上的影响
The effect of free boundary conditions on the Ising model in high dimensions
论文作者
论文摘要
我们在$ \ Mathbb {z}^d $中使用$ d \ geq4 $的有限域中的自由边界条件研究了关键的ISING模型。在假设的情况下，到目前为止，仅对高$ d $完全证明，关键的无限体积两点函数是$ | x-y |^{ - （d-2）} $对于大$ | x-y | $，我们证明，只要$ x，y $不太接近边界，我们就会证明在具有自由边界条件的大有限有限立方体上相同。这证明了物理文献中的数值预测，表明具有免费边界条件的线性尺寸$ l $的有限域中的关键易感性为$ l^2 $ as $ l \ rightarrow \ rightarrow \ infty $。我们还证明，具有自由边界条件的近临界（小外部）ISING磁化场的缩放极限是高斯，其协方差与临界缩放极限相同，因此相关性不会呈指数衰减。这与低$ d $的情况或预期行为在高$ d $中的情况大不相同。
We study the critical Ising model with free boundary conditions on finite domains in $\mathbb{Z}^d$ with $d\geq4$. Under the assumption, so far only proved completely for high $d$, that the critical infinite volume two-point function is of order $|x-y|^{-(d-2)}$ for large $|x-y|$, we prove the same is valid on large finite cubes with free boundary conditions, as long as $x, y$ are not too close to the boundary. This confirms a numerical prediction in the physics literature by showing that the critical susceptibility in a finite domain of linear size $L$ with free boundary conditions is of order $L^2$ as $L\rightarrow\infty$. We also prove that the scaling limit of the near-critical (small external field) Ising magnetization field with free boundary conditions is Gaussian with the same covariance as the critical scaling limit, and thus the correlations do not decay exponentially. This is very different from the situation in low $d$ or the expected behavior in high $d$ with bulk boundary conditions.
