三维整数格点$\boldsymbol{\text{Z}^3}$的二聚体常数的一个新上界
A new upper bound on the dimer constant of $\mathbb{Z}^3$
浏览论文内容
中文总结 AI 辅助
该研究针对三维整数格点$\text{Z}^3$的二聚体常数，采用对角转移层、Csikvári不等式及压缩方法等，将其上界从0.457547优化至0.452130。
中文摘要 AI 辅助
我们证明三维立方格点的二聚体常数上界为0.452130，改进了Lundow（2001）利用Ciucu（1998）的转移矩阵方法得到的此前最优上界0.457547。我们的构造采用对角转移层和Csikvári（2017）的一个不等式，将所得矩阵的谱半径与二聚体常数关联，而非利用矩形环面特有的对称性；我们采用与Lundow类似的压缩方法降低转移矩阵的维度；最后，我们使用源自Friedland–Schneider（1980）渐近结果的Collatz–Wielandt公式变体，得到收敛到真实谱半径的单调上界序列，每个上界无需构造完整矩阵即可计算。
英文摘要
We prove that the dimer constant of the cubic lattice is bounded above by $0.452130$, improving the previous best upper bound of $0.457547$ obtained by Lundow (2001) using the transfer-matrix method of Ciucu (1998). Our construction instead uses diagonal transfer layers and an inequality of Csikvári (2017) to relate the spectral radius of the resulting matrix to the dimer constant, rather than exploiting symmetry special to the rectangular torus. We reduce the dimension of the transfer matrix using a compression method similar to Lundow's. Finally, we use a variant of the Collatz--Wielandt formula, derived from an asymptotic result of Friedland--Schneider (1980), to obtain a monotone sequence of upper bounds converging to the true spectral radius, each of which can be computed without forming the full matrix.
