不传递骰子内核：$\frac{\mathbf{1}_{x\ge y}-\mathbf{1}_{x\le y}}{4} - \frac{3(x-y)(1+xy)}{8}$
The intransitive dice kernel: $\frac{\mathbf{1}_{x\ge y}-\mathbf{1}_{x\le y}}{4} - \frac{3(x-y)(1+xy)}{8}$
摘要：
通过对Conrey，Gabbard，Grant，Liu和Morrison的一对问题的讨论，证明了从多集模型中抽取的三个骰子的概率为$1/4+o(1)$且不传递，并且对于一个明确定义的常数$α$，随机骰子对的概率趋向于$αn^{-1}$.这扩展和锐化Polymath关于平衡序列模型的最新结果。我们进一步表明，在这两个模型中，较大的比赛的分布收敛到一个普遍的tournamenton。这个极限自然产生于某个反对称算子的离散谱（由标题中作用于$L^2([-1,1])$的核给出）。该极限表现出一定程度的对称性，并可用于证明，例如，对于$1\le i\le 4$，$A_i$击败$A_{i+1}$，以及$A_5$击败$A_1$的极限概率是$1/32+o(1)$。此外，限制tournamenton具有包含在离散集合$\{0,1\}$中的范围。这证明了相关的tournamenton是非准随机的戏剧性的方式，极大地扩展了工作的Cornacchia和Hza关于连续模拟的平衡序列模型。证明是基于减少到条件中心极限定理（与Polymath的工作有关），使用“泊松化”风格的方法来减少到独立随机变量的计算，以及系统地使用基于切换的参数来在建立局部极限类型估计时提取傅立叶估计中的抵消。
英文摘要：
Answering a pair of questions of Conrey, Gabbard, Grant, Liu, and Morrison, we prove that a triplet of dice drawn from the multiset model are intransitive with probability $1/4+o(1)$ and the probability a random pair of dice tie tends toward $αn^{-1}$ for an explicitly defined constant $α$. This extends and sharpens the recent results of Polymath regarding the balanced sequence model. We further show the distribution of larger tournaments converges to a universal tournamenton in both models. This limit naturally arises from the discrete spectrum of a certain skew-symmetric operator (given by the kernel in the title acting on $L^2([-1,1])$). The limit exhibits a degree of symmetry and can be used to prove that, for instance, the limiting probability that $A_i$ beats $A_{i+1}$ for $1\le i\le 4$ and that $A_5$ beats $A_1$ is $1/32+o(1)$. Furthermore, the limiting tournamenton has range contained in the discrete set $\{0,1\}$. This proves that the associated tournamenton is non-quasirandom in a dramatic fashion, vastly extending work of Cornacchia and Hązła regarding the continuous analogue of the balanced sequence model. The proof is based on a reduction to conditional central limit theorems (related to work of Polymath), the use of a "Poissonization" style method to reduce to computations with independent random variables, and the systematic use of switching-based arguments to extract cancellation in Fourier estimates when establishing local limit-type estimates.
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