The intransitive dice kernel: $$\frac{\mathbbm {1}_{x\ge y}-\mathbbm {1}_{x\le y}}{4} - \frac{3(x-y)(1+xy)}{8}$$

Abstract

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 $$1/4+o(1)$$ 1 / 4 + o ( 1 ) and the probability a random pair of dice tie tends toward $$\alpha n^{-1}$$ α n - 1 for an explicitly defined constant $$\alpha $$ 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 $$L^2([-1,1])$$ L 2 ( [ - 1 , 1 ] ) ). The limit exhibits a degree of symmetry and can be used to prove that, for instance, the limiting probability that $$A_i$$ A i beats $$A_{i+1}$$ A i + 1 for $$1\le i\le 4$$ 1 ≤ i ≤ 4 and that $$A_5$$ A 5 beats $$A_1$$ A 1 is $$1/32+o(1)$$ 1 / 32 + o ( 1 ) . Furthermore, the limiting tournamenton has range contained in the discrete set $$\{0,1\}$$ { 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 cancellations in Fourier estimates when establishing local limit-type estimates.