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.