The intransitive dice kernel: 1ₗ ₘ- {1ₗ ₘ}4 - 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 n^-1 α n - 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 L² (-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 Aᵢ A i beats A₈+₁ A i + 1 for 1 i 4 1 ≤ i ≤ 4 and that A₅ A 5 beats A₁ 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.