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 \(\alpha n^{-1}\) for an explicitly defined constant \(\alpha \) . 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 cancellations in Fourier estimates when establishing local limit-type estimates.
