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

TL;DR

This paper investigates the intransitivity properties of random dice in specific models, revealing their limiting behavior, spectral structure, and non-quasirandom nature through advanced probabilistic and spectral analysis.

Contribution

It proves the intransitivity probability for triplets of dice, characterizes the limiting tournament structure via spectral methods, and demonstrates non-quasirandomness in the limit, extending prior work on balanced sequence models.

Findings

Triplet intransitivity probability approaches 1/4

Distribution of larger tournaments converges to a universal tournamenton

The tournamenton range is contained in {0,1}, indicating non-quasirandomness

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 $1/4+o(1)$ 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 H\k{a}z{\l}a 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.