The probability that a random triple of dice is transitive

TL;DR

The paper analyzes transitivity in the beating relation among random $n$-sided dice within the balanced sequences model. It reduces the problem to a two-dimensional local central limit theorem for a derived pair $(U,V)$ tied to a fixed die and proves a quantitative LCLT via Fourier analysis and concentration bounds, yielding that ties are rare and the conditional beat probability is asymptotically fair, i.e., $\frac{1}{2}+o(1)$. The approach establishes a discrete Gaussian approximation for sums of pivotal random variables and uses it to derive the main theorems, aligning with Conrey et al.'s conjectures in this model. The results illuminate how quasi-random tournament behavior emerges from random dice and provide a rigorous probabilistic-analytic framework for studying random tournaments arising from dice-with-weights models.

Abstract

An $n$-sided die is an $n$-tuple of positive integers. We say that a die $(a_1,\dots,a_n)$ beats a die $(b_1,\dots,b_n)$ if the number of pairs $(i,j)$ such that $a_i>b_j$ is greater than the number of pairs $(i,j)$ such that $a_i<b_j$. We show that for a natural model of random $n$-sided dice, if $A, B$ and $C$ are three random dice then the probability that $A$ beats $C$ given that $A$ beats $B$ and $B$ beats $C$ is approximately 1/2. In other words, the information that $A$ beats $B$ and $B$ beats $C$ has almost no effect on the probability that $A$ beats $C$. This proves a statement that was conjectured by Conrey, Gabbard, Grant, Liu and Morrison for a different model.

Theorems & Definitions (44)

• Conjecture 1.1
• Conjecture 1.2
• Conjecture 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Claim 3.1
• Lemma 5.1
• proof