The intransitive dice kernel: 1x ≥ y-1x ≤y/4-3(x-y) (1+xy)/8

• 成果类型: Article
• 署名作者: Sah, Ashwin; Sawhney, Mehtaab
• 署名单位: Massachusetts Institute of Technology (MIT)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01270-8
• 发表日期: 2024
• 页码: 1073-1128
• 关键词: logarithmic sobolev inequalities; Optimal transportation

摘要

Answering a pair of questions of Conrey, Gabbard, Grant, Liu, and Morrison, we prove that a triplet of dice drawn from the multiset modelare 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 <= i <= 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 Hazla 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