The intransitive dice kernel: $$\frac{\mathbbm {1}_{x\ge y}-\mathbbm {1}_{x\le y}}{4} - \frac{3(x-y)(1+xy)}{8}$$
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440_2024_Article_1270.pdf
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Author(s)
Sah, Ashwin
Sawhney, Mehtaab
Date Issued
March 30, 2024
Journal
Probability Theory and Related Fields
Publisher
Springer Science and Business Media LLC
Citation
Sah, Ashwin and Sawhney, Mehtaab. 2024. "The intransitive dice kernel: $$\frac{\mathbbm {1}_{x\ge y}-\mathbbm {1}_{x\le y}}{4} - \frac{3(x-y)(1+xy)}{8}$$." Probability Theory and Related Fields.
Version
Final published version
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
$$\alpha n^{-1}$$
α
n
-
1
for an explicitly defined constant
$$\alpha $$
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
$$L^2([-1,1])$$
L
2
(
[
-
1
,
1
]
)
). The limit exhibits a degree of symmetry and can be used to prove that, for instance, the limiting probability that
$$A_i$$
A
i
beats
$$A_{i+1}$$
A
i
+
1
for
$$1\le i\le 4$$
1
≤
i
≤
4
and that
$$A_5$$
A
5
beats
$$A_1$$
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.
Subjects
Statistics, Probability and Uncertainty
Statistics and Probability
Analysis
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s00440-024-01270-8
