The intransitive dice kernel: 1x≥y-1x≤y4-3(x-y)(1+xy)8
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 α n - 1 for an explicitly defined constant α . This e...
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Veröffentlicht in: | Probability theory and related fields 2024, Vol.189 (3-4), p.1073-1128 |
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Sprache: | eng |
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Zusammenfassung: | 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
α
n
-
1
for an explicitly defined constant
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
]
)
). 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 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. |
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ISSN: | 0178-8051 1432-2064 |
DOI: | 10.1007/s00440-024-01270-8 |