Author:
Sah Ashwin,Sawhney Mehtaab
Abstract
AbstractAnswering 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.
Funder
Massachusetts Institute of Technology
Publisher
Springer Science and Business Media LLC