Large deviations in random Latin squares
Abstract
In this note, we study large deviations of the number $\mathbf{N}$ of intercalates ($2\times2$ combinatorial subsquares which are themselves Latin squares) in a random $n\times n$ Latin square. In particular, for constant $\delta>0$ we prove that $\Pr(\mathbf{N}\le(1-\delta)n^{2}/4)\le\exp(-\Omega(n^{2}))$ and $\Pr(\mathbf{N}\ge(1+\delta)n^{2}/4)\le\exp(-\Omega(n^{4/3}(\log n)^{2/3}))$, both of which are sharp up to logarithmic factors in their exponents. As a consequence, we deduce that a typical order-$n$ Latin square has $(1+o(1))n^{2}/4$ intercalates, matching a lower bound due to Kwan and Sudakov and resolving an old conjecture of McKay and Wanless.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2021
- DOI:
- 10.48550/arXiv.2106.11932
- arXiv:
- arXiv:2106.11932
- Bibcode:
- 2021arXiv210611932K
- Keywords:
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- Mathematics - Combinatorics;
- Mathematics - Probability
- E-Print:
- 15 pages