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Large deviations in random Latin squares
- Publication Year :
- 2021
-
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.<br />Comment: 15 pages
- Subjects :
- Mathematics - Combinatorics
Mathematics - Probability
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2106.11932
- Document Type :
- Working Paper