Large deviations in random Latin squares

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.
Comment: 15 pages