  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.
