Large deviations in random Latin squares

TL;DR

This paper investigates the probability of large deviations in the number of intercalates within random Latin squares, establishing sharp bounds and confirming that typical Latin squares have approximately a quarter of their 2x2 subsquares as intercalates, resolving a longstanding conjecture.

Contribution

The paper provides sharp probabilistic bounds on large deviations of intercalates in random Latin squares, confirming the typical count and resolving an old conjecture.

Findings

Typical Latin squares have about n^2/4 intercalates.

Large deviation probabilities decay exponentially with n^2 and n^{4/3} log^{2/3} factors.

Confirmed the conjecture on the typical number of intercalates in 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.