Large deviations in random latin squares

Abstract

In this note, we study large deviations of the number 𝐍 of intercalates (2 × 2 combinatorial subsquares which are themselves Latin squares) in a random 𝑛 × 𝑛 Latin square. In particular, for constant 𝛿 > 0 weAs 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. M S C ( 2 0 2 0 ) 05B15, 05C80, 60F10 (primary)1 † To see the analogy to permutation matrices, note that a Latin square can equivalently, and more symmetrically, be viewed as an 𝑛 × 𝑛 × 𝑛 zero-one array such that every axis-aligned line sums to exactly 1.

Transversals in quasirandom latin squares

“…However, using a recent result of Kwan, Sah, Sawhney, and Simkin [7] we are indeed able to establish that a random latin square is $$\mathcal {A}$$‐quasirandom with parameter $$o(1)$$, with high probability, and we can thus prove Theorem 1.1 as a consequence of Theorem 1.2. Theorem Let $$\mathsf {L}$$ be a uniformly random $$n \times n$$ latin square.…”

“…If $$\mathsf {L}$$ is the multiplication table of a group $$G$$ we compute the entire spectrum of $$\mathcal {A}$$ and find $$\rho = 1/D$$ where $$D$$ is the minimal dimension of a nontrivial representation of $$G$$, which shows that our notion of quasirandomness is equivalent to the usual one due to Gowers [5] in the case of groups. For genuinely random latin squares we use recent work of Kwan, Sah, Sawhney, and Simkin [7] to show that $$\operatorname{tr}\mathcal {A}^6 = 1 + o(1)$$ with high probability, and this implies that $$\rho = o(1)$$.…”

Cycles of quadratic Latin squares and antiperfect 1‐factorisations

“…However, this construction only gives a small number of $${N}_{2}$$ Latin squares, in comparison to the total number of $${N}_{2}$$ Latin squares. Kwan, Sah, Sawhney and Simkin [25] have used a probabilistic argument to show that there are at least $${({e}^{-9\unicode{x02215}4}n-o(n))}^{{n}^{2}}$$ Latin squares of order $$n$$ which are devoid of intercalates. Theorem 1.2 tells us that quadratic Latin squares of order $$q={p}^{d}$$ will not be useful for constructing perfect 1‐factorisations unless $$d$$ is small.…”

“…It is known [22, 23, 31, 37] that an $${N}_{2}$$ Latin square of order $$n$$ exists if and only if $$n\notin \{2,4\}$$. Such squares are also known to be rare [24, 30] and can be used to construct disjoint Steiner triple systems [22]. We completely characterise when a quadratic Latin square is $${N}_{2}$$.…”

Cycles of quadratic Latin squares and anti-perfect $1$-factorisations

“…It is known [20,29,21,35] that an N 2 Latin square of order n exists if and only if n ∈ {2, 4}. Such squares are also known to be rare [28,22] and can be used to construct disjoint Steiner triple systems [20]. We completely characterise when a quadratic Latin square is N 2 .…”