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 we prove that Pr(N ≤ (1 − δ)n 2 /4) ≤ exp(−Ω(n 2 )) and Pr(N ≥ (1 + δ)n 2 /4) ≤ exp(−Ω(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. Kwan was supported by NSF grant DMS-1953990. Sah and Sawhney were supported by NSF Graduate Research Fellowship Program DGE-1745302.1 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 1.2 Jacobson and Matthews [30] and Pittenger [48] designed Markov chains that converge to the uniform distribution, but it is not known whether these Markov chains mix rapidly.

Substructures in Latin squares

“…Theorem 1.3 closes the gap between lower and upper bounds recently proved by Kwan, Sah, and Sawhney in [31]. Actually, we are able to prove an even sharper large deviation inequality for intercalates in random Latin rectangles, in terms of a certain extremal function; see Theorem 2.2.…”

“…It is a classical fact that for all orders except 2 and 4 there exist Latin squares with no intercalates [28,29,41] (such Latin squares are said to have property "N 2 "). As our first result, we obtain the first nontrivial lower bound on the number of order-n Latin squares with this property (upper bounds have previously been proved in [31,40]).…”

“…So, Theorem 1.1 can be interpreted as the fact that a random order-n Latin square is intercalate-free with probability at least e −n 2 /4−o(n 2 ) . Resolving a conjecture of McKay and Wanless [40] (see also [9,34]), Kwan, Sah, and Sawhney [31] recently proved 4 that the expected number of intercalates in a random order-n Latin square is n 2 /4 + o(n 2 ), so writing N for the number of intercalates in a random order-n Latin square, Theorem 1.1 corresponds to the Poisson-type inequality…”

Note on random Latin squares and the triangle removal process

“…The purpose of this note is to record complete proofs of various lemmas about random Latin squares, which are analogues of the lemmas in [5]. In particular, these lemmas are ingredients in our recent paper on large deviations on random Latin squares [6].…”

“…We emphasise that the proofs in this note are almost exactly the same as the proofs of corresponding lemmas in [5]; the goal of this note is completeness, not new ideas. Also, we refer the reader to [5,6] for further references, motivation and background on this topic.…”