Large Deviations and Ratio Limit Theorems for Pattern-Avoiding Permutations

For a fixed permutation τ, let$\mathcal{S}_N(\tau)$be the set of permutations onNelements that avoid the pattern τ. Madras and Liu (2010) conjectured that$\lim_{N\rightarrow\infty}\frac{|\mathcal{S}_{N+1}(\tau)|}{ |\mathcal{S}_N(\tau)|}$exists; if it does, it must equal the Stanley–Wilf limit. We prove the conjecture for every permutation τ of length 5 or less, as well as for some longer cases (including 704 of the 720 permutations of length 6). We also consider permutations drawn at random from$\mathcal{S}_N(\tau)$, and we investigate properties of their graphs (viewing permutations as functions on {1,. . .,N}) scaled down to the unit square [0,1]2. We prove exact large deviation results for these graphs when τ has length 3; it follows, for example, that it is exponentially unlikely for a random 312-avoiding permutation to have points above the diagonal strip |y−x| < ε, but not unlikely to have points below the strip. For general τ, we show that some neighbourhood of the upper left corner of [0,1]2is exponentially unlikely to contain a point of the graph if and only if τ starts with its largest element. For patterns such as τ=4231 we establish that this neighbourhood can be extended along the sides of [0,1]2to come arbitrarily close to the corner points (0,0) and (1,1), as simulations had suggested.