The Infinite limit of random permutations avoiding patterns of length three

For τ ∈ S₃, let μ^τ_n denote the uniformly random probability measure on the set of τ-avoiding permutations in S_n. Let N^∗ = N ∪ {∞} with an appropriate metric and denote by S(N, N^∗) the compact metric space consisting of functions σ = {σ_i}^∞_i=₁ from N to N^∗ which are injections when restricted to σ⁻¹(N); that is, if σ_i = σ_j, i = j, then σ_i = ∞. Extending permutations σ ∈ S_n by defining σ_j = j, for j > n, we have S_n ⊂ S(N, N^∗). For each τ ∈ S₃, we study the limiting behaviour of the measures {μ^τ_n}^∞_n=₁ on S(N, N^∗).