For $τ\in S_3$, let $μ_n^τ$ denote the uniformly random probability measure on the set of $τ$-avoiding permutations in $S_n$. Let $\mathbb{N}^*=\mathbb{N}\cup\{\infty\}$ with an appropriate metric and denote by $S(\mathbb{N},\mathbb{N}^*)$ the compact metric space consisting of functions $σ=\{σ_i\}_{ i=1}^\infty$ from $\mathbb{N}$ to $\mathbb{N}^*$ which are injections when restricted to $σ^{-1}(\mathbb{N})$\rm; that is, if $σ_i=σ_j$, $i\neq j$, then $σ_i=\infty$. Extending permutations $σ\in S_n$ by defining $σ_j=j$, for $j>n$, we have $S_n\subset S(\mathbb{N},\mathbb{N}^*)$. For each $τ\in S_3$, we study the limiting behavior of the measures $\{μ_n^τ\}_{n=1}^\infty$ on $S(\mathbb{N},\mathbb{N}^*)$.