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For $\tau\in S_3$, let $\mu_n^{\tau}$ denote the uniformly random probabilitymeasure on the set of $\tau$-avoiding permutations in $S_n$. Let$\mathbb{N}^*=\mathbb{N}\cup\{\infty\}$ with an appropriate metric and denoteby  $S(\mathbb{N},\mathbb{N}^*)$ the compact metric space consisting of functions$\sigma=\{\sigma_i\}_{ i=1}^\infty$ from $\mathbb{N}$ to $\mathbb{N}^*$ whichare injections when restricted to $\sigma^{-1}(\mathbb{N})$\rm; that is, if$\sigma_i=\sigma_j$, $i\neq j$, then $\sigma_i=\infty$. Extending permutations$\sigma\in S_n$ by defining $\sigma_j=j$, for $j>n$, we have $S_n\subsetS(\mathbb{N},\mathbb{N}^*)$. For each $\tau\in S_3$, we study the limitingbehavior of the measures $\{\mu_n^{\tau}\}_{n=1}^\infty$ on$S(\mathbb{N},\mathbb{N}^*)$.

The Infinite Limit of Random Permutations Avoiding Patterns of Length Three