  For $\tau\in S_3$, let $\mu_n^{\tau}$ denote the uniformly random probability
measure on the set of $\tau$-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
$\sigma=\{\sigma_i\}_{ i=1}^\infty$ from $\mathbb{N}$ to $\mathbb{N}^*$ which
are 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 $S_n\subset
S(\mathbb{N},\mathbb{N}^*)$. For each $\tau\in S_3$, we study the limiting
behavior of the measures $\{\mu_n^{\tau}\}_{n=1}^\infty$ on
$S(\mathbb{N},\mathbb{N}^*)$.
