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 $$\sigma {\rm{ = }}\{ {\sigma _i}\} _{i = 1}^\infty {\rm{ }}$$ from $$\mathbb {N}$$ to $${\mathbb {N}^ * }$$ which are injections when restricted to $${\sigma ^{ - 1}}(\mathbb {N})$$; that is, if $${\sigma _i}{\rm{ = }}{\sigma _j}$$, $$i \ne j$$, then $${\sigma _i} = \infty $$. Extending permutations $$\sigma \in {S_n}$$ by defining $${\sigma _j} = j$$, for $$j \gt n$$, we have $${S_n} \subset S({\mathbb{N}},{{\mathbb{N}}^*})$$. For each $$\tau \in {S_3}$$, we study the limiting behaviour of the measures $$\{ \mu _n^\tau \} _{n = 1}^\infty $$ on $$S({\mathbb{N}},{\mathbb{N}^*})$$. We obtain partial results for the permutation $$\tau = 321$$ and complete results for the other five permutations $$\tau \in {S_3}$$.

中文翻译：

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避免长度为三的模式的随机排列的无限极限

为了$$\tau \in {S_3}$$， 让$$\亩_n^\tau $$表示集合上的均匀随机概率测度$$\tau $$- 避免排列$${S_n}$$. 让$${\mathbb {N}^*} = {\mathbb {N}} \cup \{ \infty \} $$具有适当的度量并表示为$$S({\mathbb{N}},{\mathbb{N}^*})$$由函数组成的紧度量空间$$\sigma {\rm{ = }}\{ {\sigma _i}\} _{i = 1}^\infty {\rm{ }}$$从$$\mathbb {N}$$到$${\mathbb {N}^ * }$$当被限制为注射时$${\sigma ^{ - 1}}(\mathbb {N})$$; 也就是说，如果$${\sigma _i}{\rm{ = }}{\sigma _j}$$,$$i \ne j$$， 然后$${\sigma _i} = \infty $$. 扩展排列$$\sigma \in {S_n}$$通过定义$${\sigma _j} = j$$， 为了$$j \gt n$$， 我们有$${S_n} \subset S({\mathbb{N}},{{\mathbb{N}}^*})$$. 对于每个$$\tau \in {S_3}$$，我们研究措施的限制行为$$\{ \mu _n^\tau \} _{n = 1}^\infty $$在$$S({\mathbb{N}},{\mathbb{N}^*})$$. 我们获得了置换的部分结果$$\tau = 321$$以及其他五个排列的完整结果$$\tau \in {S_3}$$.