置换的局部收敛性和一致$ρ$的局部极限--用$|ρ|=3$避免置换
Local convergence for permutations and local limits for uniform $ρ$-avoiding permutations with $|ρ|=3$
摘要：
我们为排列建立了一个新的局部收敛概念，并证明了\emph{consecutive}模式出现比例的特征。我们还描述了随机限制对象的这种新的拓扑结构引入了一个概念的“平移不变”属性（对应的概念的随机图的单模性）。然后，我们研究了两个模型的框架内的随机模式，避免排列。当置换的大小趋于无穷大时，我们计算了$|ρ|=3,$的一致$ρ$ -避免置换的局部极限。该论点的核心部分是描述任何给定模式的连续出现次数的渐近性。对于这个结果，我们使用双射之间的$ρ$ -避免置换和有根有序树，局部极限结果的高尔顿-沃森树，二阶矩方法和奇异性分析。
英文摘要：
We set up a new notion of local convergence for permutations and we prove a characterization in terms of proportions of \emph{consecutive} pattern occurrences. We also characterize random limiting objects for this new topology introducing a notion of "shift-invariant" property (corresponding to the notion of unimodularity for random graphs). We then study two models in the framework of random pattern-avoiding permutations. We compute the local limits of uniform $ρ$-avoiding permutations, for $|ρ|=3,$ when the size of the permutations tends to infinity. The core part of the argument is the description of the asymptotics of the number of consecutive occurrences of any given pattern. For this result we use bijections between $ρ$-avoiding permutations and rooted ordered trees, local limit results for Galton--Watson trees, the Second moment method and singularity analysis.
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