论文标题
随机排列的大偏差原理
Large deviation principle for random permutations
论文作者
论文摘要
我们得出了由单位正方形（称为Permuton的概率度量）引起的随机排列的大偏差原理。这些排列称为$μ$ - 随机排列。我们还介绍并研究了一个新的随机排列模型的通用类别，即Gibbs置换模型，该模型结合并概括了$ $ $ random的排列和著名的挂号木匠模型。我们的大多数结果都在Gibbs置换模型的一般环境中。 我们将开发的工具应用于$μ$ - 随机排列的情况，条件具有非典型的模式。在特定的反转情况下，使几个结果变得更具体。例如，我们证明存在至少一个相变的绿色模型的通用版本，其中基本度量是不均匀的。这与Starr（2009，2018）在（标准）摩洛斯模型上的结果相反，在该模型中，没有相变（即相位唯一性）证明。 我们的结果自然会导致我们研究了一个新的定位概念，称为有条件恒定的定位物，该插入物概述了避开模式和模式包装的置换子。我们描述了有条件恒定置换子相对于反转的一些属性。对一般模式的有条件恒定定位的研究似乎是一个具有挑战性的问题。
We derive a large deviation principle for random permutations induced by probability measures of the unit square, called permutons. These permutations are called $μ$-random permutations. We also introduce and study a new general class of models of random permutations, called Gibbs permutation models, which combines and generalizes $μ$-random permutations and the celebrated Mallows model for permutations. Most of our results hold in the general setting of Gibbs permutation models. We apply the tools that we develop to the case of $μ$-random permutations conditioned to have an atypical proportion of patterns. Several results are made more concrete in the specific case of inversions. For instance, we prove the existence of at least one phase transition for a generalized version of the Mallows model where the base measure is non-uniform. This is in contrast with the results of Starr (2009, 2018) on the (standard) Mallows model, where the absence of phase transition, i.e., phase uniqueness, was proven. Our results naturally lead us to investigate a new notion of permutons, called conditionally constant permutons, which generalizes both pattern-avoiding and pattern-packing permutons. We describe some properties of conditionally constant permutons with respect to inversions. The study of conditionally constant permutons for general patterns seems to be a challenging problem.
