$(1+1)$维中的变形多核生长
Deformed Polynuclear Growth in $(1+1)$ Dimensions
摘要：
本文引入并研究了$(1+1)$维多核生长（PNG）的单参数形变，称之为$t$ -PNG模型.它的定义是，当两个扩张的岛合并时，它们有可能在合并位置的顶部产生另一个岛，即$t$。在$t=0$，这成为标准（非变形）PNG模型，在液滴几何形状中，可以通过均匀随机排列的最长递增序列或通过称为耐心排序的算法来重新制定。就后者而言，$t$ -PNG模型允许在排序算法中出现概率为$t$的错误。我们证明了$t$ -PNG模型对任意固定的$t\in [0,1)$都具有一点Tracy-Widom GUE渐近性，并且当$t$趋于$1$时，模型一点收敛于Kardar-Parisi-Zhang（KPZ）方程的窄楔形解.我们进一步构建了一个外部源，很可能会导致拜克-本Arous-Peche型相变的分布。证明是基于可解的随机顶点模型和它们的连接所产生的Schur措施分区的行列式点过程。
英文摘要：
We introduce and study a one parameter deformation of the polynuclear growth (PNG) in $(1+1)$-dimensions, which we call the $t$-PNG model. It is defined by requiring that, when two expanding islands merge, with probability $t$ they sprout another island on top of the merging location. At $t=0$, this becomes the standard (non-deformed) PNG model that, in the droplet geometry, can be reformulated through longest increasing subsequences of uniformly random permutations or through an algorithm known as patience sorting. In terms of the latter, the $t$-PNG model allows errors to occur in the sorting algorithm with probability $t$. We prove that the $t$-PNG model exhibits one-point Tracy-Widom GUE asymptotics at large times for any fixed $t\in [0,1)$, and one-point convergence to the narrow wedge solution of the Kardar-Parisi-Zhang (KPZ) equation as $t$ tends to $1$. We further construct distributions for an external source that are likely to induce Baik-Ben Arous-Peche type phase transitions. The proofs are based on solvable stochastic vertex models and their connection to the determinantal point processes arising from Schur measures on partitions.
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