Growth models, random matrices and Painlevé transcendents
Growth models, random matrices and Painlevé transcendents
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增长模型、随机矩阵和 Painlevé 超越数
DOI：
10.1088/0951-7715/16/6/201
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发表时间：
2003
期刊：
影响因子：
--
通讯作者：
P. Forrester
中科院分区：
文献类型：
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作者：
P. Forrester
Hammersley 过程涉及从 (0, 0) 到 (1, 1) 的单位正方形内连接给定密度的随机点的所有向上/向右路径的最大长度的统计特性。这个过程也可以用多核生长模型的高度或随机排列中最长增加子序列的长度来解释。最长路径长度的累积分布可以用单一组的平均值来表示。 Hammersley 过程的各个版本（其中点被限制为具有一定的正方形对称性）允许类似的公式。回顾了这些公式的推导。将原始模型推广为沿正方形的两个边界具有点源，并适当缩放参数给出了 Kardar-Parisi-Zhang 普适性类中的模型。继 Baik 和 Rains、Prahofer 和 Spohn 的工作之后，我们回顾了缩放累积分布的计算，其中特定的 Painleve II 超越函数发挥了重要作用。
The Hammersley process relates to the statistical properties of the maximum length of all up/right paths connecting random points of a given density in the unit square from (0, 0) to (1, 1). This process can also be interpreted in terms of the height of the polynuclear growth model, or the length of the longest increasing subsequence in a random permutation. The cumulative distribution of the longest path length can be written in terms of an average over the unitary group. Versions of the Hammersley process in which the points are constrained to have certain symmetries of the square allow similar formulae. The derivation of these formulae is reviewed. Generalizing the original model to have point sources along two boundaries of the square, and appropriately scaling the parameters gives a model in the Kardar–Parisi–Zhang universality class. Following works of Baik and Rains, and Prahofer and Spohn, we review the calculation of the scaled cumulative distribution, in which a particular Painleve II transcendent plays a prominent role.