We consider a discrete polynuclear growth (PNG) process and prove a functioal limit theorem for its convergrence to the Airy process. This generalizes previous results by Pr"ahofer and Spohn. The result enables us to express the GOE largest eigenvalue (Tracy-Widom) distribution in terms of the Airy process. We also show some results and give a conjecture about the transversal fluctuations in a point to line last passage percolation problem.
This article studies the inhomogeneous geometric polynuclear growth model, the distribution of which is related to Schur functions. We explain a method to derive its distribution functions in both space-like and time-like directions, focusing on the two-time distribution. Asymptotics of the two-time distribution in the KPZ-scaling limit is then considered, extending to two times several single-time distributions in the KPZ universality class.
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 sortin…
The $t$-PNG model introduced in Aggarwal, Borodin, and Wheeler (2021) is a deformed version of the polynuclear growth (PNG) model. In this paper, we prove the hydrodynamic limit of the model using soft techniques. One key element of the proof is the construction of a colored version of the $t$-PNG model, which allows us to apply the superadditive ergodic theorem and obtain the hydrodynamic limit, albeit without identifying the limiting constant. We then find this constant by proving a law of large numbers for the $α$-points, which generalizes Groeneboom (2001). Along the way, we construct the …
由Prähofer和Spohn引入的Airy过程A（t）是多核增长模型的极限平稳过程。Adler和van Moerbeke发现了变量s_1、s_2和t中的偏微分方程，其概率为A（0）<s_1和A（t）<s_2。利用这一点，假设某个猜想是正确的，并且具有适当的一致性，他们就能够得到这个概率在t->无穷大时的渐近展开的前几项，其中s_1和s_2是固定的。我们将证明，展开式可以通过使用概率的弗雷德霍姆行列式表示来获得。主要成分是作者和C. A. Tracy在推导Painlevé II表示的分布函数F_2加上其他几个以同样的方式获得。
Product approximations for uniform spanning trees and determinantal processes
HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）；Einstein Institute of Mathematics, Hebrew University of Jerusalem（耶路撒冷希伯来大学爱因斯坦数学研究所）
In this note we study a natural measure on plane partitions giving rise to a certain discrete-time Muttalib-Borodin process (MBP): each time-slice is a discrete version of a Muttalib-Borodin ensemble (MBE). The process is determinantal with explicit time-dependent correlation kernel. Moreover, in the $q \to 1$ limit, it converges to a continuous Jacobi-like MBP with Muttalib-Borodin marginals supported on the unit interval. This continuous process is also determinantal with explicit correlation kernel. We study its hard-edge scaling limit (around 0) to obtain a discrete-time-dependent generali…
