Cube root fluctuations for the corner growth model associated to the exclusion process

促进排斥过程

Facilitated exclusion process

本研究针对一维Facilitated TASEP粒子系统，通过半象限最后通过渗流映射与Pfaffian Schur过程求解，证明其最右粒子位置服从GSE统计，揭示了带密度间断的排斥过程的立方根涨落特征及模型依赖的统计分布，并探讨了三类分布的二维交叉现象。

The $q$-Hahn asymmetric exclusion process

We introduce new integrable exclusion and zero-range processes on the one-dimensional lattice that generalize the $q$-Hahn TASEP and the $q$-Hahn Boson (zero-range) process introduced in [Pov13] and further studied in [Cor14], by allowing jumps in both directions. Owing to a Markov duality, we prove moment formulas for the locations of particles in the exclusion process. This leads to a Fredholm determinant formula that characterizes the distribution of the location of any particle. We show that the model-dependent constants that arise in the limit theorems predicted by the KPZ scaling theory …

Percolation, first-passage percolation, and covering times for Richardson's model on the n-cube

Percolation with edge-passage probability p and first-passage percolation are studied for the n-cube B_n ={0,1}^n with nearest neighbor edges. For oriented and unoriented percolation, p=e/n and p=1/n are the respective critical probabilities. For oriented first-passage percolation with i.i.d. edge-passage times having a density of 1 near the origin, the percolation time (time to reach the opposite corner of the cube) converges in probability to 1 as n->infty. This resolves a conjecture of David Aldous. When the edge-passage distribution is standard exponential, the (smaller) percolation time f…

Crystal Growth Inside an Octant

We study crystal growth inside an infinite octant on a cubic lattice. The growth proceeds through the deposition of elementary cubes into inner corners. After re-scaling by the characteristic size, the interface becomes progressively more deterministic in the long-time limit. Utilizing known results for the crystal growth inside a two-dimensional corner, we propose a hyperbolic partial differential equation for the evolution of the limiting shape. This equation is interpreted as a Hamilton-Jacobi equation which helps in finding an analytical solution. Simulations of the growth process are in e…

最后通行渗流中最可能的测地线

On the most likely geodesic in last passage percolation

本研究探讨最后通行渗流中最可能的测地线问题，确定三种利于极端角路径的机制，在指数最后通行渗流中证明角路径近乎典型，并推测测地线概率单调性在一般情况下成立。

Weakly asymmetric bridges and the KPZ equation

We consider the corner growth dynamics on discrete bridges from $(0,0)$ to $(2N,0)$, or equivalently, the weakly asymmetric simple exclusion process with $N$ particles on $2N$ sites. We take an asymmetry of order $N^{-α}$ with $α\in (0,1)$ and provide a complete description of the asymptotic behaviour of this model. In particular, we show that the hydrodynamic limit of the density of particles is given by the inviscid Burgers equation with zero-flux boundary condition. When the interface starts from the flat initial profile, we show that KPZ fluctuations occur whenever $α\in (0,1/3]$. In the p…