AI 中文总结
研究维度 \(d = 1\)、\(2\)、\(3\) 中狄利克雷随机环境下随机游走，通过数值研究其与 KPZ 增长的关系，验证 \(d = 1\) 和 \(d = 2\) 时的兼容性，确认 \(d = 3\) 时的相变并获下界，发现不同无序相特点及可计算样本间方差。
AI 中文摘要
时间依赖随机环境中随机游走的非典型行为最近与 Kardar-Parisi-Zhang (KPZ) 增长相关。虽然在空间维度 \(d = 1\) 时已被充分理解，但在 \(d>1\) 维度中还需进一步研究。本文针对 \(d = 1\)、\(2\) 和 \(3\) 对该问题进行数值研究，聚焦具有狄利克雷分布转移概率的离散模型。验证了 \(d = 1\) 和 \(d = 2\) 时点对点概率对数方差的增长与 KPZ 增长兼容。在 \(d = 3\) 时，确认了随着 \(\vert x\vert /t\) 增加存在相变并基于精确二阶矩计算得到下界。发现弱无序相中点对点概率具有重尾分布，强无序相中其对数累积量随时间增长。还表明可精确计算热平均 \(\overline{ \langle x \rangle^2}\) 的样本间方差且与最近引入的极端扩散系数有关。
英文摘要
The atypical behaviour of random walks in time-dependent random environment was recently related to Kardar-Parisi-Zhang (KPZ) growth. While this is now well-understood in spatial dimension $d=1$, further efforts are necessary to better understand these connections in dimensions $d>1$. In this paper, we study this problem numerically for $d=1, 2$ and $3$, focusing on a discrete model with Dirichlet distributed transition probabilities. This model is a generalization of an integrable model in $d=1$, and it has the advantage of admitting an explicit, product-form, stationary measure. We verify that the growth of the variance of the logarithm of point-to-point probabilities, namely from the origin to position $x$ in time $t$, is compatible with KPZ growth in dimension $d=1$ and $d=2$. In spatial dimension $d=3$, we confirm the existence of a phase transition as the angle $\vert x\vert /t$ increases and we obtain a lower bound based on an exact second moment calculation. We find that in the weak disorder phase the point-to-point probability acquires a heavy tailed distribution, and that in the strong disorder phase the cumulants of its logarithm grow with time. Further, we show that for this model, we can compute exactly the sample to sample variance of the thermal average $\overline{ \langle x \rangle^2}$ and that it is related to the extreme diffusion coefficient introduced recently.
Comments27 pages, 13 figures. Data gnuplot files for plots available at https://dare.uol.de/dataset.xhtml?persistentId=doi:10.57782/G0HPXH
