论文标题
对数时间尺度上各向异性增长的缩放限制
Scaling limits of anisotropic growth on logarithmic time-scales
论文作者
论文摘要
我们研究Hastings-Levitov模型AHL $（ν）$的各向异性版本。先前的结果表明，在有界的时间尺度上，群集边界上的谐波度量在天粒子极限以确定性的普通微分方程的溶液中收敛。我们考虑谐波测度在时间尺度上的演变，随着粒径收敛到零，对数生长，并表明在这个时间尺度上，谐波测量的领先顺序行为变为随机。具体而言，我们表明存在一个关键的对数时间窗口，其中谐波测量流程从不稳定的固定点开始，从不稳定点向稳定的固定点开始随机移动，并且我们表明可以以单个高斯随机变量来表征完整的轨迹。
We study the anisotropic version of the Hastings-Levitov model AHL$(ν)$. Previous results have shown that on bounded time-scales the harmonic measure on the boundary of the cluster converges, in the small-particle limit, to the solution of a deterministic ordinary differential equation. We consider the evolution of the harmonic measure on time-scales which grow logarithmically as the particle size converges to zero and show that, over this time-scale, the leading order behaviour of the harmonic measure becomes random. Specifically, we show that there exists a critical logarithmic time window in which the harmonic measure flow, started from the unstable fixed point, moves stochastically from the unstable point towards a stable fixed point, and we show that the full trajectory can be characterised in terms of a single Gaussian random variable.
