局部有限偏序集上的晶体生长
Crystal Growth on Locally Finite Partially Ordered Sets
AI总结：
研究了偏序集上的马尔可夫生长过程，基于独立指数分布权重的最后通过渗透模型，推导了生长时间的非渐近界及极限形状定理。
摘要：
我们考虑了一个偏序集Λ上的马尔可夫生长过程，等价于具有独立（不一定是相同）指数分布权重的最后通过渗透（LPP）模型。此类过程包括欧几里得晶格N_0^d上的非均匀指数LPP。我们给出了生长任意集合A⊆Λ的时间τ_A的均值、方差及更高矩的非渐近界，以A的特性为依据。我们还给出了当Λ配备独立法则结构时的极限形状定理。方法涉及利用马尔可夫演变的逆方程和相对于时间反转生成元的比较不等式。
英文摘要：
We consider a Markovian growth process on a partially ordered set $Λ$, equivalent to last passage percolation (LPP) with independent (not necessarily identical) exponentially distributed weights on the elements of $Λ$. Such a process includes inhomogeneous exponential LPP on the Euclidean lattice $\mathbb{N}_0^d$. We give non-asymptotic bounds on the mean and variance, as well as higher, central, and exponential moments of the passage time $τ_A$ to grow any set $A \subseteq Λ$ in terms of characteristics of $A$. We also give a limit shape theorem when $Λ$ is equipped with a monoid structure. Methods involve making use of the backward equation associated to the Markovian evolution and comparison inequalities with respect to the time-reversed generator.
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