标题： 有向景观中测地线的分支长度与生长的空间结构种群中的突变模式 显示英文标题

标题： Branch lengths for geodesics in the directed landscape and mutation patterns in growing spatially structured populations

摘要： 考虑一个在二维空间中扩展的种群。 假设我们从整个种群或种群外边界附近的个体中随机抽取样本收集数据。 在群体遗传学中一个感兴趣的数量是位点频率谱，即出现在 $k$ 个被采样的 $n$ 个个体中的突变数，其中 $k = 1, \dots, n-1$。 只要突变率恒定，这个数量大致与所有基因树分支的总长度成正比，这些分支位于 $k$ 个被采样的个体的祖先线上。 尽管严格意义上的文献主要关注没有空间结构的模型，但在许多自然环境中，比如肿瘤或细菌菌落，生长受到空间限制的影响。 大量此类二维生长模型预计会落入 KPZ 普适类，表现出与 Kardar-Parisi-Zhang 方程类似的特性。 本文采用的观点是，对于属于 KPZ 普适类的群体模型，基因树可以通过由 \cite{dov22} 中构造的定向景观中的无限向上测地线形成的树来近似，这些测地线起始于 $n$ 个随机选择的点。 基于测地线合并，我们证明了关于这些测地线长度的新渐近结果，这些测地线是与从$n$中采样的点中的$k$个祖先相关的，并由此得到了驱动位点频率谱的指数，正如在\cite{fgkah16}中所预测的那样。 证明中的一个重要组成部分是对三个无限向上测地线在时间$t$内保持不相交的概率进行了新的紧密估计，即对著名的 N3G 问题的一个尖锐定量版本，这具有独立的兴趣。 显示英文摘要

摘要： Consider a population that is expanding in two-dimensional space. Suppose we collect data from a sample of individuals taken at random either from the entire population, or from near the outer boundary of the population. A quantity of interest in population genetics is the site frequency spectrum, which is the number of mutations that appear on $k$ of the $n$ sampled individuals, for $k = 1, \dots, n-1$. As long as the mutation rate is constant, this number will be roughly proportional to the total length of all branches in the genealogical tree that are on the ancestral line of $k$ sampled individuals. While the rigorous literature has primarily focused on models without any spatial structure, in many natural settings, such as tumors or bacteria colonies, growth is dictated by spatial constraints. A large number of such two dimensional growth models are expected to fall in the KPZ universality class exhibiting similar features as the Kardar-Parisi-Zhang equation. In this article we adopt the perspective that for population models in the KPZ universality class, the genealogical tree can be approximated by the tree formed by the infinite upward geodesics in the directed landscape, a universal scaling limit constructed in \cite{dov22}, starting from $n$ randomly chosen points. Relying on geodesic coalescence, we prove new asymptotic results for the lengths of the portions of these geodesics that are ancestral to $k$ of the $n$ sampled points and consequently obtain exponents driving the site frequency spectrum as predicted in \cite{fgkah16}. An important ingredient in the proof is a new tight estimate of the probability that three infinite upward geodesics stay disjoint up to time $t$, i.e., a sharp quantitative version of the well studied N3G problem, which is of independent interest.