论文标题
单面树的局部限制
Local limits of one-sided trees
论文作者
论文摘要
高度$ h $的有限\ emph {单方面树}定义为通过将分支在一侧握把的分支（例如，脊柱的右侧），即长度$ h $始于根部的线性路径，从而使结果树在长度大于$ h $的根本上没有简单的路径，从而获得了根的平面树。我们考虑一组单方面树的分布$τ_n$ $ t $ n $，因此$ t $的重量与$ e^{ - μh（t）} $成正比，其中$μ$是一个真正的常数，$ h（t）$表示$ t $的高度。我们表明，对于$ n $，$τ_n$的限制较弱，作为在无限单方面树上支持的概率度量。极限度量$τ$对$μ$的依赖性显示出$μ_0= - \ ln 2 $从单个脊柱相位的过渡，以$μ\ leqμ_0$ $ us>μ_0$，以$μ\ leqμ_0$对多脊柱相位。相应地，在根周围的球的体积增长率是$μ<μ_0$的半径函数的函数，以$μ=μ_0$的二次增长，以及以$μ>μ_0$的Qubic Growth。
A finite \emph{one-sided tree} of height $h$ is defined as a rooted planar tree obtained by grafting branches on one side, say the right, of a spine, i.e. a linear path of length $h$ starting at the root, such that the resulting tree has no simple path starting at the root of length greater than $h$. We consider the distribution $τ_N$ on the set of one-sided trees $T$ of fixed size $N$, such that the weight of $T$ is proportional to $e^{-μh(T)}$, where $μ$ is a real constant and $h(T)$ denotes the height of $T$. We show that, for $N$ large, $τ_N$ has a weak limit as a probability measure supported on infinite one-sided trees. The dependence of the limit measure $τ$ on $μ$ shows a transition at $μ_0=-\ln 2$ from a single spine phase for $μ\leq μ_0$ to a multi-spine phase for $μ> μ_0$. Correspondingly, there is a transition in the volume growth rate of balls around the root as a function of radius from linear growth for $μ<μ_0$, to quadratic growth at $μ=μ_0$, and to qubic growth for $μ> μ_0$.
