运动种群内波前传播速度的一个上界
An upper bound for front propagation velocities inside moving populations
摘要：
我们考虑两种类型的（红色和蓝色或$R$和$B$）粒子的人口发展的$d$维晶格根据一些反应-扩散过程$R+B\to 2R$，并开始于一个单一的红色粒子和密度$ρ$的蓝色粒子。对于两类模型，我们给出了一个上界的传播速度的红色粒子的正面与$ρ$显式依赖。在第一类模型中，红色和蓝色粒子分别以扩散常数$D_R=1$和可能随时间变化的跳跃率$D_B \geq 0$演化--更一般地说，蓝色粒子遵循一些独立的双随机过程，这也包括具有漂移和各种确定性过程的长程随机游走。然后，我们在所有维度上得到一个$\max(ρ,\sqrtρ)$阶的上界，它只依赖于$ρ$和$d$，而不依赖于蓝色粒子所遵循的特定过程，特别是不依赖于$D_B$。我们认为，对于$d \geq 2$或$ρ\geq 1$，这个界限可以是最佳的（在$ρ$中），而对于最简单的情况下，$d=1$和$ρ< 1$被称为青蛙模型，我们给出了一个更好的界阶$ρ$。在第二类模型中，粒子在一个具有周期性边界条件的大二维盒子内以排斥和可能的吸引力演化，根据川崎动力学（当吸引力设置为零时，这变成简单的排斥）。在低密度区域，我们得到了$\sqrtρ$阶的上界。这证明了在这种低密度状态下动力学事件的长程去相关。
英文摘要：
We consider a two type (red and blue or $R$ and $B$) particle population that evolves on the $d$-dimensional lattice according to some reaction-diffusion process $R+B\to 2R$ and starts with a single red particle and a density $ρ$ of blue particles. For two classes of models we give an upper bound on the propagation velocity of the red particles front with explicit dependence on $ρ$. In the first class of models red and blue particles respectively evolve with a diffusion constant $D_R=1$ and a possibly time dependent jump rate $D_B \geq 0$ -- more generally blue particles follow some independent bistochastic process and this also includes long range random walks with drift and various deterministic processes. We then get in all dimensions an upper bound of order $\max(ρ,\sqrtρ)$ that depends only on $ρ$ and $d$ and not on the specific process followed by blue particles, in particular that does not depend on $D_B$. We argue that for $d \geq 2$ or $ρ\geq 1$ this bound can be optimal (in $ρ$), while for the simplest case with $d=1$ and $ρ< 1$ known as the frog model, we give a better bound of order $ρ$. In the second class of models particles evolve with exclusion and possibly attraction inside a large two-dimensional box with periodic boundary conditions according to Kawasaki dynamics (that turns into simple exclusion when the attraction is set to zero.) In a low density regime we then get an upper bound of order $\sqrtρ$. This proves a long-range decorrelation of dynamical events in this low density regime.
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