Dynamical pruning of rooted trees with applications to 1D ballistic annihilation

Abstract

We introduce generalized dynamical pruning on rooted binary trees with edge lengths. The pruning removes parts of a tree $T$, starting from the leaves, according to a pruning function defined on subtrees within $T$. The generalized pruning encompasses a number of discrete and continuous pruning operations, including the tree erasure and Horton pruning. The main result is invariance of a finite critical binary Galton-Watson tree with exponential edge lengths with respect to the generalized dynamical pruning for an arbitrary admissible pruning function. The second part of the paper examines the continuum 1-D ballistic annihilation model $A+A \rightarrow \emptyset$ for a constant particle density and initial velocity that alternates between the values of $\pm$1. The model evolution is equivalent to a generalized dynamical pruning of the shock tree that represents dynamics of sinks (points of particle annihilation), with the pruning function equal to the total tree length. The shock tree is isometric to the level set tree of the model potential (integral of velocity). This equivalence allows us to construct a complete probabilistic description of the annihilation dynamics for the initial velocity that alternates between the values of $\pm$1 at the epochs of a stationary Poisson process. Finally, we discuss several real tree representations of the ballistic annihilation model, closely connected to the shock wave tree.