One-dimensional Poisson growth models with random and asymmetric growth
The following growth model is considered. There is a Poisson point process of germs on the line. Starting at time zero, a grain begins to grow from each germ point \(X_i\) at rates \(A_i\) to the left and \(B_i\) to the right where \(A_i, B_i\) are random variables. A grain stops growing when it touches another grain. Under various assumptions on the pairs \((A_i, B_i)\), several characteristics of this coverage process are obtained such as the fraction of the uncovered line, clumps' length distribution, etc. These results follow from similar theorems for growth processes on a circle. Additionally, simulation case study is performed.
scientific article; zbMATH DE number 1917926