Percolation model for the existence of a mitochondrial Eve

We look at the process of inheritance of mitochondrial DNA as a percolation model on trees equivalent to the Galton-Watson process. The model is exactly solvable for its percolation threshold pc and percolation probability critical exponent. In the approximation of small percolation probability, and assuming limited progeny number, we are also able to find the maximum and minimum percolation probabilities over all probability distributions for the progeny number constrained to a given pc. As a consequence, we can relate existence of a mitochondrial Eve to quantitative knowledge about demographic evolution of early mankind. In particular, we show that a mitochondrial Eve may exist even in an exponentially growing population, provided that the average number of children per individual is constrained to a small range depending on the probability p that a newborn child is a female.