The Galton–Watson process is one of the most fundamental branching models, and branching is one of the most fundamental characteristics of life. This chapter looks closely at a measure with little coverage in the literature: the limiting conditional non-extinct mean of a subcritical Galton–Watson process. Put simply, if the process is still alive, what is its expected size? At late times that mean tends to a constant, 1/A.

We wanted a formula for A(p) in terms of p. Months of different approaches brought no luck, until eventually, with the help of ChatGPT, it was determined that no such closed form exists. In short, A(p) is defined by a limit on the survival probability, which satisfies a quadratic iterated map equivalent to the logistic map after a change of variables. So A(p) can be written in terms of the Koenigs linearisation function for the logistic map, and that function is provably hypertranscendental in the relevant parameter range: it cannot be expressed in finitely many terms of p.

On this page

Galton Watson branching and Yaglom's limit: the process and its limiting conditional law.
Logistic map classic and Julia set visualiser: the map behind A(p) and the fractals it draws.
Genetic algorithm and evolutionary function finders: searches for a closed form for A(p).