22 - A dynamical-system picture of a simple branching-process phase transition

Summary

Abstract

This paper proves certain results from the ‘appetizer for non-linear Wiener–Hopf theory’ [5]. Like that paper, it considers only the simplest possible case in which the underlying Markov process is a two-state Markov chain. Key generating functions provide solutions of a simple two-dimensional dynamical system, and the main interest is in the way in which Probability Theory and ODE theory complement each other. No knowledge of either ODE theory or Wiener–Hopf theory is assumed. Theorem 1.1 describes one aspect of a phase transition which is more strikingly conveyed by Figures 4.1 and 4.2.

AMS subject classification (MSC2010) 60J80, 34A34

Introduction

This paper is a development of something I mentioned briefly in talks I gave at Bristol, when John Kingman was in the audience, and at the Waves conference in honour of John Toland at Bath. I thanked both John K and John T for splendid mathematics and for their wisdom and kindness.

The main point of the paper is to prove Theorem 1.1 and related results in a way which emphasizes connections with a simple dynamical system. The phase transition between Figures 4.1 and 4.2 looks more dramatic than the famous 1-dimensional result we teach to all students.

The model studied here is a special case of the model introduced in Williams [5]. I called that paper, which contained no proofs, an ‘appetizer’; but before writing a fuller version, I became caught up in Jonathan Warren's enthusiasm for the relevance of complex dynamical systems (in ℂ²).