Galton-Watson branching: the criticality threshold
Each individual produces $\mu$ children on average. Below $\mu = 1$: certain extinction. Above $\mu = 1$: positive survival probability. The boundary is everywhere in finance — default cascades, viral order flow, market-maker hedging chains all live on it.
A branching process tracks a population where each individual produces a random number of offspring with mean $\mu$. The classical fact, due to Galton and Watson, is that the behaviour of the population is governed entirely by whether $\mu$ is greater than, equal to, or less than $1$. This is the cleanest example in probability of a phase transition: a one-parameter family with two qualitatively different regimes joined by a critical boundary. Once you see the threshold, you start seeing it in default cascades (each default infects $\mu$ counterparties), market-maker inventory spirals (each trade triggers $\mu$ hedging trades), and information cascades.
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Worked example
An individual produces either $0$ children (with probability $1-p$) or $2$ children (with probability $p$); call this the Bernoulli$\{0, 2\}$ offspring distribution. For $p = 0.6$ (so $\mu = 1.2$), what is the probability that the population eventually goes extinct, starting from a single founding individual?
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Reflection
Default cascades in credit-risk are formally a Galton-Watson process where $\mu$ is the average number of counterparties one default infects. Why does this make $\mu = 1$ a *systemic-risk threshold* that supervisors care about, and what real-world quantities could move $\mu$ across that line?