Let $V$ be a random variable supported on the nonnegative integers (including $\infty$) and $f(x) = \mathbf E x^V$ be the probability generating function. In our model $V$ is the number of visits to the root of a graph by an interacting particle system. We prove $V = \infty$ a.s. by showing that $f$ satisfies a recurrence relation

$$f(x) =\frac{x+2}{3}f\Bigl(\frac{x+1}{2}\Bigr)^2 +\frac{x+1}{3}f\Bigl(\frac{x}{2}\Bigr)\biggl(1 -f\Bigl(\frac{x+1}{2}\Bigr)\biggr)$$

which through analytic methods we prove can only be satisfied when $f \equiv 0$ on $[0,1)$. Though our technique works we are somewhat baffled and are hoping to, in our upcoming paper, give some context by providing examples of this type of argument occurring in probability literature.

So, the question is are there other examples of proving a r.v. is a.s. infinite by proving the generating function is identically zero?