# Student Probability Seminar

#### Interpretability of Fixed Points of Recursive Distributional Equations

Speaker: Moumanti Podder

Location: Warren Weaver Hall 905

Date: Monday, February 13, 2017, 11 a.m.

Synopsis:

Consider the property $$A$$ that there is a complete binary tree, as a subtree, starting at the root. We wish to find $$P_{\lambda}(A) = P[T_{\lambda} \text{ satisfies } A]$$, where $$T_{\lambda}$$ is the Galton-Watson process with $$Poisson(\lambda)$$ offspring distribution. This probability is given by a fixed point of the function (recursive distribution equation): $$\Psi(x) = 1 - e^{-\lambda x} (1 + \lambda x)$$. But for $$\lambda$$ bigger than a critical $$\lambda_{0}$$, this function has $$3$$ fixed points: the $$0$$ solution, the true probability $$P_{\lambda}(A)$$, and a third fixed point $$q_{\lambda}$$. We are interested in finding an interpretation for $$q_{\lambda}$$. In other words, we wish to find if there is any non-analytic reason why this fixed point appears.