# 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.