Student Probability Seminar

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Voting models for branching random walks

Speaker: Xaver Kriechbaum, Weizmann Institute of Science

Location: Warren Weaver Hall 517

Date: Monday, October 30, 2023, 12:30 p.m.

Synopsis:

Reaction-diffusion equations take the form $\partial_t u = \Delta u + g(u)$ ($g$ is the "reaction" term). They have broad applications in the natural sciences, in major part due to their solutions taking the form of "travelling waves". The F-KPP equation $\partial_t u = \Delta u + u(1-u)$ is a particularly well-studied case because its solution can be described in terms of binary branching Brownian motion (BBM) [McKean '75]. Recently, it has been shown that for arbitrary polynomials $g$, the reaction-diffusion equation can be linked to a voting model on the genealogical tree of a BBM, though this probabilistic model has not yet yielded new PDE results.

Motivated by this, we'll talk about voting models for branching random walks. Instead of PDEs, these have a connection with recursion equations of the form $u_{n+1} = g(u_n\ast q)$. We will describe for which polynomials $g$ the $(u_n)_{n\in\mathbb{N}}$ can be represented via the voting models and use this connection to describe the geometry of the $(u_n)$: we will show that on a macroscopic level, $u_n$ has finitely many jumps and is otherwise flat, and describe a class of models for which the location of these jumps can be isolated to an interval of finite-in-$n$ length (which corresponds to tightness of a certain random variable).