Probability and Mathematical Physics Seminar

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Asymptotics of the Resistance of the Critical Series-Parallel Graph via Parabolic PDE Theory

Speaker: Peter Morfe, Max Planck Institute

Location: Warren Weaver Hall 1302

Date: Friday, October 10, 2025, 11:10 a.m.

Synopsis:

Hambly and Jordan (2004) introduced the series-parallel graph, a random hierarchical lattice that is easy to define: Start with the graph consisting of one edge connecting two terminal nodes.  At each subsequent step of the construction, perform independent coin flips for each edge of the graph, and replace the edge by two edges in series if the coin is heads-up or two edges in parallel if tails.  This results in a sequence of random graphs, which can be interpreted as a resistor network.  Hambly and Jordan showed that the logarithm of the resistance grows linearly if the coins are biased to land more often heads-up.  In this talk, I will discuss what happens in the critical case when fair coins are used.  Starting with a new recursive distributional equation (RDE) observed by Gurel-Gurevich, I develop a framework for analyzing RDE's based on parabolic PDE theory and use this to characterize the asymptotic behavior of the log. resistance.  In the sub- or supercritical case (where the coins are biased), I discuss a tantalizing connection to the Fisher-KPP equation and front propagation.