Analysis of a class of recursive distributional equations including the resistance of the series-parallel graph

TL;DR

This paper studies recursive distributional equations related to the resistance of series-parallel graphs, deriving limit theorems and PDE scaling limits, especially for the critical bias parameter p=1/2.

Contribution

It introduces a PDE scaling limit for the RDEs and confirms a conjecture about the distributional limit of the resistance logarithm at p=1/2.

Findings

When p=1/2, N^{-1/3} log R^{(N)} converges to a nondegenerate limit.

Derived a PDE scaling limit for the distributional equations.

Confirmed the conjecture by Addario-Berry et al. regarding the resistance of series-parallel graphs.

Abstract

This paper analyzes a class of recursive distributional equations (RDE's) proposed by Gurel-Gurevich [17] and involving a bias parameter $p$, which includes the logarithm of the resistance of the series-parallel graph. A discrete-time evolution equation resembling a quasilinear Fisher-KPP equation is derived to describe the CDF's of solutions. When the bias parameter $p = \frac{1}{2}$, this equation is shown to have a PDE scaling limit, from which distributional limit theorems for the RDE are derived. Applied to the series-parallel graph, the results imply that $N^{-1/3} \log R^{(N)}$ has a nondegenerate limit when $p = \frac{1}{2}$, as conjectured by Addario-Berry, Cairns, Devroye, Kerriou, and Mitchell [1].