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

Peter S. Morfe

TL;DR

The paper develops a unified PDE-based framework to analyze a broad class of recursive distributional equations (RDEs), including the log-resistance of the series-parallel graph, by tracking the evolution of CDFs through a nonlinear, nonlocal operator. At the critical bias $p=\tfrac{1}{2}$, it proves a rigorous scaling limit: the CDFs converge to viscosity solutions of either a Burgers-type equation $\partial_t F - \sigma|\partial_x F|^2=0$ (when $\sigma\neq0$) or a porous-medium-type equation $\partial_t F - a|\partial_x F|\partial_{xx}^2 F=0$ (when $\sigma=0$), with the corresponding $X^{(n)}$-limits linked to Beta distributions (Beta$(2,1)$ or Beta$(2,2)$, depending on the regime). By applying an extended Barles–Souganidis approach to monotone semigroups in the space of CDFs, the authors derive distributional limit theorems for several RDEs, including the series-parallel graph resistance (confirming the conjecture of Addario-Berry et al.) and related models such as min-plus binary trees, hipster random walks, and cooperative motion. The work elegantly connects RDE asymptotics to nonlinear PDE scaling limits and demonstrates how monotonicity and consistency drive convergence to well-known Brownian- or Barenblatt-type limiting distributions. Overall, the results give a rigorous, PDE-based pathway to understand distributional limits for a broad family of RDEs and their manifestations in random hierarchical structures.

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 nonlinear, fractional 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].

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

TL;DR

The paper develops a unified PDE-based framework to analyze a broad class of recursive distributional equations (RDEs), including the log-resistance of the series-parallel graph, by tracking the evolution of CDFs through a nonlinear, nonlocal operator. At the critical bias , it proves a rigorous scaling limit: the CDFs converge to viscosity solutions of either a Burgers-type equation (when ) or a porous-medium-type equation (when ), with the corresponding -limits linked to Beta distributions (Beta or Beta, depending on the regime). By applying an extended Barles–Souganidis approach to monotone semigroups in the space of CDFs, the authors derive distributional limit theorems for several RDEs, including the series-parallel graph resistance (confirming the conjecture of Addario-Berry et al.) and related models such as min-plus binary trees, hipster random walks, and cooperative motion. The work elegantly connects RDE asymptotics to nonlinear PDE scaling limits and demonstrates how monotonicity and consistency drive convergence to well-known Brownian- or Barenblatt-type limiting distributions. Overall, the results give a rigorous, PDE-based pathway to understand distributional limits for a broad family of RDEs and their manifestations in random hierarchical structures.

Abstract

This paper analyzes a class of recursive distributional equations (RDE's) proposed by Gurel-Gurevich [17] and involving a bias parameter , which includes the logarithm of the resistance of the series-parallel graph. A discrete-time evolution equation resembling a nonlinear, fractional Fisher-KPP equation is derived to describe the CDF's of solutions. When the bias parameter , 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 has a nondegenerate limit when , as conjectured by Addario-Berry, Cairns, Devroye, Kerriou, and Mitchell [1].

Paper Structure

This paper contains 40 sections, 23 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Resistance of the Series-Parallel Graph
  3. General Setting
  4. Evolution of the CDF
  5. Assumptions
  6. Main Result
  7. Idea of Proof
  8. Variants of the Fisher-KPP Equation when $p \neq \frac{1}{2}$
  9. Related Literature
  10. Organization of the Paper
  11. Preliminaries
  12. Extended-Real-Valued Random Variables
  13. Symmetry
  14. Monotonicity
  15. Continuity
  16. ...and 25 more sections

Key Result

Theorem 1

Assume that $p = \frac{1}{2}$, $\mathbf{P}\{ f_{+},f_{-} \in \mathcal{S} \} = 1$, and E: decay holds. If $\{ F_{n} \}$ is the sequence of CDF's associated to some solution $\{X^{(n)}\}$ of E: main RDE, then, independent of the initial CDF $F_{0}$, if $F_{\epsilon}$ is the rescaled CDF defined by then $F_{\epsilon} \to F$ locally uniformly in $\mathbb{R} \times (0,\infty)$, where $F$ is determined

Theorems & Definitions (41)

  • Theorem 1
  • Corollary 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Remark 1
  • ...and 31 more