Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hipster random walks, random series-parallel graph and random homogeneous systems

Xinxing Chen, Thomas Duquesne, Zhan Shi

TL;DR

This work develops a unified framework for random 1-homogeneous, monotone recursions driven by random functions $\mathbf{F}\in\mathscr{H}$, unifying two canonical models—critical random series-parallel graph resistances and hipster random walks. The authors build a general Lambda-condition-based method to obtain sharp lower bounds, and under suitable moment conditions prove Derrida-type results: after scaling by $ (c_* n)^{1/3}$, $\log X_n$ converges in distribution to the density $\frac{3}{4} (1-x^2) \mathbf{1}_{\{|x|<1\}}$, with $c_* = \tfrac{9}{4} \mathbb{E}[\Gamma_{\mathbf{F}}^{(0,2)} + 2\Gamma_{\mathbf{F}}^{(1,1)}]$. This yields the limiting law for the effective resistance in the critical series-parallel graph and recovers the hipster-walk result, while applying to a broad family of random homogeneous systems. The constant is expressed via Gamma-function moments, tying the limit to intrinsic features of the random function field, and the paper suggests broader functional-limit prospects alongside connections to PDE approaches. Overall, the work confirms Derrida’s conjecture in the $n^{1/3}$-scaling regime and provides a versatile probabilistic route to universal limiting laws for recursive random systems.

Abstract

We study a class of random homogeneous systems. Our main result says that under suitable general assumptions, these systems converge weakly, upon a suitable normalization, to the probability distribution with density $\frac34 \, (1-x^2) \, {\bf 1}_{\{ x\in (-1, \, 1)\} }$. Two special cases are of particular interest: for the effective resistance of the critical random series-parallel graph, our result gives an affirmative answer to a conjecture of Hambly and Jordan (Adv. Appl. Probab. 2004) and further conjectures of Addario-Berry et al. (Probab.Theory Related Fields 2020) and Derrida whereas for the hipster random walk, we recover a previous result of Addario-Berry et al.~(Probab. Theory Related Fields 2020).

Hipster random walks, random series-parallel graph and random homogeneous systems

TL;DR

This work develops a unified framework for random 1-homogeneous, monotone recursions driven by random functions , unifying two canonical models—critical random series-parallel graph resistances and hipster random walks. The authors build a general Lambda-condition-based method to obtain sharp lower bounds, and under suitable moment conditions prove Derrida-type results: after scaling by , converges in distribution to the density , with . This yields the limiting law for the effective resistance in the critical series-parallel graph and recovers the hipster-walk result, while applying to a broad family of random homogeneous systems. The constant is expressed via Gamma-function moments, tying the limit to intrinsic features of the random function field, and the paper suggests broader functional-limit prospects alongside connections to PDE approaches. Overall, the work confirms Derrida’s conjecture in the -scaling regime and provides a versatile probabilistic route to universal limiting laws for recursive random systems.

Abstract

We study a class of random homogeneous systems. Our main result says that under suitable general assumptions, these systems converge weakly, upon a suitable normalization, to the probability distribution with density . Two special cases are of particular interest: for the effective resistance of the critical random series-parallel graph, our result gives an affirmative answer to a conjecture of Hambly and Jordan (Adv. Appl. Probab. 2004) and further conjectures of Addario-Berry et al. (Probab.Theory Related Fields 2020) and Derrida whereas for the hipster random walk, we recover a previous result of Addario-Berry et al.~(Probab. Theory Related Fields 2020).

Paper Structure

This paper contains 13 sections, 27 theorems, 262 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A general recipe for lower bound
  3. Preliminaries
  4. Comparison
  5. Elementary properties of $T_F$
  6. Asymptotics for $\Lambda_{\phi_n, \, F}$
  7. Proof of Theorem \ref{['p:main']}
  8. Proof of \ref{['lb:pf2_(v>0)']}: Case I (large values of $v$)
  9. Proof of \ref{['lb:pf2_(v>0)']}: Case II (positive intermediate values of $v$)
  10. Proof of \ref{['lb:pf2_(v<0)']}: Case III (negative intermediate values of $v$)
  11. Proof of \ref{['lb:pf2_(v<0)']}: Case IV (small values of $v$)
  12. Proof of Theorem \ref{['t:main']}
  13. Further remarks and problems

Key Result

Theorem 1.4

(Auffinger and Cable. auffinger-cable) Let $(\Delta_n, \, n\ge 0)$ be as in def_recurrence_eff_conduct with $p= p_c= \frac{1}{2}$. Then for all $y\in {\mathbb R}$, where $c_\mathtt{dis} :=\pi^2/6$.

Figures (1)

  • Figure 1: Graphs of $\phi_n$ (boldface and blue) and $\widetilde{\phi}_n$ (dashdotted and red). The graph of $\phi_n$ is supported in $(-\sigma_n, \, \sigma_n)$; only the part of the graph in this interval is presented in boldface and blue. The graph of $\widetilde{\phi}_n$ is supported in the larger interval $(- \widetilde{\sigma}_n, \, \widetilde{\sigma}_n)$; again only the part in the interval is dashdotted and in red.

Theorems & Definitions (40)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Conjecture 1.6
  • Conjecture 1.7
  • Example 1.8
  • Theorem 1.9
  • Conjecture 1.10
  • ...and 30 more