---
title: "Robust Golden Ratio universality as a consequence of a circulant symmetry"
authors:
  - "Luka Paljk"
  - "Johannes Schmidt"
  - "Vladislav Popkov"
year: 2026
venue: "arXiv"
arxiv: "2610.13842"
doi: null
source: "https://arxiv.org/abs/2610.13842"
topics:
  - "statistical-mechanics"
  - "exclusion-process"
  - "universality"
  - "golden-ratio"
  - "lattice-gases"
seed_rank: 2271
seed_batch: "curiosity-2026-10-13"
reviewed: "2026-10-13"
pool: "physics"
relevance_score: 7
cites:
  - title: "Robust Golden Ratio universality as a consequence of a circulant symmetry"
    url: "https://arxiv.org/abs/2610.13842"
    year: 2026
    arxiv: "2610.13842"
    doi: null
see: ["1850-two-dimensional-majority-rule-is-p-complete"]
---

# Robust Golden Ratio universality as a consequence of a circulant symmetry

## One-sentence takeaway

The golden ratio shows up as a dynamical exponent z = (1+√5)/2 generically, not by fine tuning, in Markov processes whose flux Jacobian has a circulant symmetry and whose number of conservation laws is odd and larger than three, with a parameter-free multilane exclusion process as the minimal example.

## Why it's lovely

Exclusion processes are the simplest models of particles hopping along lanes, the physics behind traffic jams and crowd flow, and it is delightful that their large-scale fluctuations can spread with exponent exactly the golden ratio. Until now that Fibonacci class needed carefully tuned parameters; here a symmetry makes it robust. It is a lovely bridge between a lattice-gas cellular automaton and number-flavoured exponents, and a reminder that multi-lane flow (units on parallel paths in an RTS) has richer statistics than the single-lane KPZ picture.

## Key ideas

- **Golden-ratio class.** Dynamical exponent z = φ = (1+√5)/2, previously reachable only with fine-tuned parameters.
- **Circulant symmetry.** A global circulant symmetry of the flux Jacobian forces GR universality when the number of conservation laws is odd and larger than three.
- **Minimal model.** A multilane exclusion process with Bernoulli stationary measure and no free parameters shows GR universality for all conserved modes, most doubly degenerate.
- **Method.** Mode-coupling theory generalized to degenerate conserved modes, checked against large-scale Monte Carlo simulations.

## Caveats

Short preprint (6+5 pages, Oct 2026). The predictions rest on mode-coupling theory plus simulation, not a rigorous proof. Bridge to games is analogy-level only.

## Links

- arXiv abstract: https://arxiv.org/abs/2610.13842
- PDF: https://arxiv.org/pdf/2610.13842