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General multi-scale estimates for Lyapunov data of Perron-Frobenius matrices. The case of diluted autocatalytic chemical reaction networks

Jeremie Unterberger

TL;DR

The paper tackles the challenge of estimating Lyapunov data for large Perron–Frobenius matrices arising from autocatalytic chemical networks with highly uncertain kinetic rates. It introduces a multi-scale renormalization framework that iteratively coarse-grains fast reaction cycles into effective nodes, producing hierarchical formulas for the Lyapunov exponent $\lambda^*$ and the right/left eigenvectors $v^*$, $v^{\dagger,*}$ that depend rationally on the rates. Central to the method are resolvent-based path expansions, dominant-SCC/DAG structures, and Doeblin-type bounds, which together yield rigorous and tractable a priori estimates and perturbative corrections. The approach remains robust to limited rate information and naturally captures resonance and transition regimes across scales, offering a practical tool for predicting autocatalytic growth in large chemical networks. The results provide a principled way to extract leading growth rates and stationary distributions from sparse kinetic data, with potential implications for understanding prebiotic chemistry and reaction networks under stiff conditions.

Abstract

Autocatalytic chemical reaction networks are dynamical systems whose linearization around zero, dX/dt = AX, is represented by a Perron-Frobenius matrix A with positive Lyapunov exponent; this exponent gives the growth rate of the species concentration vector X in the diluted regime, i.e. in a vicinity of zero. We introduce here a new, general recursive procedure providing precise quantitative information about Lyapunov data, namely, the Lyapunov eigenvalue, and left and right eigenvectors. Our estimates are based on a multi-scale algorithm inspired from Wilson's renormalization group method in quantum field theory, and Markov chain arguments introduced in (Nghe & Unterberger). They are compatible with the very scarce knowledge of kinetic rates (coefficients of A) generally available in chemistry, and take on the form of simple rational functions of the latter.

General multi-scale estimates for Lyapunov data of Perron-Frobenius matrices. The case of diluted autocatalytic chemical reaction networks

TL;DR

The paper tackles the challenge of estimating Lyapunov data for large Perron–Frobenius matrices arising from autocatalytic chemical networks with highly uncertain kinetic rates. It introduces a multi-scale renormalization framework that iteratively coarse-grains fast reaction cycles into effective nodes, producing hierarchical formulas for the Lyapunov exponent and the right/left eigenvectors , that depend rationally on the rates. Central to the method are resolvent-based path expansions, dominant-SCC/DAG structures, and Doeblin-type bounds, which together yield rigorous and tractable a priori estimates and perturbative corrections. The approach remains robust to limited rate information and naturally captures resonance and transition regimes across scales, offering a practical tool for predicting autocatalytic growth in large chemical networks. The results provide a principled way to extract leading growth rates and stationary distributions from sparse kinetic data, with potential implications for understanding prebiotic chemistry and reaction networks under stiff conditions.

Abstract

Autocatalytic chemical reaction networks are dynamical systems whose linearization around zero, dX/dt = AX, is represented by a Perron-Frobenius matrix A with positive Lyapunov exponent; this exponent gives the growth rate of the species concentration vector X in the diluted regime, i.e. in a vicinity of zero. We introduce here a new, general recursive procedure providing precise quantitative information about Lyapunov data, namely, the Lyapunov eigenvalue, and left and right eigenvectors. Our estimates are based on a multi-scale algorithm inspired from Wilson's renormalization group method in quantum field theory, and Markov chain arguments introduced in (Nghe & Unterberger). They are compatible with the very scarce knowledge of kinetic rates (coefficients of A) generally available in chemistry, and take on the form of simple rational functions of the latter.

Paper Structure

This paper contains 37 sections, 9 theorems, 283 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Aim of the article
  3. Resolvent formula
  4. Merging patterns and hierarchical structures
  5. Multi-scale method
  6. Overview of the article
  7. Main notations
  8. Renormalization: a heuristic introduction
  9. Cycles: an elementary example
  10. Heuristics for the merging procedure
  11. Description of the renormalization algorithm
  12. Description
  13. Two examples
  14. Example 1
  15. Example 2. Two coupled cycles
  16. ...and 22 more sections

Key Result

Lemma 5.1

Let Then

Figures (2)

  • Figure 1: Numerical checks for Example 1. Legend: (HF) = hierarchical formulas, see text.
  • Figure 2: Numerical checks for variant of Example 2. Legend: (HF) = hierarchical formulas, see text.

Theorems & Definitions (12)

  • Definition 3.1: weight renormalization factor of ${\cal G}$)
  • Lemma 5.1: first-order formula for Lyapunov eigenvalue
  • Lemma 5.2: $L^1$-contraction bound for $^t\tilde{\cal W}^{int}$
  • Lemma 5.3: $L^{\infty}$-contraction bound for $\tilde{\cal W}^{int}$
  • Lemma 5.4
  • Corollary 5.5
  • Corollary 5.6: adjoint Lyapunov vector
  • Lemma 5.7: exit probabilities
  • Lemma 5.8: time-continuous adjoint problem
  • Definition 5.9: renormalized generator $\underline{A}^{ren}$
  • ...and 2 more