Optimal multi-time-scale estimates for diluted autocatalytic chemical networks. (1) Introduction and $σ^*$-dominant case

TL;DR

This work develops a Renormalization-Group–inspired multi-scale framework to estimate the growth rate of diluted autocatalytic chemical networks, encapsulated by the Lyapunov exponent $\lambda^*$, from the linearized generator $A$ and its deficiency structure. Central to the method is the Resolvent Formula, which recasts $\lambda^*$ as a self-consistent threshold determined by excursion weights on a multi-scale Markov graph; scale separation ensures that dominant paths yield accurate, rational expressions in kinetic rates. The paper introduces multi-scale splitting and the notion of $\sigma^*$-dominant graphs, showing that for such graphs $\lambda^*$ can be determined from a finite generating set of loops through a distinguished vertex and providing explicit asymptotics and Lyapunov eigenvectors in several examples, including a Formose-type network. The results offer semi-quantitative, physically interpretable predictions that relate long-time behavior to simple functions of kinetic rates, and they lay the groundwork for rate inference and systematic analysis of large reaction networks. Overall, the framework connects Markov-chain theory, RG-inspired resummation, and graph-based dominance to yield tractable estimates of growth rates in complex autocatalytic networks.

Abstract

Autocatalytic chemical networks are dynamical systems whose linearization around zero has a positive Lyapunov exponent; this exponent gives the growth rate of the system in the diluted regime, i.e. for near-zero concentrations. The generator of the dynamics in the kinetic limit is then a Perron-Frobenius matrix, suggesting the use of Markov chain techniques to get long-time asymptotics. This series of works introduces a new, general procedure providing precise quantitative information about such asymptotics, based on estimates for the Lyapunov eigenvalue and eigenvector. The algorithm, inspired from Wilson's renormalization group method in quantum field theory, is based on a downward recursion on kinetic scales, starting from the fastest, and terminating with the slowest rates. Estimates take on the form of simple rational functions of kinetic rates. They are accurate under a separation of scales hypothesis, loosely stating that kinetic rates span many orders of magnitude. We provide here a brief general motivation and introduction to the method, present some simple examples, and derive a number of preliminary results, in particular the estimation of Lyapunov data for a subclass of so-called $σ^*$-dominant graphs.

Figures (2)

• Figure 1: Lyapunov exponent as a function of kinetic parameter $k'_4$
• Figure 2: RMSE errors on Lyapunov eigenvector as a function of kinetic parameter $k'_4$

Theorems & Definitions (8)

• Proposition 2.1: Resolvent formula
• Lemma 3.1: deficiency rate identity
• Theorem 3.1
• Definition 7.1: $\sigma^*$-dominant graph
• Theorem 7.1
• Theorem 7.2: threshold parameter and approximate Lyapunov exponent
• Lemma 7.2: Lyapunov weights for irreducible graphs
• Lemma 7.3