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IPS/Zeta Correspondence for the Domany-Kinzel model

Chusei Kiumi, Norio Konno, Yuki Oshima

TL;DR

The paper extends zeta-function analysis to interacting particle systems (IPS), using the Domany-Kinzel model as a representative probabilistic IPS. It defines the IPS-type zeta function $ar{oldsymbol{ aisebox{0pt}{ aisebox{-0.2pt}{ ext}}}ar{}}(Q^{(l)}, P_N,u)$ and studies the global evolution operator $Q_N^{(g)}$ on an $N$-site configuration space, deriving trace and spectral relationships that yield closed-form expressions for the configuration-count coefficients $C_r$ and the zeta function in key regimes. A central result is the hierarchical spectrum relation $ ext{Spec}(Q_N^{(g)})= ext{Spec}(Q_{N-1}^{(g)})igl(t ext{Spec}(Q_{N-1}^{(g)})igr)$ under a parameter constraint, enabling explicit computation of $C_r$ and $ar{oldsymbol{ aisebox{0pt}{ aisebox{-0.2pt}{ ext}}}ar{}}igl(Q^{(l)}, P_N,uigr)$ and revealing a QCA analogue with phase-based zeta expressions. The work advances the IPS/Zeta correspondence by linking multi-particle dynamics, spectral theory, and percolation-type phenomena in both probabilistic and quantum cellular automata contexts.

Abstract

Previous studies presented zeta functions by the Konno-Sato theorem or the Fourier analysis for one-particle models, including random walks, correlated random walks, quantum walks, and open quantum random walks. Furthermore, the zeta functions for the multi-particle model with probabilistic or quantum interactions, called the interacting particle system (IPS), were also investigated. In this paper, we focus on the zeta function for a class of IPS, including the Domany-Kinzel model, which is a typical model of the probabilistic IPS in the field of statistical mechanics and mathematical biology.

IPS/Zeta Correspondence for the Domany-Kinzel model

TL;DR

The paper extends zeta-function analysis to interacting particle systems (IPS), using the Domany-Kinzel model as a representative probabilistic IPS. It defines the IPS-type zeta function and studies the global evolution operator on an -site configuration space, deriving trace and spectral relationships that yield closed-form expressions for the configuration-count coefficients and the zeta function in key regimes. A central result is the hierarchical spectrum relation under a parameter constraint, enabling explicit computation of and and revealing a QCA analogue with phase-based zeta expressions. The work advances the IPS/Zeta correspondence by linking multi-particle dynamics, spectral theory, and percolation-type phenomena in both probabilistic and quantum cellular automata contexts.

Abstract

Previous studies presented zeta functions by the Konno-Sato theorem or the Fourier analysis for one-particle models, including random walks, correlated random walks, quantum walks, and open quantum random walks. Furthermore, the zeta functions for the multi-particle model with probabilistic or quantum interactions, called the interacting particle system (IPS), were also investigated. In this paper, we focus on the zeta function for a class of IPS, including the Domany-Kinzel model, which is a typical model of the probabilistic IPS in the field of statistical mechanics and mathematical biology.
Paper Structure (7 sections, 7 theorems, 72 equations, 3 figures, 2 tables)

This paper contains 7 sections, 7 theorems, 72 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Definition
  3. Multi-particle system
  4. DK model
  5. The IPS-type zeta function
  6. Results
  7. Summary

Key Result

Theorem 1

Figures (3)

  • Figure 1: Figures show the histograms of the eigenvalues of $Q^{(g)}_N$ in the complex plain, for oriented site percolation $q=p$ with $N=8$.
  • Figure 2: Figures show the histograms of the eigenvalues of $Q^{(g)}_N$ in the complex plain, for oriented bond percolation $q=1-(1-p)^{2}$ with $N=8$.
  • Figure 3: Figures show the histograms of the eigenvalues of $Q^{(g)}_N$ in the complex plain for $q=0$ with $N=8$.

Theorems & Definitions (13)

  • Theorem 1
  • Lemma 1
  • proof
  • Corollary 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 2
  • proof
  • ...and 3 more