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The Eyring-Kramers Law for the Extinction Time of the Contact Process on Stars

Younghun Jo

TL;DR

This work establishes a sharp Eyring-Kramers law for the extinction time of the contact process on a star graph with $N$ leaves at fixed infection rate $\lambda$, including the exact subexponential prefactor. The authors introduce a regenerative modification to handle absorbing states and develop a non-reversible potential-theoretic framework to analyze metastability, supported by explicit quasi-stationary asymptotics via special functions and refined Laplace methods. They derive detailed asymptotics for the quasi-stationary distribution, analyze the energy landscape, and perform capacity estimates through trace processes to obtain the mean extinction time as $\mathbb{E}_x\tau_{(0,0)} = \kappa_{\lambda} N^{-1/(1+2\lambda)} \left(\frac{(1+\lambda)^2}{1+2\lambda}\right)^N (1+o(1))$, with $\kappa_{\lambda}=\left(\frac{1+\lambda}{\lambda}\right)^{\frac{2}{1+2\lambda}} \Gamma\left(\frac{2(1+\lambda)}{1+2\lambda}\right)$. The exponent in the main term matches the large-deviation prediction $\lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}_x\tau = 2\log(1+\lambda)-\log(1+2\lambda)$. Collectively, the results provide a novel methodological toolkit for sharp metastability analyses in non-reversible Markov processes and yield precise extinction-time predictions for hub-spoke epidemic models.

Abstract

In this paper, we derive a precise estimate for the mean extinction time of the contact process with a fixed infection rate on a star graph with $N$ leaves. Specifically, we determine not only the exponential main factor but also the exact sub-exponential prefactor in the asymptotic expression for the mean extinction time as $N\to\infty$. Previously, such detailed asymptotic information on the mean extinction time of the contact process was available exclusively for complete graphs. To obtain our results, we first establish an accurate estimate for the stationary distribution of a modified contact process, employing special function theory and refined Laplace's method. Subsequently, we apply a recently developed potential theoretic approach for analyzing metastability in non-reversible Markov processes, enabling us to deduce the asymptotic expression. The integration of these methodologies constitutes a novel approach developed in this paper, one which has not been utilized previously in the study of the contact process.

The Eyring-Kramers Law for the Extinction Time of the Contact Process on Stars

TL;DR

This work establishes a sharp Eyring-Kramers law for the extinction time of the contact process on a star graph with leaves at fixed infection rate , including the exact subexponential prefactor. The authors introduce a regenerative modification to handle absorbing states and develop a non-reversible potential-theoretic framework to analyze metastability, supported by explicit quasi-stationary asymptotics via special functions and refined Laplace methods. They derive detailed asymptotics for the quasi-stationary distribution, analyze the energy landscape, and perform capacity estimates through trace processes to obtain the mean extinction time as , with . The exponent in the main term matches the large-deviation prediction . Collectively, the results provide a novel methodological toolkit for sharp metastability analyses in non-reversible Markov processes and yield precise extinction-time predictions for hub-spoke epidemic models.

Abstract

In this paper, we derive a precise estimate for the mean extinction time of the contact process with a fixed infection rate on a star graph with leaves. Specifically, we determine not only the exponential main factor but also the exact sub-exponential prefactor in the asymptotic expression for the mean extinction time as . Previously, such detailed asymptotic information on the mean extinction time of the contact process was available exclusively for complete graphs. To obtain our results, we first establish an accurate estimate for the stationary distribution of a modified contact process, employing special function theory and refined Laplace's method. Subsequently, we apply a recently developed potential theoretic approach for analyzing metastability in non-reversible Markov processes, enabling us to deduce the asymptotic expression. The integration of these methodologies constitutes a novel approach developed in this paper, one which has not been utilized previously in the study of the contact process.
Paper Structure (13 sections, 19 theorems, 207 equations, 2 figures)

This paper contains 13 sections, 19 theorems, 207 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model and Main Results
  3. Energy Landscape
  4. Quasi-Stationary Distribution
  5. Asymptotic Analysis
  6. Basic Properties of the Energy Landscape
  7. Potential Theory for Non-Reversible Markov Chains
  8. Basic Terminology
  9. Dirichlet and Thomson Principles
  10. Trace Processes
  11. Proof of the Eyring--Kramers Law
  12. Capacity Estimate
  13. Proof of the Main Theorem

Key Result

Theorem 2.1

Let $\varepsilon > 0$ be given. Then, for each $x \in \{0,1\}\times [\varepsilon N, N]$, we have Let $(f_N)_{N\ge 1} = (f_N(n))_{N\ge 1}$ and $(g_N)_{N\ge 1} = (g_N(n))_{N\ge 1}$ be collections of real functions in $n$. We write $f_N = O(g_N)$, $g_N = \Omega(f_N)$, or $f_N \lesssim g_N$ if there exi as $N \to \infty$, where the error term $o(1)$ is uniform in $x$ and the constant $\kappa_\lambda$

Figures (2)

  • Figure 1: Example configurations of the contact process on a star. Note that the number of infected leaves cannot increase when the hub is healthy (see the configuration on the left). A healthy leaf (denoted by $A$) can become infected only after the hub has been reinfected.
  • Figure 2: Transition rates for the contact process on a star. Dark circles denote infected vertices, and light circles denote healthy vertices.

Theorems & Definitions (36)

  • Theorem 2.1: Eyring--Kramers law
  • Proposition 3.1: $3$-term recurrence relation for the quasi-stationary distribution
  • proof
  • Proposition 3.2
  • Remark 3.3
  • proof : Proof of Proposition \ref{['prop:recurrence solution']}
  • Theorem 3.4: Asymptotic behavior of the quasi-stationary measure, high infection regime
  • Theorem 3.5: Asymptotic behavior of the quasi-stationary measure, low infection regime
  • Remark 3.6
  • Remark 3.7
  • ...and 26 more