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.
Simulated annealing from continuum to discretization: a convergence analysis via the Eyring--Kramers law
研究了连续时间模拟退火算法$(X_t; \, t \ge 0)$及其离散化算法$(x_k; \, k =0,1, \ldots)$逼近给定函数$f$全局最优值的收敛速度。我们证明了尾概率$\mathbb{P}(f(X_t) > \min f +δ)$（resp. $\mathbb{P}(f(x_k) > \min f +δ)$）随时间衰减多项式（分别为以累积步长计），并提供作为模型参数的函数的显式速率。我们的论点适用于最近的发展吉布斯措施在低温下的功能不等式-Eyring-Kramers定律。在离散情形下，我们得到了保证收敛的步长条件.
Eyring--Kramers law for the hyperbolic $ϕ^4$ model
Max Planck Institute for Mathematics in the Sciences（德国莱比锡马克斯·普朗克数理科学研究所）；Department of Mathematics and Statistics, University of Helsinki（赫尔辛基大学数学与统计系）
