论文标题
弱混合马尔可夫链的可稳定行为：可逆的，关键的零范围过程的情况
Metastable behavior of weakly mixing Markov chains: the case of reversible, critical zero-range processes
论文作者
论文摘要
我们提出了一种推导弱混合马尔可夫链的亚稳态行为的通用方法。这种方法基于分解方程的特性，可以应用于不满足Beltrán和Landim（2010,2012）或Landim ET中所需的混合条件的亚稳态动力学。 al。 （2020）。 作为应用程序，我们研究关键零范围过程的亚稳定行为。令$ r：s \ times s \ to \ bb r _+$是在有限套件上$ s $不可约的随机步行的跳转率，相对于统一度量可逆。对于$α> 0 $，令$ g：\ bb n \ to \ bb r _+$由$ g（0）= 0 $，$ g（1）= 1 $，$ g（k）= [k/（k-1）^α$，$ k \ ge 2 $。考虑$ s $的零范围流程，其中粒子从$ k $粒子占用的$ x $跳到$ y $ y $ y $ g（k）r（k）r（x，y）$。对于$α\ ge 1 $，在固定状态下，作为以$ n $表示的粒子总数，倾向于无穷大，除了一个可忽略的粒子在一个地点都积累了所有粒子。这种现象称为凝结。由于且仅当$α\ ge 1 $时发生冷凝，因此我们称CASE $α= 1 $关键。通过将文章第一部分中建立的一般方法应用于关键案例，我们表明，集中所有粒子在时间尺度的$ n^2 \ log n $中演变为随机步行$ s $，其过渡率与基础随机步行的能力成比例。
We present a general method to derive the metastable behavior of weakly mixing Markov chains. This approach is based on properties of the resolvent equations and can be applied to metastable dynamics which do not satisfy the mixing conditions required in Beltrán and Landim (2010,2012) or in Landim et. al. (2020). As an application, we study the metastable behavior of critical zero-range processes. Let $r: S\times S\to \bb R_+$ be the jump rates of an irreducible random walk on a finite set $S$, reversible with respect to the uniform measure. For $α>0$, let $g: \bb N\to \bb R_+$ be given by $g(0)=0$, $g(1)=1$, $g(k) = [k/(k-1)]^α$, $k\ge 2$. Consider a zero-range process on $S$ in which a particle jumps from a site $x$, occupied by $k$ particles, to a site $y$ at rate $g(k) r(x,y)$. For $α\ge 1$, in the stationary state, as the total number of particles, represented by $N$, tends to infinity, all particles but a negligible number accumulate at one single site. This phenomenon is called condensation. Since condensation occurs if and only if $α\ge 1$, we call the case $α=1$ critical. By applying the general method established in the first part of the article to the critical case, we show that the site which concentrates almost all particles evolves in the time-scale $N^2 \log N$ as a random walk on $S$ whose transition rates are proportional to the capacities of the underlying random walk.
