Tagged particle motion in a clustered random walk system
The formal generator of a new particle system studied in this paper is as follows: \[ \Omega f(x)=\sum_{i}\sum_{j>i}\sum_{y}c(i,j,y;x)(f(x^{(i,j),y})-f(x)), \] where \(x\in {\mathcal X}:=\{x=(x_ i)_{i\in {\mathbb{Z}}}:\) \(x_ i\in {\mathbb{Z}}\) and \(...<x_{i-1}<x_ i<...\}\), \(c(i,j,y;x)=p(y)q(j-i)\quad I(x_{i-1}<x_ i+y,\quad x_{j-1}<x_ j+y),\) I(A) is the usual indicator function of A, and \(p=p(y)=q=(q(k))\) are probabilities on \({\mathbb{Z}}\) and \({\mathbb{N}}\) respectively. For each \(x\in {\mathcal X}\), \(i<j\) and y, \(x^{(i,j),y}\in {\mathcal X}\) is defined by \((x^{(i,j),y})_ k=x_ k+y\) if \(k\in (i,j)\), and \(=x_ k\) otherwise. Assume that p is symmetric, \(p(1),q(1)>0\) and \(\sup \{| yp(y)|:\quad y\in {\mathbb{Z}}\}<\infty.\) The critical hypothesis is \[ 1- \sum^{\infty}_{k=1}q(k)\cos k\theta =| \theta |^{\alpha}L(\theta)\quad for\quad some\quad 1<\alpha \leq 2, \] where L satisfies \(\lim_{\theta \to 0}L(c\theta)/L(\theta)=1\) for all \(c>0\). Denote by \(X(t)=(X_ i(t):\) \(i\in {\mathbb{Z}})\), \(t\geq 0\), the Markov process generated by \(\Omega\). The main result says that the tagged particle \((X_ 0(t))\), \(t\geq 0\), has an asymptotically Gaussian distribution with variance \(\beta t^{1/\alpha}\).
scientific article; zbMATH DE number 4076276
scientific article