Central limit theorem for the random walk of one and two particles in a random environment, with mutual interaction.

*(English)*Zbl 0815.60097
Dobrushin, R. L. (ed.), Probability contributions to statistical mechanics. Transl. by A. S. Sossinsky. Providence, RI: American Mathematical Society. Adv. Sov. Math. 20, 21-75 (1994).

In the one-particle model, \(\{X_ t, t = 1,2,\dots\}\) is a random walk on \(\mathbb{Z}^ \nu\) for some \(\nu \geq 1\) which interacts with a random environment \(\{\xi_ t(x)\}\), in the sense that
\[
P(X_ t = z \mid X_{t - 1} = x, \xi_{t - 1} = \eta) = P_ 0(z - x) + c(z - x,\eta(x)),\tag{\(*\)}
\]
where \(P_ 0\) is the “unperturbed” jump distribution; \(X_ t\) and \(\xi_ t\) are conditionally independent given \((X_{t - 1}, \xi_{t - 1})\). A local limit theorem is proved for \(P(X_ t = x \mid X_ 0 = u)\). A similar result was proved by different means in [C. Boldrighini, I. A. Ignatyuk, V. A. Malyshev and A. Pellegrinotti, Stochastic Processes Appl. 41, No. 1, 157-177 (1992; Zbl 0758.60065)]. In the two-particle model, \((*)\) becomes
\[
\begin{split} P(X_ t^{(i)} = x^{(i)} \mid X^{(1)}_{t - 1} = z^{(1)}, X^{(2)}_{t - 1} = z^{(2)}, \xi_{t - 1} =\eta)\\ = P_ 0(x^{(i)} - z^{(i)}) + \widehat{c} (x^{(i)} - z^{(i)}, z^{(1)} - z^{(2)}, \eta(z^{(i)})),\quad i = 1,2.\end{split}
\]
Thus as well as interaction with the environment, there is interaction between the particles as well. The main result of the paper (Theorem 2) gives a central limit theorem (in integral form) for the joint distribution of the displacements of the two particles \(X^{(1)}_ t\) and \(X^{(2)}_ t\), for \(\nu \geq 3\). It shows that asymptotically in time, the two particles perform independent random walks and that each random walk is the same as if the other particle were absent.

For the entire collection see [Zbl 0802.00011].

For the entire collection see [Zbl 0802.00011].

Reviewer: M.Quine (Sydney)