Loop condensation effects in the behaviour of random walks. (English) Zbl 0814.60063

Khanin, K. M.; Mazel, A. E.; Shlosman, S. B.; Sinai, Ya. G.

Freidlin, Mark I. (ed.), The Dynkin Festschrift. Markov processes and their applications. In celebration of Eugene B. Dynkin’s 70th birthday. Boston, MA: Birkhäuser. Prog. Probab. 34, 167-184 (1994).

This paper deals with two asymptotic problems in the theory of random walks in \(\mathbb{Z}^ d\). The first one is that of the validity of Donsker’s invariance principle for random walks in a random environment. The results obtained by the authors suggest strongly that this principle does not apply. They consider a random potential \(F\) consisting of a family of i.i.d. random variables (in \(\{\pm 1\})\) indexed by the set of bonds (or sites) of the lattice \(\mathbb{Z}^ d\). This random potential is used to perturb the probability distribution of a simple random walk on \(\mathbb{Z}^ d\) by putting a weight proportional to the exponential of the action \[ S_ \lambda (\omega^{(n)}) = \sum^ n_{i=0} \lambda F \biggl( \omega^{(n)} (i) \biggr) \] in the case of site potential, and \[ S_ \lambda (\omega^{(n)}) = \sum^{n-1}_{i=0} \lambda F \bigl( \omega^{(n)} (i), \omega^{(n)} (i + 1) \bigr) \] in the case of bond potential. Here \(\omega^{(n)} = \omega^{(n)} (0), \dots, \omega^{(n)} (n)\) is a path in \(\mathbb{Z}^ d\) and \(\lambda\) some coupling constant. The main result of the authors is that, for all \(\lambda > 0\), and almost all values of \(F\), there exists a set \(C_ F\) of sites such that, for large \(n\), the path of the random walk spends most of its times in \(C_ F\), and \(F \equiv 1\) on \(C_ F\). Furthermore, it is shown that \(C_ F\) can be chosen so that \(C_ F\) consists of finite connected components only. This result strongly suggests that the limiting behaviour of the random walk in random potential is not diffusive.
The second problem is that of estimating the probability that two infinite random walks have more than \(k\) points in common. Let \(P_ \eta (k)\) be the probability that they have more than \(k\) lattice points in common, then (for \(d > 4)\) \(\text{Log} P_ \eta (k) \approx - k^{1- 2/d}\) as \(k \to \infty\). If \(P_ \xi (k)\) is the probability that there are more than \(k\) times at which they intersect, then \(\log P_ \xi (k) \approx - k^{1/2}\). It turns out that the approaches to both problems are quite similar, which is why they are studied in the same paper.
For the entire collection see [Zbl 0808.00010].

Reviewer: Ph.Biane (Paris)

MSC:

60G50	Sums of independent random variables; random walks
82D60	Statistical mechanics of polymers

Keywords:

random walk in random environment; intersection of random walks; random walks; Donsker’s invariance principle