Distance fluctuations and Lyapunov exponents.

*(English)*Zbl 0871.60088The author continues his studies of the Brownian motion in a Poissonian potential, this time truncated for simplification. In the present article he examines certain metrics depending on the random environment, which he already introduced in [Commun. Pure Appl. Math. 47, No. 12, 1655-1688 (1994; Zbl 0814.60022)]. These metrics are analogues of the point to point passage times in first passage percolation. It was shown in the cited paper that for large distances they behave as the (deterministic) Lyapunov exponents. Now upper bounds on the size of the fluctuations of the metrics around their mean are derived, which are not expected to be of the true order, but which “can be very very useful”, as mentioned by the author. Under the additional assumption of rotational invariance of the shape of the potential, upper bounds on the difference of the mean of the metrics and the Lyapunov exponents are also derived.

Reviewer: M.Mürmann (Heidelberg)

##### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82D30 | Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) |