Collet P., Eckmann J.-P.
Oscillations of Observables in 1-Dimensional Lattice Systems
(203K, Postscript)
ABSTRACT. Using, and extending, striking inequalities by V.V. Ivanov on the
down-crossings of monotone functions and ergodic sums, we give
universal bounds on the probability of finding oscillations of
observables in 1-dimensional lattice gases in infinite
volume. In particular, we study the finite volume average of the
occupation number as one runs through an
increasing sequence of boxes of size $2n$ centered at the origin.
We show that the probability to see $k$
oscillations of this average between
two values $\beta $ and $0<\alpha <\beta $ is
bounded by $C R^k$, with $R<1$, where the constants
$C$ and $R$ do {\em not} depend on any detail of the
model, nor on the state one observes, but only on the ratio $\alpha/\beta $.