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Oscillations of observables in 1-dimensional lattice systems. (English) Zbl 0897.70004
Summary: Using and extending inequalities on the down-crossings of monotone functions and ergodic sums, we give universal bounds on the probability of finding oscillations of observables in one-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 $$CR^k$$, with $$R<1$$, where the constants $$C$$ and $$R$$ do not depend on any detail of the model, nor on the state one observes, but only on the ratio $$\alpha/ \beta$$.

##### MSC:
 70-XX Mechanics of particles and systems 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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