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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 \(\alpha/ \beta\).

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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