Oscillations of Observables in 1-Dimensional Lattice Systems

TL;DR

This paper establishes universal probabilistic bounds on oscillations of observables in 1D lattice gases, extending inequalities related to monotone functions and ergodic sums, with results independent of specific models.

Contribution

It extends inequalities by Ivanov to provide universal bounds on observable oscillations in 1D lattice gases, applicable across models and states.

Findings

Probability of k oscillations bounded by C R^k with R<1

Bounds depend only on ratio α/β, not on model details

Results hold for infinite volume and increasing box sizes

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 not depend on any detail of the model, nor on the state one observes, but only on the ratio $\alpha/\beta $.