Thermodynamic formalism and large deviations for multiplication-invariant potentials on lattice spin systems. (English) Zbl 1290.82004

Chazottes, Jean-René; Redig, Frank

Electron. J. Probab. 19, Paper No. 39, 19 p. (2014).

Summary: We introduce the multiplicative Ising model and prove basic properties of its thermodynamic formalism such as existence of pressure and entropies. We generalize to one-dimensional “layer-unique” Gibbs measures for which the same results can be obtained. For more general models associated to a \(d\)-dimensional multiplicative invariant potential, we prove a large deviation theorem in the uniqueness regime for averages of multiplicative shifts of general local functions. This thermodynamic formalism is motivated by the statistical properties of multiple ergodic averages.

MSC:

82B20	Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60F10	Large deviations
11B25	Arithmetic progressions

Keywords:

Gibbs measures; multiplicative shift; multiple ergodic averages