Abstract
We consider deterministic and disordered frustrated systems in which we can show some strict inequalities with respect to related ferromagnetic systems. A case particularly interesting is the Edwards-Anderson spin-glass model in which it is possible to determine a region of uniqueness of the Gibbs measure, which is strictly larger than the region of uniqueness for the related ferromagnetic system. We analyze also deterministic systems with $|J_b| \in [J_A, J_B]$ where $0 < J_A \leq J_B < \infty$, for which we prove strict inequality for the critical points of the related FK model. The results are obtained for the Ising models but some extensions to Potts models are possible.
Highlights
The problem of proving inequalities in probabilistic models is very common; especially in statistical mechanics, there are models in which many qualitative properties, such as the existence of a phase transition, are only shown by using inequalities
Inequalities are proved between the critical points in percolation and in the Ising model and there is an extension to the Potts model and to many-body interactions in [Gr94]
These are models in which the interactions are themselves random variables, so that the Gibbs measure becomes a function of these random variables
Summary
The problem of proving inequalities in probabilistic models is very common; especially in statistical mechanics, there are models in which many qualitative properties, such as the existence of a phase transition, are only shown by using inequalities. We will prove a strict inequality between phase transitions in the ferromagnetic Ising and in the Edwards-Anderson models (see [Is25, EA75]). The main differences between our work and [Ca98, Gr99] are: a) we prove a strict inequality for the phase transition, i.e. for uniqueness of the Gibbs distribution, and for symmetry breaking; b) we show a strict inequality for the phase transitions of disordered ferromagnetic Ising models (such a result, but with a different methodology, is proved by Gandolfi [Ga98]); c) our method can be extended to Potts models and to frustrated many-body models; in a future paper we will provide these extensions (this is not explicit in [Ca98] and we do not know if it is possible to find a related Gibbs measure for the random cluster measure in [Ca98] besides the Ising model). We will prove this theorem at the end of section 4
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