Extremal decomposition for random Gibbs measures: from general metastates to metastates on extremal random Gibbs measures

Abstract

The concept of metastate measures on the states of a random spin system was introduced to be able to treat the large-volume asymptotics for complex quenched random systems, like spin glasses, which may exhibit chaotic volume dependence in the strong-coupling regime. We consider the general issue of the extremal decomposition for Gibbsian specifications which depend measurably on a parameter that may describe a whole random environment in the infinite volume. Given a random Gibbs measure, as a measurable map from the environment space, we prove measurability of its decomposition measure on pure states at fixed environment, with respect to the environment. As a general corollary we obtain that, for any metastate, there is an associated decomposition metastate, which is supported on the extremes for almost all environments, and which has the same barycenter.In this note, we consider quenched random spin models in the infinite volume on lattices or more generally countably-infinite graphs. This includes examples like the random-field Ising model and the Edwards-Anderson nearest-neighbor spin glass, but also more generally continuous spin models. We prove an extended version of the known extremal decomposition (see [Föl75] and [Geo11, Theorem 7.26]) for the infinite-volume Gibbs measures in terms of pure states, i.e., the extremal Gibbs measures, which is measurable w.r.t. the random environment, see Theorem 2.1. We also present a connection between this result and the theory of metastates via the notion of the decomposition metastate, see Corollary 2.3.What are the difficulties? Indeed, measurability w.r.t. random environments in the infinite volume, which can be taken for granted in simple situations, becomes nontrivial for general systems. The difficulty comes from the following physical phenomenon that may happen for general random systems and in particular those with frustrated interactions like spin glasses. A system is frustrated, if it has competing interactions, in the sense that it is impossible to minimize the total finite-volume

Spin glass phase at zero temperature in the Edwards-Anderson model

“…Metastates capture some aspects of the chaotic nature of spin glasses, such as the "chaotic size dependence" proved in [49], which means that the ground state in a finite region is chaotic with respect to changes in the size of the region. For some recent results and different perspectives on metastates, see [29].…”

Gibbs-Non Gibbs Transitions in Different Geometries: The Widom-Rowlinson Model Under Stochastic Spin-Flip Dynamics

“…Let us now for our graph consider a Cayley tree, which is the infinite graph which has no loops, and where each vertex has precisely k + 1 nearest neighbors. The Widom-Rowlinson model in the hard-core version, and in the soft-core version, is again defined by the specification kernels of (4) and (5). We need to start with a good understanding of the Widom-Rowlinson model in equilibrium.…”

“…Existence and extremal decomposition of proper infinite-volume measures becomes even more involved for systems with random potentials. In general, for systems like spin-glasses, the construction of infinite-volume states by non-random sequences of volumes which exhaust the whole lattice is problematic, and for such systems the higher-level notion of a metastate (a measure on infinite-volume Gibbs measures) is useful [34,2,27,1,5].…”

The roles of random boundary conditions in spin systems

“…The distributional limit behaviour can be described in terms of "metastates", objects which were introduced by Aizenman-Wehr [2] and Newman-Stein [46,49,50] via different constructions, which then were shown to be equivalent, see also [19].…”

“…We remark moreover that it follows from [19] that a metastate supported on pure states also exists; however, its construction will have to be different than just imposing independent random boundary conditions (possibly by making use of a maximizing procedure, or by considering highly correlated boundary conditions). For the ferromagnet this is immediate, for the Mattis version of our models less so.…”