By Arnaud Le Ny
Summary. during this monograph, we survey a few key problems with the speculation of
Gibbsian and non-Gibbsian measures in finite-spin lattice platforms. While
non-Gibbsian measures are really in simple terms the thing of the final bankruptcy, the
material of the 1st chapters is chosen with generalized Gibbs measures
in brain. the subjects of Gibbsian concept are then selected for his or her foun-
dational or contrasting function with appreciate to the measures analyzed in the
¯nal bankruptcy, together with particularly extra designated elements as e.g. the proof
of the Choquet decomposition of Gibbs measures in bankruptcy 2, an evidence of
the Kozlov theorem in bankruptcy three, less than a marginally novel presentation that
serves to introduce a telescoping process wanted for generalized Gibbs
measures in bankruptcy five, and a cautious dialogue of the variational princi-
ple in bankruptcy four. This monograph covers additionally the contents of mini-courses
given in 2007 on the universities UFMG (Belo Horizonte) and UFRGS
(Porto Alegre), whose target was once to express, in a comparatively brief quantity of
lectures, the center of the idea had to comprehend Gibbsianness and
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Extra info for Introduction to (generalized) Gibbs measures
10 (Potential). A potential is a family Φ = (ΦA )A∈S of functions ΦA : Ω −→ R indexed by the set S of finite subsets of S, such that ∀A ∈ S, ΦA is FA measurable. Our infinite-volume formalism incorporates the finite-volume one by considering free (or empty) boundary conditions, to extend Hamiltonians from ΩΛ to Ω in a well defined way by considering finite sums in the following1 1 This is true only when Φ is bounded for all A ∈ S. g. [29, 86]. 11 (Hamiltonian with free boundary condition). Consider a potential Φ.
A ∈ F. 45) holds for all A and for all n. Remark that the uniformity in n implies that Ωµ is a tail-event. t. the tail σalgebra F∞ . ” and “∀A ∈ F”. 28) to introduce appropriate objects for a decomposition of DLR measures when applied on the countable family of cylinders first, to derive some (tail-) measurability properties using countability, and thereafter to extend in a standard way the latter objects onto probability kernels that are in some sense extreme “µ-almost surely”. In fact, the starting point of the decomposition is one of the defining properties of versions of conditional probability with respect to the tail σ-algebra: ∀µ ∈ M+ 1 (Ω), µ(·) = µ[·|F∞ ](ω) dµ(ω).
37 (Uniform non-nullness). t. 0 < αΛ ρ(A) ≤ γΛ (A | ω) ≤ βΛ ρ(A), ∀ω ∈ Ω, ∀A ∈ F. 38) Simple non-nullness, in the sense that ρ(A) > 0 =⇒ γΛ (A | ω) > 0, for all ω ∈ Ω, is equivalent to uniform non-nullness when γ is quasilocal, . It is also-called the finite-energy condition in percolation circles. A measure µ is then said to be (uniformly) non-null if there exists a (uniformly) non-null specification γ such that µ ∈ G(γ). 33. For the purpose of this theorem, the inverse temperature β has been incorporated in the potential.