zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Completely analytical Gibbs fields. (English) Zbl 0569.46043
Statistical physics and dynamical systems. Rigorous results, Pap. 2nd Colloq. Workshop, Köszeg/Hung. 1984, Prog. Phys. 10, 371-403 (1985).
[For the entire collection see Zbl 0557.00018.] The set of completely analytical interactions is defined by thirteen (!) properties (in the revised version of the article). The striking point here is the fact that all of them are equivalent - and that alone shows the class to be natural. The corresponding Gibbs field is unique and possesses all properties of high temperature fields: exponential decay of correlations, analyticity of free energy, local limit theorem and so on. The proofs of these properties to hold is much easier for CA interactions compared with the usual approach based on cluster expansion, and it seems that CA interactions usually include those for which cluster expansion converges. The above conditions are of four types. We give below some examples. The interaction $U\in CA$ if: 1. The partition function $Z(\tilde U;V,{\bar \sigma})\ne 0$ for all complex perturbation $\tilde U$ of U such that $\Vert U-\tilde U\Vert <\epsilon$ for some $\epsilon >0$, uniformly in volume V and boundary condition ${\bar \sigma}$; or 2. The conditional Gibbs distribution in V, being projected into the subvolume $W\subset V$, depends exponentially week on the values of the corresponding boundary condition ${\bar \sigma}$ in points $t\in \partial V$, which are far from W; or 3. The semiinvariants of conditional Gibbs distribution in V decay exponentially, with the estimate uniform in V, ${\bar \sigma}$; or 4. The logarithm of the partition function $Z(V,{\bar\sigma})$ has the asymptotic decomposition into volume and boundary terms - -PLUS the following restriction: U can be connected with zero interaction by a path U(t), $0\le t\le 1$, such that the same condition holds for all interactions U(t) as well. (For zero interaction they are easy.)

46N99Miscellaneous applications functional analysis
82B05Classical equilibrium statistical mechanics (general)