Abstract:Gibbs field is one of the central objects of the modern probability, mathematical statistical physics and euclidean quantum field theory. Here we define and study its natural generalization for the case when the space, where the random field is defined is itself random. Moreover, this randomness is not given apriori and independently of the configuration, but rather they depend on each other, and both are given by Gibbs procedure; We call the resulting object a Gibbs family because it parametrizes Gibbs fields… Show more
“…We will show (Lemma 3.1) that the family of conditional distributions constructed from the introduced above Gibbs family is consistent. This gives a ground for the Dobrushin-Lanford-Ruelle construction of Gibbs measure on T ∞ (see [17]).…”
Section: Towards Constructing a Measure On Infinite Triangulationsmentioning
confidence: 99%
“…One can find in Malyshev [17] a general outline of the theory of Gibbs families on spin graphs. We follow his approach and develop it here for a class of planar triangulations.…”
Section: Gibbs Family On Spin-graphsmentioning
confidence: 99%
“…Malyshev [17] gave a solid mathematical ground for such models, developing a general theory of Gibbs fields on random spaces where matter (represented by the configurations of spins) is naturally coupled with the gravity (represented by the graph).…”
We investigate a Gibbs (annealed) probability measure defined on Ising spin configurations on causal triangulations of the plane. We study the region where such measure can be defined and provide bounds on the boundary of this region (critical line). We prove that for any finite random triangulation the magnetization of the central spin is sensitive to the boundary conditions. Furthermore, we show that in the infinite volume limit, the magnetization of the central spin vanishes for values of the temperature high enough.
“…We will show (Lemma 3.1) that the family of conditional distributions constructed from the introduced above Gibbs family is consistent. This gives a ground for the Dobrushin-Lanford-Ruelle construction of Gibbs measure on T ∞ (see [17]).…”
Section: Towards Constructing a Measure On Infinite Triangulationsmentioning
confidence: 99%
“…One can find in Malyshev [17] a general outline of the theory of Gibbs families on spin graphs. We follow his approach and develop it here for a class of planar triangulations.…”
Section: Gibbs Family On Spin-graphsmentioning
confidence: 99%
“…Malyshev [17] gave a solid mathematical ground for such models, developing a general theory of Gibbs fields on random spaces where matter (represented by the configurations of spins) is naturally coupled with the gravity (represented by the graph).…”
We investigate a Gibbs (annealed) probability measure defined on Ising spin configurations on causal triangulations of the plane. We study the region where such measure can be defined and provide bounds on the boundary of this region (critical line). We prove that for any finite random triangulation the magnetization of the central spin is sensitive to the boundary conditions. Furthermore, we show that in the infinite volume limit, the magnetization of the central spin vanishes for values of the temperature high enough.
“…jointly with a field on it) is called random graph grammars [2, 5, 1] and [21,22]. It appears to be quite natural in connection with the emerging new physical theories [16,17,25] and social networks [20]. Namely, if eventually the local space-time appears to be discrete, then the most natural language for it is a graph with some physical fields on it.…”
Section: Dynamics Of Graphs and Of Marked Graphmentioning
confidence: 99%
“…There are many results-examples (by physicists and mathematicians) related to "quantum gravity", see for example [17,25] and references therein.…”
Network (as a general notion) is not a mathematical object -there is no even any definition. However, there is a lot of good rigorous mathematics for well-defined classes of networks. In sections 1-3 we give a short overview of classes of networks which interested the authors for some time. In section 4 we consider in detail a new class of networks, related to markets with many agents.
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