Uniqueness of Gibbs fields with unbounded random interactions on unbounded degree graphs
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Uniqueness of Gibbs fields with unbounded random interactions on unbounded degree graphs 1 · Yuri Kozitsky1 Dorota Kepa-Maksymowicz ¸
Received: 31 October 2019 / Revised: 3 May 2020 / Accepted: 18 June 2020 © The Author(s) 2020
Abstract Gibbs fields with continuous spins are studied, the underlying graphs of which can be of unbounded vertex degree and the spin–spin pair interaction potentials are random and unbounded. A high-temperature uniqueness of such fields is proved to hold under the following conditions: (a) the vertex degree is of tempered growth, i.e., controlled in a certain way; (b) the interaction potentials Wx y are such that Wx y = supσ,σ |Wx y (σ, σ )| are independent (for different edges x, y), identically distributed and exponentially integrable random variables. Keywords DLR equation · Specification · Quenched state · Unbounded disorder · High-temperature uniqueness · Animal Mathematics Subject Classification 60K35 · 82B20
1 Introduction and setup In this work, we continue studying quenched Gibbs fields with unbounded disorder [1] focusing on their high-temperature uniqueness. Permanent interest to this problem may arise from the fact that—even in the simplest case of an Ising model with unbounded random interactions—due to so-called Griffiths’ singularities [2–4] at arbitrarily high temperatures there may exist arbitrarily large subsets of the underlying lattice, in which the spins are strongly correlated. Our work can be considered as a continuation of the research performed in [2,5–8]. Novel aspects here are: (a) instead of finite-valued spins we allow them to take values in arbitrary Polish spaces; (b) instead of regular underlying graphs (like Zd ) we employ graphs of unbounded vertex degree, cf. [9,10].
Supported in part by National Science Centre (NCN), Poland, Grant 2017/25/B/ST1/00051.
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Yuri Kozitsky [email protected] Institute of Mathematics, Maria Curie-Sklodowska University, 20-031 Lublin, Poland
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D. K¸epa-Maksymowicz, Y. Kozitsky
Markov random fields on such underlying graphs [11] naturally appear in the following physical applications: (a) random (also quantum) fields on Riemannian manifolds, see e.g., [12]; (b) thermodynamic states of interacting oscillators based on networks— so-called oscillating networks [13]; (c) thermodynamic states of continuous systems with spins, like ferrofluids, where (random) geometric graphs are used as underlying graphs [14,15]. We refer the interested reader to [10] for more details on this matter. Let G = (V, E) be a countably infinite graph with vertex and edge sets V and E, respectively. We assume that G is connected, has no loops and multiple edges, and ∀x ∈ V
n(x) := #{y ∈ V : y ∼ x} < ∞,
(1.1)
where x ∼ y denotes adjacency. In the latter case, we write the corresponding edge as x, y. Let S be a Polish space—a complete and separable metric space. By B(S) we denote the corresponding Borel σ -field, whereas F will stand for the smallest σ -field of subsets of S V such that the maps S V σ → σ (x) ∈ S are measu
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