# A model with uncountable set of spin values on a Cayley tree: phase transitions

A model with uncountable set of spin values on a Cayley tree: phase transitions In this paper we consider a model with nearest-neighbor interactions and with the set [0,1] of spin values, on a Cayley tree of order two. This model depends on two parameters $$n\in \mathbb N$$ n ∈ N and $$\theta \in [0,1)$$ θ ∈ [ 0 , 1 ) . We prove that if $$0 \le \theta \le \frac{2n+3}{2(2n+1)}$$ 0 ≤ θ ≤ 2 n + 3 2 ( 2 n + 1 ) , then for the model there exists a unique translational-invariant Gibbs measure; If $$\frac{2n+3}{2(2n+1)}< \theta <1$$ 2 n + 3 2 ( 2 n + 1 ) < θ < 1 , then there are three translational-invariant Gibbs measures (i.e. phase transition occurs). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# A model with uncountable set of spin values on a Cayley tree: phase transitions

, Volume 21 (3) – Sep 10, 2016
7 pages

/lp/springer_journal/a-model-with-uncountable-set-of-spin-values-on-a-cayley-tree-phase-ioFE3mkkP9
Publisher
Springer International Publishing
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-016-0445-x
Publisher site
See Article on Publisher Site

### Abstract

In this paper we consider a model with nearest-neighbor interactions and with the set [0,1] of spin values, on a Cayley tree of order two. This model depends on two parameters $$n\in \mathbb N$$ n ∈ N and $$\theta \in [0,1)$$ θ ∈ [ 0 , 1 ) . We prove that if $$0 \le \theta \le \frac{2n+3}{2(2n+1)}$$ 0 ≤ θ ≤ 2 n + 3 2 ( 2 n + 1 ) , then for the model there exists a unique translational-invariant Gibbs measure; If $$\frac{2n+3}{2(2n+1)}< \theta <1$$ 2 n + 3 2 ( 2 n + 1 ) < θ < 1 , then there are three translational-invariant Gibbs measures (i.e. phase transition occurs).

### Journal

PositivitySpringer Journals

Published: Sep 10, 2016

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations