Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gibbs measures and free energies of the Ising-Vannimenus Model on the
  Cayley tree   In this paper, we consider Ising-Vannimenus model on a Cayley tree for order
two with competing nearest-neighbor, prolonged next-nearest neighbor
interactions. We stress that the mentioned model was investigated only
numerically, without rigorous (mathematical) proofs. One of the main point of
this paper is to propose a measure-theoretical approach the considered model.
We find certain conditions for the existence of Gibbs measures corresponding to
the model. Then we establish the existence of the phase transition. Moreover,
the free energies of the found Gibbs measures are calculated.

7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
8
The categories of this paper are: hep-ex
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
16
The categories of this paper are: stat
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
24
The categories of this paper are: dg-ga
35
The categories of this paper are: mtrl-th
18
The categories of this paper are: econ
21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc