Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On $P$-adic Ising-Vannimenus model on an arbitrary order Cayley tree   In this paper, we continue an investigation of the $p$-adic Ising-Vannimenus
model on the Cayley tree of an arbitrary order $k$ $(k\geq 2$). We prove the
existence of $p$-adic quasi Gibbs measures by analyzing fixed points of
multi-dimensional $p$-adic system of equations. We are also able to show the
uniqueness of translation-invariant $p$-adic Gibbs measure. Finally, it is
established the existence of the phase transition for the Ising-Vannimenus
model depending on the order $k$ of the Cayley tree and the prime $p$. Note
that the methods used in the paper are not valid in the real setting, since all
of them are based on $p$-adic analysis and $p$-adic probability measures.

36
The categories of this paper are: cmp-lg
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The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
19
The categories of this paper are: eess
12
The categories of this paper are: nucl-th
11
The categories of this paper are: hep-lat
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
25
The categories of this paper are: patt-sol
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The categories of this paper are: chem-ph
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int
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The categories of this paper are: comp-gas
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc