Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The 1-2 model   The current paper is a short review of rigorous results for the 1-2 model.
The 1-2 model on the hexagonal lattice is a model of statistical mechanics in
which each vertex is constrained to have degree either 1 or 2. It was proposed
in a study by Schwartz and Bruck of constrained coding systems, and is strongly
connected to the dimer model on a decoration of the lattice, and to an enhanced
Ising model and an associated polygon model on the graph derived from the
hexagonal lattice by adding a further vertex in the middle of each edge.
  The general 1-2 model possesses three parameters $a$, $b$, $c$. The
fundamental technique is to represent probabilities of interest as ratios of
counts of dimer coverings of certain associated graphs, and to apply the
Pfaffian method of Kasteleyn, Fisher, and Temperley.
  Of special interest is the existence (or not) of phase transitions. It turns
out that all clusters of the infinite-volume limit are almost surely finite. On
the other hand, the existence (with strictly positive probability) of infinite
`homogeneous' clusters, containing vertices of given type, depends on the
values of the parameters.
  A further type of phase transition emerges in the study of the two-edge
correlation function, and in this case the critical surface may be found
explicitly. For instance, when $a \ge b \ge c > 0$, the surface given by $\sqrt
a = \sqrt b + \sqrt c$ is critical.

33
The categories of this paper are: plasm-ph
36
The categories of this paper are: cmp-lg
26
The categories of this paper are: bayes-an
4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
8
The categories of this paper are: hep-ex
27
The categories of this paper are: comp-gas
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
17
The categories of this paper are: q-fin
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
29
The categories of this paper are: funct-an
0
The categories of this paper are: hep-ph
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con