Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The $Z$-invariant massive Laplacian on isoradial graphs   We introduce a one-parameter family of massive Laplacian operators
$(\Delta^{m(k)})_{k\in[0,1)}$ defined on isoradial graphs, involving elliptic
functions. We prove an explicit formula for the inverse of $\Delta^{m(k)}$, the
massive Green function, which has the remarkable property of only depending on
the local geometry of the graph, and compute its asymptotics. We study the
corresponding statistical mechanics model of random rooted spanning forests. We
prove an explicit local formula for an infinite volume Boltzmann measure, and
for the free energy of the model. We show that the model undergoes a second
order phase transition at $k=0$, thus proving that spanning trees corresponding
to the Laplacian introduced by Kenyon are critical. We prove that the massive
Laplacian operators $(\Delta^{m(k)})_{k\in(0,1)}$ provide a one-parameter
family of $Z$-invariant rooted spanning forest models. When the isoradial graph
is moreover $\mathbb{Z}^2$-periodic, we consider the spectral curve of the
characteristic polynomial of the massive Laplacian. We provide an explicit
parametrization of the curve and prove that it is Harnack and has genus $1$. We
further show that every Harnack curve of genus $1$ with
$(z,w)\leftrightarrow(z^{-1},w^{-1})$ symmetry arises from such a massive
Laplacian.

31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
25
The categories of this paper are: patt-sol
36
The categories of this paper are: cmp-lg
17
The categories of this paper are: q-fin
27
The categories of this paper are: comp-gas
20
The categories of this paper are: acc-phys
26
The categories of this paper are: bayes-an
16
The categories of this paper are: stat
18
The categories of this paper are: econ
15
The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
30
The categories of this paper are: q-alg
1
The categories of this paper are: cs
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con