Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Convergence of the Probabilistic Interpretation of Modulus   Given a Jordan domain $\Omega\subset\mathbb{C}$ and two disjoint arcs $A, B$
on $\partial\Omega$, the modulus $m$ of the curve family connecting $A$ and $B$
in $\Omega$ is equal to the corresponding modulus in the rectangle
$R=[0,1]\times[0,m]$, and $m>0$ has the property that there is a conformal map
$\phi$ mapping $\Omega$ to ${\rm int}(R)$ so that $\phi$ extends continuously
to a homeomorphism of $\partial \Omega$ onto $\partial R$ and the arcs $A$ and
$B$ are sent to the vertical sides of $R$. Moreover, in the case of the
rectangle the family of horizontal segments connecting the two sides has the
same modulus as the entire connecting family. Pulling these segments back to
$\Omega$ via $\phi$ yields a family of extremal curves (also known as
horizontal trajectories) connecting $A$ to $B$ in $\Omega$. In this paper, we
show that these extremal curves can be approximated by some discrete paths
arising from an orthodiagonal approximation of $\Omega$. Moreover, we show that
there is a natural probability mass function (pmf) on these paths, deriving
from the theory of discrete modulus, which converges to the transverse measure
on the set of extremal curves. The key ingredient is an algorithm that, for an
embedded planar graph, takes the current flow between two sets of nodes $A$ and
$B$, and produces a unique path decomposition with non-crossing paths.
Moreover, some care was taken to adapt recent results for harmonic convergence
on orthodiagonal maps, due to Gurel-Gurevich, Jerison, and Nachmias, to our
context. Finally, we generalize a result of N. Alrayes from the square grid
setting to the orthodiagonal setting, and prove that the discrete modulus of
the approximating non-overlapping paths converges to the continuous modulus.

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The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
16
The categories of this paper are: stat
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
28
The categories of this paper are: alg-geom
29
The categories of this paper are: funct-an
32
The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
19
The categories of this paper are: eess
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
25
The categories of this paper are: patt-sol
1
The categories of this paper are: cs
7
The categories of this paper are: hep-th