Convergence of the probabilistic interpretation of modulus

Given a Jordan domain $\Omega\subset\mathbb{C}$ and two arcs $A, B$ on $\partial\Omega$, the modulus of the curve family connecting $A$ and $B$ in $\Omega$ is famously related, via the conformal map $\phi$ mapping $\Omega$ to a rectangle $R=[0,L]\times[0,1]$ so that $A$ and $B$ are sent to the vertical sides, to the corresponding modulus in $R$. 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 curves arising from an orthodiagonal approximation of $\Omega$. Moreover, we show that these curves carry a natural probability mass function (pmf) deriving from the theory of discrete modulus and that these pmf's converge to the uniform distribution 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.