Convergence of the Probabilistic Interpretation of Modulus

TL;DR

This paper demonstrates that extremal curves related to the modulus of curve families in Jordan domains can be approximated by discrete paths from orthodiagonal maps, with convergence of associated probability measures and discrete modulus to their continuous counterparts.

Contribution

It introduces an algorithm for path decomposition in planar graphs and proves the convergence of discrete modulus and probability measures to their continuous analogs in orthodiagonal approximations.

Findings

Discrete extremal curves approximate continuous extremal curves.

Probability mass functions on discrete paths converge to transverse measures.

Discrete modulus converges to continuous modulus in orthodiagonal settings.

Abstract

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 modulus of the curve family connecting the vertical sides in the rectangle $R=[0,1]\times[0,m]$. Also, $m>0$ is the unique value such that there is a conformal map $\psi$ mapping $\Omega$ to ${\rm int}(R)$ so that $\psi$ 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 $\psi$ 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, 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-crossing paths converges to the continuous modulus.