Convergence of the Probabilistic Interpretation of Modulus

Abstract

Given a Jordan domain $Ω\subset\mathbb{C}$ and two disjoint arcs $A, B$ on $\partialΩ$, the modulus $m$ of the curve family connecting $A$ and $B$ in $Ω$ 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 $ψ$ mapping $Ω$ to ${\rm int}(R)$ so that $ψ$ extends continuously to a homeomorphism of $\partial Ω$ 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 $Ω$ via $ψ$ yields a family of extremal curves (also known as horizontal trajectories) connecting $A$ to $B$ in $Ω$. In this paper, we show that these extremal curves can be approximated by some discrete paths arising from an orthodiagonal approximation of $Ω$. 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.

Figures (9)

• Figure 1: The graph for Example \ref{['RotatedRectangle']} with $L=2$, $n=3$, the vertices of $A$ shown in red, and the vertices of $B$ shown in blue.
• Figure 2: Top left: An example of an orthodiagonal map with the bipartation of the vertices. Top right: The topological boundary of the map decomposed into four paths: $S_1, S_2$ (solid) and $T_1,T_2$ (dashed). Bottom left: The primal graph $G^\bullet$ (black/red vertices and solid edges) with $\partial V^\bullet$ shown in red and the dual graph $G^\circ$ (white vertices and dashed edges). Bottom right: Finite volume interpretation of the edge weights assigned to an orthodiagonal map.
• Figure 3: This is an example of Remark \ref{['rem:rect-pack']} in the case of orthodiagonal maps. For the map on the left, the primal graph (in blue) is given by the nodes $A,B,H,J,K,N,P,U$. Here we compute the modulus $M$ of the family of curves connecting the set $\{A,H\}$ to the set $\{U\}$. In this case the quantity $\tilde{h}(e)$ coincides with the harmonic function defined on the dual graph with $\tilde{h}=0$ on the set $\{C,L,T\}$ and $\tilde{h}=M$ on the set $\{I,R\}$.
• Figure 4: Left: a Delaunay triangulation of a cross domain; Center: the triangulation shown together with its dual Voronoi graph; Right: the corresponding orthodiagonal map.
• Figure 5: Left: non-crossing paths for a triangulation of the cross domain; Center: the corresponding curvess as computed by the Schwarz-Christoffel Toolbox in Matlab; Right: the superposition of both sets of curves.

Theorems & Definitions (75)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6: Theorem 2 of acfpc:ampa2019
• Corollary 7
• proof
• Proposition 8
• proof