Random Triangulations and Emergent Conformal Structure Discrete - PowerPoint PPT Presentation

Random Triangulations and Emergent Conformal Structure Discrete Differential Geometry, Berlin, July 2007 Ken Stephenson, University of Tennessee . – p.1/67

Outline Background on Circle Packing: Giving geometry to combinatorics Classical Conformal Geometry — and Companion Notions Discrete Conformal Geometry Emergent Conformal Geometry Applications . – p.2/67

Outline Background on Circle Packing: Giving geometry to combinatorics Classical Conformal Geometry — and Companion Notions Discrete Conformal Geometry Emergent Conformal Geometry Applications . – p.2/67

1. Circle Packing Basics . – p.3/67

Circle Packing Structures Def: A circle packing P is a configuration of circles with a specified pattern of tangencies. (Initiated by Koebe, Andreev, and (principally) Bill Thurston.) . – p.4/67

Circle Packing Structures Def: A circle packing P is a configuration of circles with a specified pattern of tangencies. (Initiated by Koebe, Andreev, and (principally) Bill Thurston.) The pattern of P is given by a (simplicial) complex K which triangulates an oriented topological surface. The configuration P has a circle C v for each vertex v ∈ K . When � u, v � is an edge of K , then C u and C v are tangent. When � u, v, w � is an oriented face of K , then � C u , C v , C w � is an oriented triple of mutually tangent circles. The radii are given in a label R . (Computing R is where the work goes; compatibility depends on angle sums — centers are secondary.) Typical operation: given K − → compute R − → lay out P . – p.4/67

Packing Plasticity The theory has been extended with boundary conditions and branching (not pertinent here) to give amazing plasticity. . – p.5/67

Packing Plasticity The theory has been extended with boundary conditions and branching (not pertinent here) to give amazing plasticity. Common Combinatorics K Specified boundary radii Disc Specified Boundary angles Sphere "Maximal" packing P_K . – p.5/67

Thurston’s Excellent Idea Ω . – p.6/67

Thurston’s Excellent Idea Ω . – p.7/67

Thurston’s Excellent Idea Ω . – p.8/67

Thurston’s Excellent Idea f Ω . – p.9/67

Thurston’s Excellent Idea f Ω Thurston’s Conjecture: If increasingly fine hexagonal circle packings P n are used in Ω and the maps f n are appropriately normalized, then f n converges uniformly on compact subsets of D to the classical conformal mapping F : D − → Ω . . – p.9/67

f 1 Ω P P K 1 f 2n P P K 2n 2n f 4n P 4n P K 4n . – p.10/67

Enabling Theory Koebe-Andreev-Thurston Theorem: For any triangulation K of a sphere, there exists an associated univalent circle packing e P of the Riemann sphere, unique up to Möbius transformations . – p.11/67

Enabling Theory Koebe-Andreev-Thurston Theorem: For any triangulation K of a sphere, there exists an associated univalent circle packing e P of the Riemann sphere, unique up to Möbius transformations Thurston’s Conjecture on convergence to conformal mapping: proved by Burt Rodin and Dennis Sullivan using quasiconformal mapping theory. . – p.11/67

Enabling Theory Koebe-Andreev-Thurston Theorem: For any triangulation K of a sphere, there exists an associated univalent circle packing e P of the Riemann sphere, unique up to Möbius transformations Thurston’s Conjecture on convergence to conformal mapping: proved by Burt Rodin and Dennis Sullivan using quasiconformal mapping theory. Convergence extended by various authors to more general combinatorics, still using quasiconformal theory . – p.11/67

Enabling Theory Koebe-Andreev-Thurston Theorem: For any triangulation K of a sphere, there exists an associated univalent circle packing e P of the Riemann sphere, unique up to Möbius transformations Thurston’s Conjecture on convergence to conformal mapping: proved by Burt Rodin and Dennis Sullivan using quasiconformal mapping theory. Convergence extended by various authors to more general combinatorics, still using quasiconformal theory Culminating in a theorem of Zheng-Xu He and Oded Schramm that removes the quasiconformal theory, implying: . – p.11/67

Enabling Theory Koebe-Andreev-Thurston Theorem: For any triangulation K of a sphere, there exists an associated univalent circle packing e P of the Riemann sphere, unique up to Möbius transformations Thurston’s Conjecture on convergence to conformal mapping: proved by Burt Rodin and Dennis Sullivan using quasiconformal mapping theory. Convergence extended by various authors to more general combinatorics, still using quasiconformal theory Culminating in a theorem of Zheng-Xu He and Oded Schramm that removes the quasiconformal theory, implying: The Koebe-Andreev-Thurston Theorem is equivalent to the Riemann Mapping Theorem for plane domains. . – p.11/67

And ... . – p.12/67

And ... Circle packings, refinements, and convergence results are extended to Riemann surfaces . – p.12/67

And ... Circle packings, refinements, and convergence results are extended to Riemann surfaces With notion of branch points, circle packings provide wide ranging “discrete analytic functions”: discrete rational maps, inner functions, entire functions, covering maps, etc. . – p.12/67

And ... Circle packings, refinements, and convergence results are extended to Riemann surfaces With notion of branch points, circle packings provide wide ranging “discrete analytic functions”: discrete rational maps, inner functions, entire functions, covering maps, etc. Indeed, a fairly comprehensive theory of discrete analytic functions emerges: . – p.12/67

And ... Circle packings, refinements, and convergence results are extended to Riemann surfaces With notion of branch points, circle packings provide wide ranging “discrete analytic functions”: discrete rational maps, inner functions, entire functions, covering maps, etc. Indeed, a fairly comprehensive theory of discrete analytic functions emerges: Circle Packing: “quantum” complex analysis, classical in the limit. . – p.12/67

And ... Circle packings, refinements, and convergence results are extended to Riemann surfaces With notion of branch points, circle packings provide wide ranging “discrete analytic functions”: discrete rational maps, inner functions, entire functions, covering maps, etc. Indeed, a fairly comprehensive theory of discrete analytic functions emerges: Circle Packing: “quantum” complex analysis, classical in the limit. Important to our story: the existence of practical (and provable) algorithms for computing circle packings and software CirclePack for manipulating them. . – p.12/67

2. Classical Conformal Structure and Companion Notions Conformal maps Brownian motion Harmonic measure Extremal length . – p.13/67

. – p.14/67

f harmonic measure . – p.15/67

f g L H extremal length = L/H harmonic measure . – p.16/67

3. Discrete Conformal Structure — Discrete Companions Discrete conformal maps Random walks Discrete harmonic measure Discrete extremal length . – p.17/67

. – p.18/67

. – p.19/67

f . – p.20/67

f g . – p.21/67

Discrete Conformal Mappings Definition: A discrete conformal mapping is a map f : Q − → P between univalent circle packings associated with the same complex K . K Q P f . – p.22/67

Discrete Conformal Mappings Definition: A discrete conformal mapping is a map f : Q − → P between univalent circle packings associated with the same complex K . K Q P f Proposal: A discrete conformal structure for an oriented topological surface S is a simplicial complex K which triangulates S . . – p.22/67

4. Emergent Conformal Structure A random idea Experimental support Intuition What is a “random” triangulation . – p.23/67

Packing Triangulations . – p.24/67

Packing Triangulations − → Random Triangulations . – p.25/67

Random Triangulations — and Companions Random discrete maps Random walks Discrete harmonic measure Discrete extremal length . – p.26/67

f g . – p.27/67

Emergent Conformal Structure Setting: Let Ω be a bounded simply connected plane domain, z 1 , z 2 ∈ Ω , and let F : Ω − → D be the unique conformal mapping with F ( z 1 ) = 0 , F ( z 2 ) > 0 . . – p.28/67

Emergent Conformal Structure Setting: Let Ω be a bounded simply connected plane domain, z 1 , z 2 ∈ Ω , and let F : Ω − → D be the unique conformal mapping with F ( z 1 ) = 0 , F ( z 2 ) > 0 . Random Maps: For n >> 1 , define a “random” map f n : Ω − → D as follows: Select a random triangulation K n of Ω having n vertices Compute the maximal circle packing P n for K n (in D ) Define f n : K n − → carrier ( P n ) (An appropriate φ ∈ Auto ( D ) applied to P n ensures f n ( z 1 ) = 0 , f n ( z 2 ) > 0 ) . – p.28/67