Scaling limits of random dissections Igor Kortchemski (work with N. - PowerPoint PPT Presentation

Scaling limits of random dissections Igor Kortchemski (work with N. Curien and B. Haas) DMA – École Normale Supérieure SIAM Discrete Mathematics - June 2014

Motivation Definition: Boltzmann dissections Theorem Applications Proof Outline O. Motivation I. Definition II. Theorem III. Application IV. Proof Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 2 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof O. Motivation I. Definition II. Theorem III. Application IV. Proof Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 3 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof Scaling limits of random maps Goal: choose a random map M n of size n and study its global geometry as n → ∞ . Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 4 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof Scaling limits of random maps Goal: choose a random map M n of size n and study its global geometry as n → ∞ . If possible, show that a continuous limit exists (Chassaing & Schaeffer ’04). Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 4 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof Scaling limits of random maps Goal: choose a random map M n of size n and study its global geometry as n → ∞ . If possible, show that a continuous limit exists (Chassaing & Schaeffer ’04). Problem (Schramm at ICM ’06): Let T n be a random uniform triangulation of the sphere with n vertices. Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 4 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof Scaling limits of random maps Goal: choose a random map M n of size n and study its global geometry as n → ∞ . If possible, show that a continuous limit exists (Chassaing & Schaeffer ’04). Problem (Schramm at ICM ’06): Let T n be a random uniform triangulation of the sphere with n vertices. View T n as a compact metric space. Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 4 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof Scaling limits of random maps Goal: choose a random map M n of size n and study its global geometry as n → ∞ . If possible, show that a continuous limit exists (Chassaing & Schaeffer ’04). Problem (Schramm at ICM ’06): Let T n be a random uniform triangulation of the sphere with n vertices. View T n as a compact metric space. Show that n − 1 / 4 · T n converges towards a random compact metric space (the Brownian map) Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 4 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof Scaling limits of random maps Goal: choose a random map M n of size n and study its global geometry as n → ∞ . If possible, show that a continuous limit exists (Chassaing & Schaeffer ’04). Problem (Schramm at ICM ’06): Let T n be a random uniform triangulation of the sphere with n vertices. View T n as a compact metric space. Show that n − 1 / 4 · T n converges towards a random compact metric space (the Brownian map), in distribution for the Gromov–Hausdorff topology. Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 4 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof Scaling limits of random maps Goal: choose a random map M n of size n and study its global geometry as n → ∞ . If possible, show that a continuous limit exists (Chassaing & Schaeffer ’04). Problem (Schramm at ICM ’06): Let T n be a random uniform triangulation of the sphere with n vertices. View T n as a compact metric space. Show that n − 1 / 4 · T n converges towards a random compact metric space (the Brownian map), in distribution for the Gromov–Hausdorff topology. Solved by Le Gall in 2011. Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 4 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof Scaling limits of random maps Goal: choose a random map M n of size n and study its global geometry as n → ∞ . If possible, show that a continuous limit exists (Chassaing & Schaeffer ’04). Problem (Schramm at ICM ’06): Let T n be a random uniform triangulation of the sphere with n vertices. View T n as a compact metric space. Show that n − 1 / 4 · T n converges towards a random compact metric space (the Brownian map), in distribution for the Gromov–Hausdorff topology. Solved by Le Gall in 2011. (see Le Gall’s proceeding at ICM ’14 for more information and references) Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 4 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof A simulation of the Brownian CRT Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 5 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof Random maps having the CRT as a scaling limit Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 6 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof Random maps having the CRT as a scaling limit ◮ Albenque & Marckert (’07): Uniform stack triangulations with 2 n faces Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 6 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof Random maps having the CRT as a scaling limit ◮ Albenque & Marckert (’07): Uniform stack triangulations with 2 n faces Figure : Figure by Albenque & Marckert Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 6 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof Random maps having the CRT as a scaling limit ◮ Albenque & Marckert (’07): Uniform stack triangulations with 2 n faces ◮ Janson & Steffánsson (’12): Boltzmann-type bipartite maps with n edges, having a face of macroscopic degree. Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 6 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof Random maps having the CRT as a scaling limit ◮ Albenque & Marckert (’07): Uniform stack triangulations with 2 n faces ◮ Janson & Steffánsson (’12): Boltzmann-type bipartite maps with n edges, having a face of macroscopic degree. Figure : Figure by Janson & Steffánsson Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 6 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof Random maps having the CRT as a scaling limit ◮ Albenque & Marckert (’07): Uniform stack triangulations with 2 n faces ◮ Janson & Steffánsson (’12): Boltzmann-type bipartite maps with n edges, having a face of macroscopic degree. ◮ Bettinelli (’11): Uniform quadrangulations with n faces with fixed boundary ≫ √ n . Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 6 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof O. Motivation I. Definition: Boltzmann dissections II. Theorem III. Application IV. Proof Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 7 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof Dissections 2 i πj n ( j = 0, 1, . . . , n − 1 ) . Let P n be the polygon whose vertices are e Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 8 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof Dissections 2 i πj n ( j = 0, 1, . . . , n − 1 ) . Let P n be the polygon whose vertices are e A dissection of P n is the union of the sides P n and of a collection of noncrossing diagonals. Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 8 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof Dissections 2 i πj n ( j = 0, 1, . . . , n − 1 ) . Let P n be the polygon whose vertices are e A dissection of P n is the union of the sides P n and of a collection of noncrossing diagonals. � We will view dissections as compact metric spaces . Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 8 / 475

Motivation Definition: Boltzmann dissections Theorem Applications Proof Dissections à la Boltzmann Let D n be the set of all dissections of P n . Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 9 / 147

Motivation Definition: Boltzmann dissections Theorem Applications Proof Dissections à la Boltzmann Let D n be the set of all dissections of P n . Fix a sequence ( µ i ) i � 2 of nonnegative real numbers. Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 9 / 147

Motivation Definition: Boltzmann dissections Theorem Applications Proof Dissections à la Boltzmann Let D n be the set of all dissections of P n . Fix a sequence ( µ i ) i � 2 of nonnegative real numbers. Associate a weight π ( ω ) with every dissection ω ∈ D n : � π ( ω ) = µ deg ( f )− 1 . f faces of ω Igor Kortchemski (ÉNS Paris) Scaling limits of random dissections 9 / 147