First-passage percolation in random triangulations Jean-François Le Gall (joint with Nicolas Curien) Université Paris-Sud Orsay and Institut universitaire de France Random Geometry and Physics, Paris 2016 Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 1 / 30
To define a canonical random geometry in two dimensions (motivations from physics: 2D quantum gravity) Replace the sphere S 2 by a discretization, namely a graph drawn on the sphere (= planar map). Choose such a planar map uniformly at random in a suitable class and equip its vertex set with the graph distance. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 2 / 30
To define a canonical random geometry in two dimensions (motivations from physics: 2D quantum gravity) Replace the sphere S 2 by a discretization, namely a graph drawn on the sphere (= planar map). Choose such a planar map uniformly at random in a suitable class and equip its vertex set with the graph distance. Let the size of the graph tend to infinity and pass to the limit after rescaling to get a random metric space: the Brownian map. This convergence is robust: it still holds if we make local modifications of the graph distance. (Assign random lengths to the edges: first-passage percolation distance.) Goal of the lecture: Explain this robustness property. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 2 / 30
1. The geometry of large random planar maps Definition A planar map is a proper embedding of a finite connected graph into the two-dimensional sphere (considered up to orientation-preserving homeomorphisms of the sphere). Loops and multiple edges allowed. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 3 / 30
1. The geometry of large random planar maps Definition A planar map is a proper embedding of a finite connected graph into the two-dimensional sphere (considered up to orientation-preserving homeomorphisms of the sphere). Loops and multiple edges allowed. Faces = connected components of root vertex the complement of edges root p -angulation: edge each face is incident to p edges p = 3: triangulation p = 4: quadrangulation Rooted map: distinguished oriented edge A rooted triangulation with 20 faces Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 3 / 30
A large triangulation of the sphere Can we get a continuous model out of this ? Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 4 / 30
Planar maps as metric spaces 1 0 1 2 M planar map 4 V ( M ) = set of vertices of M d gr graph distance on V ( M ) 1 2 ( V ( M ) , d gr ) is a (finite) metric space 3 2 In blue : distances from root Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 5 / 30
Planar maps as metric spaces 1 0 1 2 M planar map 4 V ( M ) = set of vertices of M d gr graph distance on V ( M ) 1 2 ( V ( M ) , d gr ) is a (finite) metric space 3 2 In blue : distances from root M p n = { rooted p − angulations with n faces } M p n is a finite set ( finite number of possible “shapes” ) Choose M n uniformly at random in M p n . Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 5 / 30
Planar maps as metric spaces 1 0 1 2 M planar map 4 V ( M ) = set of vertices of M d gr graph distance on V ( M ) 1 2 ( V ( M ) , d gr ) is a (finite) metric space 3 2 In blue : distances from root M p n = { rooted p − angulations with n faces } M p n is a finite set ( finite number of possible “shapes” ) Choose M n uniformly at random in M p n . View ( V ( M n ) , d gr ) as a random variable with values in K = { compact metric spaces, modulo isometries } which is equipped with the Gromov-Hausdorff distance. (A sequence ( E n ) of compact metric spaces converges if one can embed all E n ’s isometrically in the same big space E so that they converge for the Hausdorff metric on compact subsets of E .) Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 5 / 30
The Brownian map M p n = { rooted p − angulations with n faces } M n uniform over M p n , V ( M n ) vertex set of M n , d gr graph distance Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 6 / 30
The Brownian map M p n = { rooted p − angulations with n faces } M n uniform over M p n , V ( M n ) vertex set of M n , d gr graph distance Theorem (LG 2013, Miermont 2013) Suppose that either p = 3 (triangulations) or p ≥ 4 is even. Set 9 � 1 / 4 � c 3 = 6 1 / 4 , c p = if p is even. p ( p − 2 ) Then, ( d ) ( V ( M n ) , c p n − 1 / 4 d gr ) n →∞ ( m ∞ , D ∗ ) − → in the Gromov-Hausdorff sense. The limit ( m ∞ , D ∗ ) is a random compact metric space that does not depend on p (universality) and is called the Brownian map (after Marckert-Mokkadem). Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 6 / 30
The Brownian map M p n = { rooted p − angulations with n faces } M n uniform over M p n , V ( M n ) vertex set of M n , d gr graph distance Theorem (LG 2013, Miermont 2013) Suppose that either p = 3 (triangulations) or p ≥ 4 is even. Set 9 � 1 / 4 � c 3 = 6 1 / 4 , c p = if p is even. p ( p − 2 ) Then, ( d ) ( V ( M n ) , c p n − 1 / 4 d gr ) n →∞ ( m ∞ , D ∗ ) − → in the Gromov-Hausdorff sense. The limit ( m ∞ , D ∗ ) is a random compact metric space that does not depend on p (universality) and is called the Brownian map (after Marckert-Mokkadem). Remarks • The case p = 3 (triangulations) solves a question of Schramm (2006) • Extensions to other classes of random planar maps: Abraham, Addario-Berry-Albenque, Beltran-LG, Bettinelli-Jacob-Miermont, etc. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 6 / 30
Two properties of the Brownian map Theorem (Hausdorff dimension) dim ( m ∞ , D ∗ ) = 4 a.s. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 7 / 30
Two properties of the Brownian map Theorem (Hausdorff dimension) dim ( m ∞ , D ∗ ) = 4 a.s. Theorem (topological type) Almost surely, ( m ∞ , D ∗ ) is homeomorphic to the 2 -sphere S 2 . Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 7 / 30
Construction of the Brownian map The Brownian map ( m ∞ , D ∗ ) is constructed as a quotient space of Aldous’ Brownian continuum random tree (the CRT), for an equivalence relation defined in terms of Brownian labels assigned to the vertices of the CRT. Two points a and b of the CRT are glued if they have the same label and if one can go from a to b around the tree (clockwise or counterclockwise) meeting only points with greater label. A simulation of the CRT Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 8 / 30
Recent progress Miller, Sheffield (2015-2016) have developed a program aiming to relate the Brownian map with Liouville quantum gravity: new construction of the Brownian map via the random growth process called Quantum Loewner Evolution (an analog of the celebrated SLE processes studied by Lawler, Schramm and Werner) this construction makes it possible to equip the Brownian map with a conformal structure, and in fact to show that this conformal structure is determined by the Brownian map. Still open: To show that this conformal structure also arises in the limit of conformal embeddings of discrete graphs (e.g. triangulations) that can be produced via circle packings or the uniformization theorem of the theory of Riemann surfaces. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 9 / 30
2. Modifications of distances on random planar maps (joint work with Nicolas Curien, arXiv:1511.04264) Assign i.i.d. random weights (lengths) w e to the edges of a (random) planar map M . Define the weight w ( γ ) of a path γ as the sum of the weights of the edges it contains. The first passage percolation distance d FPP is defined on the vertex set V ( M ) by d FPP ( v , v ′ ) = inf { w ( γ ) : γ path from v to v ′ } . Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 10 / 30
2. Modifications of distances on random planar maps (joint work with Nicolas Curien, arXiv:1511.04264) Assign i.i.d. random weights (lengths) w e to the edges of a (random) planar map M . Define the weight w ( γ ) of a path γ as the sum of the weights of the edges it contains. The first passage percolation distance d FPP is defined on the vertex set V ( M ) by d FPP ( v , v ′ ) = inf { w ( γ ) : γ path from v to v ′ } . Goal: In large scales, d FPP behaves like the graph distance d gr (asymptotically, balls for d FPP are close to balls for d gr ). This is not expected to be true in deterministic lattices such as Z d , but random planar maps are in a sense more isotropic. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 10 / 30
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