First-passage percolation in random triangulations

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1 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 / 30

2 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 / 30

3 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 / 30

4 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 / 30

5 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. root vertex root edge A rooted triangulation with 20 faces Faces = connected components of the complement of edges p-angulation: each face is incident to p edges p = 3: triangulation p = 4: quadrangulation Rooted map: distinguished oriented edge Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

6 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 / 30

7 Planar maps as metric spaces M planar map V (M) = set of vertices of M d gr graph distance on V (M) (V (M), d gr ) is a (finite) metric space In blue : distances from root Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

8 Planar maps as metric spaces M planar map V (M) = set of vertices of M d gr graph distance on V (M) (V (M), d gr ) is a (finite) metric space 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 / 30

9 Planar maps as metric spaces M planar map V (M) = set of vertices of M d gr graph distance on V (M) (V (M), d gr ) is a (finite) metric space 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 /

10 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 / 30

11 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 ( c 3 = 6 1/4 9 ) 1/4, c p = if p is even. p(p 2) Then, (V (M n ), c p n 1/4 (d) d gr ) (m, D ) n 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 / 30

12 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 ( c 3 = 6 1/4 9 ) 1/4, c p = if p is even. p(p 2) Then, (V (M n ), c p n 1/4 (d) d gr ) (m, D ) n 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 / 30

13 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 / 30

14 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 / 30

15 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 / 30

16 Recent progress Miller, Sheffield ( ) 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 / 30

17 2. Modifications of distances on random planar maps (joint work with Nicolas Curien, arxiv: ) 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 / 30

18 2. Modifications of distances on random planar maps (joint work with Nicolas Curien, arxiv: ) 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 / 30

19 2. Modifications of distances on random planar maps (joint work with Nicolas Curien, arxiv: ) 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. Consequence: The scaling limit of the metric space associated with d FPP will again be the Brownian map! (Universality of the limit!) Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

20 The Uniform Infinite Planar Triangulation (UIPT) Let n be uniformly distributed over {triangulations with n faces}. For every r 1, let B r ( n ) be the ball of radius r in n, defined as the union of all faces incident to a vertex at graph distance strictly less than r from the root vertex ρ. One can prove (Angel-Schramm 2003, Stephenson 2014) that n (d) n in the local limit sense, where is a (rooted) infinite random triangulation called the UIPT for Uniform Infinite Planar Triangulation. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

21 The Uniform Infinite Planar Triangulation (UIPT) Let n be uniformly distributed over {triangulations with n faces}. For every r 1, let B r ( n ) be the ball of radius r in n, defined as the union of all faces incident to a vertex at graph distance strictly less than r from the root vertex ρ. One can prove (Angel-Schramm 2003, Stephenson 2014) that n (d) n in the local limit sense, where is a (rooted) infinite random triangulation called the UIPT for Uniform Infinite Planar Triangulation. The convergence holds in the sense of local limits: for every r and for every fixed planar map M, P(B r ( n ) = M) n P(B r ( ) = M). This is very different from the Gromov-Hausdorff convergence: Here we do no rescaling and thus the limit is a non-compact (infinite) random lattice. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

22 An artistic representation of the UIPT (artist: N. Curien) Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

23 First-passage percolation in the UIPT Assign i.i.d. weights w e with common distribution ν to the edges of the UIPT and consider the associated first-passage percolation distance d FPP. Assume ν is supported on [c, C], where 0 < c C <. For every real r 0, let B FPP r ( ) be the ball of radius r for d FPP. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

24 First-passage percolation in the UIPT Assign i.i.d. weights w e with common distribution ν to the edges of the UIPT and consider the associated first-passage percolation distance d FPP. Assume ν is supported on [c, C], where 0 < c C <. For every real r 0, let B FPP r ( ) be the ball of radius r for d FPP. Theorem There exists a constant c 0 with c c 0 C, such that, for every ε > 0, we have ( lim P sup d FPP (x, y) c 0 d gr (x, y) ) > εr = 0. r x,y B r ( ) In particular, B (1 ε)r/c0 ( ) B FPP r ( ) B (1+ε)r/c0 ( ) with probability tending to 1 as r. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

25 First-passage percolation in the UIPT Assign i.i.d. weights w e with common distribution ν to the edges of the UIPT and consider the associated first-passage percolation distance d FPP. Assume ν is supported on [c, C], where 0 < c C <. For every real r 0, let B FPP r ( ) be the ball of radius r for d FPP. Theorem There exists a constant c 0 with c c 0 C, such that, for every ε > 0, we have ( lim P sup d FPP (x, y) c 0 d gr (x, y) ) > εr = 0. r x,y B r ( ) In particular, B (1 ε)r/c0 ( ) B FPP r ( ) B (1+ε)r/c0 ( ) with probability tending to 1 as r. The ball of radius r for the FPP distance is asymptotically close to the ball of radius r/c 0 for the graph distance. Remark. In general one cannot compute the constant c 0, except in special cases. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

26 The Eden model The Eden growth model corresponds to first-passage percolation with exponential edge weights on the dual map of the UIPT. At rate 1 for each edge of the boundary, one reveals the triangle incident to this edge (the chosen edge is thus uniform on the boundary). Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

27 The Eden model The Eden growth model corresponds to first-passage percolation with exponential edge weights on the dual map of the UIPT. At rate 1 for each edge of the boundary, one reveals the triangle incident to this edge (the chosen edge is thus uniform on the boundary). Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

28 The Eden model The Eden growth model corresponds to first-passage percolation with exponential edge weights on the dual map of the UIPT. At rate 1 for each edge of the boundary, one reveals the triangle incident to this edge (the chosen edge is thus uniform on the boundary). This is an instance of the so-called peeling process (studied by Angel and others) and one can use known results for the volume of the discovered region at step n (Curien-LG) to compute c 0 = 2 3 (the set discovered at time r in the Eden model is close to the ball of radius r/(2 3) for the graph distance on the UIPT) Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

29 First-passage percolation in finite triangulations n is uniformly distributed over {triangulations with n faces} d FPP first-passage percolation distance on V ( n ) defined using weights i.i.d. according to ν (same assumption on ν). Theorem (V ( n ), 6 1/4 n 1/4 d FPP ) (d) n (m, c 0 D ) in the Gromov-Hausdorff sense. Here c 0 is the same constant as in the UIPT case, and (m, D ) is the Brownian map. Idea of the proof: Use absolute continuity arguments to relate large (finite) triangulations to the UIPT, and then apply the theorem about the UIPT. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

30 Other (potential) applications of the method (Without edge weights) can compare the graph distances on a large triangulation (or the UIPT) and on its dual map. In large scales, they are proportional, with a scaling constant c 0 = FPP on quadrangulations (similar techniques should apply). Tutte s bijection relates quadrangulations to general planar maps: compare the graph distances on a quadrangulation and on the associated general planar map. Here one expects c 0 = 1 Tutte s bijection A general planar map The associated quadrangulation (red edges) Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

31 3. Geometric ingredients: Balls and hulls in the UIPT In view of studying the first-passage percolation distance on the UIPT, one needs more information about its geometry. The hull of radius r, denoted by B r ( ), is obtained by filling in the holes in the ball B r ( ). ρ The shaded part is the ball B 2 ( ) (all triangles that contain a vertex at distance 1 from ρ) Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

32 3. Geometric ingredients: Balls and hulls in the UIPT In view of studying the first-passage percolation distance on the UIPT, one needs more information about its geometry. The hull of radius r, denoted by B r ( ), is obtained by filling in the holes in the ball B r ( ). ρ The hull B 2 ( ) is the union of B 2 ( ) and the two holes. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

33 Layers in the UIPT related to Krikun (2005) B l For k < l, the successive layers between B k and B l are the sets B k B j \B j 1 for k < j l. (Here 3 layers) Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

34 Downward triangles in the layers k < l fixed B l For each edge of B l consider the unique triangle (called a downward triangle) containing this edge whose third vertex is on B l 1 B k (We do not get all triangles in the layer B l \B l 1, only those that have an edge in the exterior boundary of the layer) Do the same for the layer B l 1 \B l 2 And so on. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

35 Downward triangles in the layers B l B k Remove the edges not on the downward (colored) triangles. This creates white slots. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

36 The forest coding downward triangles e e B l Can represent the configuration of downward triangles by a forest of trees whose vertices are the edges of B j for all k j l. B k An edge e of B j is the parent of an edge e of B j 1 if the white slot whose boundary contains e is bounded on its right by the downward triangle associated with e. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

37 The forest coding downward triangles B l The forest representing the structure of layers between B k and B l. The roots of trees in the forest are all edges of B l. B k To reconstruct B l \B k one only needs the forest coding the configuration of downward triangles, the triangulations (with boundaries) filling in the white slots. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

38 The Galton-Watson structure Let T 1, T 2,..., T Pl be the forest coding the configuration of downward triangles between B k and B l. Here k < l, P l, resp. P k, is the size of B l, resp. B k. τ 1,..., τ p deterministic forest with height l k and q vertices at height l k. Write V (τ i ) for all vertices of τ i except those at height l k. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

39 The Galton-Watson structure Let T 1, T 2,..., T Pl be the forest coding the configuration of downward triangles between B k and B l. Here k < l, P l, resp. P k, is the size of B l, resp. B k. τ 1,..., τ p deterministic forest with height l k and q vertices at height l k. Write V (τ i ) for all vertices of τ i except those at height l k. Proposition (related to Krikun (2005)) ( ) P (T 1, T 2,..., T Pl ) = (τ 1,..., τ p ) P k = q = h(p) h(q) where n v is the number of offspring of v; v V (τ 1 ) V (τ p) θ is the critical offspring distribution with generating function ( θ(n) x n 1 ) 2; = x h(p) = 4 p (2p)! (p!) 2, for p 1 is the stationary distribution for the Galton-Watson process with offspring distribution θ. θ(n v ) Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

40 4. The half-plane model Construct a triangulation L of the lower half-plane as follows. Each horizontal edge on the line Z { k} belongs to a downward triangle whose third vertex is on the line Z { k 1}. 0 ( 1, 0) (0, 0) (1, 0) The trees characterizing the configuration of downward triangles are independent Galton-Watson trees with offspring distribution θ. Slots are filled in with free triangulations with a boundary (probab. of a given triangul. with n inner vertices is K (12 3) n ). Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

41 First-passage percolation in the half-plane model Assign i.i.d. weights w e to the edges of L, with common distribution ν. Consider the associated first-passage percolation distance d FPP. Proposition Let ρ = (0, 0) be the root and for every k 0, let L k be the horizontal line at vertical coordinate k. Then 1 k d FPP(ρ, L k ) a.s. k c 0 [c, C]. Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

42 First-passage percolation in the half-plane model Assign i.i.d. weights w e to the edges of L, with common distribution ν. Consider the associated first-passage percolation distance d FPP. Proposition Let ρ = (0, 0) be the root and for every k 0, let L k be the horizontal line at vertical coordinate k. Then 1 k d FPP(ρ, L k ) a.s. k c 0 [c, C]. Proof: Kingman s subadditive ergodic theorem. 0 k k l v k ρ geodesic from ρ to L k geodesic from v k to L k+l (in region below L k ) d FPP (ρ, L k+l ) d FPP (ρ, L k )+Z k,l where (d) Z k,l = d FPP (ρ, L l ) and Z k,l is independent of d FPP (ρ, L k ). Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

43 5. A few ideas of the proof for FPP on the UIPT Locally (say below the boundary of the hull of radius r), the UIPT looks like the half-plane model. distance from ρ r r r 1 r 2 r 3 ρ Can use the result in the half-plane model to estimate the FPP distance between a typical point of B r ( ) and B (1 ε)r ( ). Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

44 Ideas of the proof II distance from ρ We consider slices of the UIPT corresponding to sets of the form B r ( )\B (1 ε)r ( ) For each slice, one can control the distance between an arbitrary point of the upper boundary and the lower boundary of the slice. ρ Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

45 Ideas of the proof II distance from ρ ρ We consider slices of the UIPT corresponding to sets of the form B r ( )\B (1 ε)r ( ) For each slice, one can control the distance between an arbitrary point of the upper boundary and the lower boundary of the slice. Main technical issue: In order to compare with the half-plane model, one should exclude the possibility that FPP geodesics move too fast in the horizontal direction (turn around the slice) Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30

46 A few references BENJAMINI: Random planar metrics. Proc. ICM BOUTTIER, DI FRANCESCO, GUITTER: Planar maps as labeled mobiles. Electr. J. Combinatorics (2004) BOUTTIER, GUITTER: The three-point function of planar quadrangulations. J. Stat. Mech. Theory Exp. (2008) CURIEN, LE GALL: First-passage percolation and local modifications of distances in random triangulations (2015) LE GALL: Geodesics in large planar maps... Acta Math. (2010) LE GALL: Uniqueness and universality of the Brownian map. Ann. Probab. (2013) MILLER, SHEFFIELD: An axiomatic characterization of the Brownian map (2015) MILLER, SHEFFIELD: Liouville quantum gravity and the Brownian map I, II, III ( ) MIERMONT: The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. (2013) SCHRAMM: Conformally invariant scaling limits. Proc. ICM Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP, Paris / 30