Firstpassage percolation on random planar maps
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Transcript of Firstpassage percolation on random planar maps
 1. Probability on trees and planar graphs, Ban, Canada, 15092014 Firstpassage percolation on random planar maps Timothy Budd Niels Bohr Institute, Copenhagen. [email protected], http://www.nbi.dk/~budd/ Partially based on arXiv:1408.3040
 2. Firstpassage percolation on a graph Random i.i.d. edge weights w(e) with mean 1. Passage time v1 v2 T = min :v1v2 e w(e)
 3. Firstpassage percolation on a graph Random i.i.d. edge weights w(e) with mean 1. Passage time v1 v2 T = min :v1v2 e w(e)
 4. Firstpassage percolation on a graph Random i.i.d. edge weights w(e) with mean 1. Passage time v1 v2 T = min :v1v2 e w(e)
 5. Firstpassage percolation on a graph Random i.i.d. edge weights w(e) with mean 1. Passage time v1 v2 T = min :v1v2 e w(e)
 6. Firstpassage percolation on a graph Random i.i.d. edge weights w(e) with mean 1. Passage time v1 v2 T = min :v1v2 e w(e)
 7. Firstpassage percolation on a graph Random i.i.d. edge weights w(e) with mean 1. Passage time v1 v2 T = min :v1v2 e w(e)
 8. Firstpassage percolation on a graph Random i.i.d. edge weights w(e) with mean 1. Passage time v1 v2 T = min :v1v2 e w(e)
 9. Firstpassage percolation on a graph Random i.i.d. edge weights w(e) with mean 1. Passage time v1 v2 T = min :v1v2 e w(e) Exponential distribution: equivalent to an Eden model.
 10. Firstpassage percolation on a graph Random i.i.d. edge weights w(e) with mean 1. Passage time v1 v2 T = min :v1v2 e w(e) Exponential distribution: equivalent to an Eden model.
 11. Firstpassage percolation on a graph Random i.i.d. edge weights w(e) with mean 1. Passage time v1 v2 T = min :v1v2 e w(e) Exponential distribution: equivalent to an Eden model.
 12. Firstpassage percolation on a graph Random i.i.d. edge weights w(e) with mean 1. Passage time v1 v2 T = min :v1v2 e w(e) Exponential distribution: equivalent to an Eden model.
 13. Firstpassage percolation on a graph Random i.i.d. edge weights w(e) with mean 1. Passage time v1 v2 T = min :v1v2 e w(e) Exponential distribution: equivalent to an Eden model.
 14. Firstpassage percolation on a graph Random i.i.d. edge weights w(e) with mean 1. Passage time v1 v2 T = min :v1v2 e w(e) Exponential distribution: equivalent to an Eden model.
 15. Firstpassage percolation on a graph Random i.i.d. edge weights w(e) with mean 1. Passage time v1 v2 T = min :v1v2 e w(e) Exponential distribution: equivalent to an Eden model.
 16. Firstpassage percolation on a graph Random i.i.d. edge weights w(e) with mean 1. Passage time v1 v2 T = min :v1v2 e w(e) Exponential distribution: equivalent to an Eden model. How does T compare to the graph distance d (asymptotically)?
 17. Firstpassage percolation on a graph Random i.i.d. edge weights w(e) with mean 1. Passage time v1 v2 T = min :v1v2 e w(e) Exponential distribution: equivalent to an Eden model. How does T compare to the graph distance d (asymptotically)? Hop count h is the number of edges in shortest time path.
 18. Firstpassage percolation on a graph Random i.i.d. edge weights w(e) with mean 1. Passage time v1 v2 T = min :v1v2 e w(e) Exponential distribution: equivalent to an Eden model. How does T compare to the graph distance d (asymptotically)? Hop count h is the number of edges in shortest time path. Asymptotically T < d < h.
 19. Time constants For many plane graphs (Z2 , PoissonVoronoi, Random Geometric) one can show that the time constants limd T/d and limd h/d exist almost surely using Kingmans subadditive ergodic theorem.
 20. Time constants For many plane graphs (Z2 , PoissonVoronoi, Random Geometric) one can show that the time constants limd T/d and limd h/d exist almost surely using Kingmans subadditive ergodic theorem. However, very few exact time constants known (ladder graph,. . . ?).
 21. Time constants For many plane graphs (Z2 , PoissonVoronoi, Random Geometric) one can show that the time constants limd T/d and limd h/d exist almost surely using Kingmans subadditive ergodic theorem. However, very few exact time constants known (ladder graph,. . . ?). Numerically, many models seem to have d right in the middle of the bounds (T < d < h). Explanation?
 22. Time constants For many plane graphs (Z2 , PoissonVoronoi, Random Geometric) one can show that the time constants limd T/d and limd h/d exist almost surely using Kingmans subadditive ergodic theorem. However, very few exact time constants known (ladder graph,. . . ?). Numerically, many models seem to have d right in the middle of the bounds (T < d < h). Explanation?
 23. Time constants For many plane graphs (Z2 , PoissonVoronoi, Random Geometric) one can show that the time constants limd T/d and limd h/d exist almost surely using Kingmans subadditive ergodic theorem. However, very few exact time constants known (ladder graph,. . . ?). Numerically, many models seem to have d right in the middle of the bounds (T < d < h). Explanation? Let us look at random planar maps: assuming existence of the time constants we can compute them!
 24. Multipoint functions of general planar maps FPP on a random cubic map with exponentially distributed edge weights shows up as a byproduct of the study of distances on a random general planar map.
 25. Multipoint functions of general planar maps FPP on a random cubic map with exponentially distributed edge weights shows up as a byproduct of the study of distances on a random general planar map. Bijection between quadrangulations and planar maps preserving distance. [Miermont, 09] [Ambjorn,TB, 13]
 26. Multipoint functions of general planar maps FPP on a random cubic map with exponentially distributed edge weights shows up as a byproduct of the study of distances on a random general planar map. Bijection between quadrangulations and planar maps preserving distance. [Miermont, 09] [Ambjorn,TB, 13] Random planar map with E edges converges to Brownian map as E . [Bettinelli,Jacob,Miermont,13]
 27. Multipoint functions of general planar maps FPP on a random cubic map with exponentially distributed edge weights shows up as a byproduct of the study of distances on a random general planar map. Bijection between quadrangulations and planar maps preserving distance. [Miermont, 09] [Ambjorn,TB, 13] Random planar map with E edges converges to Brownian map as E . [Bettinelli,Jacob,Miermont,13] Also allowed to derive various distance statistics, while controlling both the number E of edges and the number F of faces: Twopoint function (rooted) [Ambjorn, TB, 13]
 28. Multipoint functions of general planar maps FPP on a random cubic map with exponentially distributed edge weights shows up as a byproduct of the study of distances on a random general planar map. Bijection between quadrangulations and planar maps preserving distance. [Miermont, 09] [Ambjorn,TB, 13] Random planar map with E edges converges to Brownian map as E . [Bettinelli,Jacob,Miermont,13] Also allowed to derive various distance statistics, while controlling both the number E of edges and the number F of faces: Twopoint function (rooted) [Ambjorn, TB, 13] Twopoint function (unrooted) [Bouttier, Fusy, Guitter, 13]
 29. Multipoint functions of general planar maps FPP on a random cubic map with exponentially distributed edge weights shows up as a byproduct of the study of distances on a random general planar map. Bijection between quadrangulations and planar maps preserving distance. [Miermont, 09] [Ambjorn,TB, 13] Random planar map with E edges converges to Brownian map as E . [Bettinelli,Jacob,Miermont,13] Also allowed to derive various distance statistics, while controlling both the number E of edges and the number F of faces: Twopoint function (rooted) [Ambjorn, TB, 13] Twopoint function (unrooted) [Bouttier, Fusy, Guitter, 13] Threepoint function [Fusy, Guitter, 14]
 30. Multipoint functions of general planar maps FPP on a random cubic map with exponentially distributed edge weights shows up as a byproduct of the study of distances on a random general planar map. Bijection between quadrangulations and planar maps preserving distance. [Miermont, 09] [Ambjorn,TB, 13] Random planar map with E edges converges to Brownian map as E . [Bettinelli,Jacob,Miermont,13] Also allowed to derive various distance statistics, while controlling both the number E of edges and the number F of faces: Twopoint function (rooted) [Ambjorn, TB, 13] Twopoint function (unrooted) [Bouttier, Fusy, Guitter, 13] Threepoint function [Fusy, Guitter, 14] Consider the limit E keeping F xed. In terms of . . . . . . quadrangulation: Generalized Causal Dynamical Triangulations (GCDT) . . . general planar maps: after removal of trees cubic maps with edges of random length.
 31. From general maps to weighted cubic maps Take a planar map with marked vertices and weight x per edge.
 32. From general maps to weighted cubic maps Take a planar map with marked vertices and weight x per edge. Remove dangling edges .
 33. From general maps to weighted cubic maps Take a planar map with marked vertices and weight x per edge. Remove dangling edges and combine neighboring edges while keeping track of length.
 34. From general maps to weighted cubic maps Take a planar map with marked vertices and weight x per edge. Remove dangling edges and combine neighboring edges while keeping track of length.
 35. From general maps to weighted cubic maps Take a planar map with marked vertices and weight x per edge. Remove dangling edges and combine neighboring edges while keeping track of length. Each subedge comes with weight x((1 1 4x)/x)2 , hence length is geometrically distributed.
 36. From general maps to weighted cubic maps Take a planar map with marked vertices and weight x per edge. Remove dangling edges and combine neighboring edges while keeping track of length. Each subedge comes with weight x((1 1 4x)/x)2 , hence length is geometrically distributed. Scaling limit x 1/4 and edge lengths (x) := 4 16x: lengths are exponentially distributed with mean 1. Maximal number of edges preferred: almost surely cubic and univalent marked vertices.
 37. Twopoint function 2point function for general planar maps with weight x per edge and z per face and distance t between marked vertices: [Bouttier et al, 13] Gz,x (t) = log (1 a t+1 )3 (1 a t+2)3 (1 a t+3 ) (1 a t) , (1) where a = a(z, x) and = (z, x) solution of algebraic equations.
 38. Twopoint function 2point function for general planar maps with weight x per edge and z per face and distance t between marked vertices: [Bouttier et al, 13] Gz,x (t) = log (1 a t+1 )3 (1 a t+2)3 (1 a t+3 ) (1 a t) , (1) where a = a(z, x) and = (z, x) solution of algebraic equations. Scaling z = (x)3 g, t = T/ (x), x 1/4: G (1,1) g,2 (T) = 3 T log( cosh T + sinh T), := 3 2 2 1 8 , and is solution to 3 /4 + g = 0.
 39. Twopoint function 2point function for general planar maps with weight x per edge and z per face and distance t between marked vertices: [Bouttier et al, 13] Gz,x (t) = log (1 a t+1 )3 (1 a t+2)3 (1 a t+3 ) (1 a t) , (1) where a = a(z, x) and = (z, x) solution of algebraic equations. Scaling z = (x)3 g, t = T/ (x), x 1/4: G (1,1) g,2 (T) = 3 T log( cosh T + sinh T), := 3 2 2 1 8 , and is solution to 3 /4 + g = 0. Bivalent marked vertices: G (2,2) g,2 (T) = 1 4 (1 + T )2 G (1,1) g,2 (T)
 40. Twopoint function 2point function for general planar maps with weight x per edge and z per face and distance t between marked vertices: [Bouttier et al, 13] Gz,x (t) = log (1 a t+1 )3 (1 a t+2)3 (1 a t+3 ) (1 a t) , (1) where a = a(z, x) and = (z, x) solution of algebraic equations. Scaling z = (x)3 g, t = T/ (x), x 1/4: G (1,1) g,2 (T) = 3 T log( cosh T + sinh T), := 3 2 2 1 8 , and is solution to 3 /4 + g = 0. Bivalent marked vertices: G (2,2) g,2 (T) = 1 4 (1 + T )2 G (1,1) g,2 (T) Completely cubic: G (3,3) g,2 (T) = 1 9 (2 + T )2 G (2,2) g,2 (T)
 41. Twopoint function 2point function for general planar maps with weight x per edge and z per face and distance t between marked vertices: [Bouttier et al, 13] Gz,x (t) = log (1 a t+1 )3 (1 a t+2)3 (1 a t+3 ) (1 a t) , (1) where a = a(z, x) and = (z, x) solution of algebraic equations. Scaling z = (x)3 g, t = T/ (x), x 1/4: G (1,1) g,2 (T) = 3 T log( cosh T + sinh T), := 3 2 2 1 8 , and is solution to 3 /4 + g = 0. Scaling limit: g 1 12 3 , i.e. 1 12 and 0. Renormalizing T := T gives G (1,1) g,2 (T)3 2 cosh T sinh3 T .
 42. Twopoint function 2point function for general planar maps with weight x per edge and z per face and distance t between marked vertices: [Bouttier et al, 13] Gz,x (t) = log (1 a t+1 )3 (1 a t+2)3 (1 a t+3 ) (1 a t) , (1) where a = a(z, x) and = (z, x) solution of algebraic equations. Scaling z = (x)3 g, t = T/ (x), x 1/4: G (1,1) g,2 (T) = 3 T log( cosh T + sinh T), := 3 2 2 1 8 , and is solution to 3 /4 + g = 0. Scaling limit: g 1 12 3 , i.e. 1 12 and 0. Renormalizing T := T gives G (1,1) g,2 (T)3 2 cosh T sinh3 T . Agrees with distanceprole of the Brownian map.
 43. Threepoint function Three marked vertices with pairwise distances d12, d13 and d23.
 44. Threepoint function Three marked vertices with pairwise distances d12, d13 and d23.
 45. Threepoint function Three marked vertices with pairwise distances d12, d13 and d23.
 46. Threepoint function Three marked vertices with pairwise distances d12, d13 and d23. Reparametrize d12 = s + t, d13 = s + u, d23 = u + t.
 47. Threepoint function Three marked vertices with pairwise distances d12, d13 and d23. Reparametrize d12 = s + t, d13 = s + u, d23 = u + t. Even case: s, t, u Z. Odd case: s, t, u Z + 1 2 .
 48. Threepoint function Three marked vertices with pairwise distances d12, d13 and d23. Reparametrize d12 = s + t, d13 = s + u, d23 = u + t. Even case: s, t, u Z. Odd case: s, t, u Z + 1 2 .
 49. Threepoint function Three marked vertices with pairwise distances d12, d13 and d23. Reparametrize d12 = s + t, d13 = s + u, d23 = u + t. Even case: s, t, u Z. Odd case: s, t, u Z + 1 2 . Explicit expressions Geven z,x (s, t, u), Godd z,x (s, t, u) known. [Fusy, Guitter, 14]
 50. Threepoint function Three marked vertices with pairwise distances d12, d13 and d23. Reparametrize d12 = s + t, d13 = s + u, d23 = u + t. Even case: s, t, u Z. Odd case: s, t, u Z + 1 2 . Explicit expressions Geven z,x (s, t, u), Godd z,x (s, t, u) known. [Fusy, Guitter, 14] Scaling as before: G (1) g,3(S, T, U) = Geven g,3 (S, T, U) + Godd g,3 (S, T, U).
 51. Threepoint function Three marked vertices with pairwise distances d12, d13 and d23. Reparametrize d12 = s + t, d13 = s + u, d23 = u + t. Even case: s, t, u Z. Odd case: s, t, u Z + 1 2 . Explicit expressions Geven z,x (s, t, u), Godd z,x (s, t, u) known. [Fusy, Guitter, 14] Scaling as before: G (1) g,3(S, T, U) = Geven g,3 (S, T, U) + Godd g,3 (S, T, U). Scaling again g 1 12 3 , T := T, etc., gives G (1) g,3(S, T, U)2 1 12 ST U sinh2 S sinh2 T sinh2 U sinh2 (S + T + U) sinh2 (S + T ) sinh2 (T + U) sinh2 (U + S) , which is the Brownian map 3point function.
 52. Threepoint function Three marked vertices with pairwise distances d12, d13 and d23. Reparametrize d12 = s + t, d13 = s + u, d23 = u + t. Even case: s, t, u Z. Odd case: s, t, u Z + 1 2 . Explicit expressions Geven z,x (s, t, u), Godd z,x (s, t, u) known. [Fusy, Guitter, 14] Scaling as before: G (1) g,3(S, T, U) = Geven g,3 (S, T, U) + Godd g,3 (S, T, U). Unless cycle vanishes, i.e. completely conuent, even and odd occur with equal probability. Hence Gconf g,3 = Geven g,3 Godd g,3
 53. Threepoint function Three marked vertices with pairwise distances d12, d13 and d23. Reparametrize d12 = s + t, d13 = s + u, d23 = u + t. Even case: s, t, u Z. Odd case: s, t, u Z + 1 2 . Explicit expressions Geven z,x (s, t, u), Godd z,x (s, t, u) known. [Fusy, Guitter, 14] Scaling as before: G (1) g,3(S, T, U) = Geven g,3 (S, T, U) + Godd g,3 (S, T, U). Unless cycle vanishes, i.e. completely conuent, even and odd occur with equal probability. Hence Gconf g,3 = Geven g,3 Godd g,3
 54. Threepoint function Three marked vertices with pairwise distances d12, d13 and d23. Reparametrize d12 = s + t, d13 = s + u, d23 = u + t. Even case: s, t, u Z. Odd case: s, t, u Z + 1 2 . Explicit expressions Geven z,x (s, t, u), Godd z,x (s, t, u) known. [Fusy, Guitter, 14] Scaling as before: G (1) g,3(S, T, U) = Geven g,3 (S, T, U) + Godd g,3 (S, T, U). Unless cycle vanishes, i.e. completely conuent, even and odd occur with equal probability. Hence Gconf g,3 = Geven g,3 Godd g,3 Bivalent marked vertices G (2) g,3 =1 8 (1 + S )(1 + T )(1 + U )G (1) g,3(S, T, U) + G (2) g,2(T + U)(S) + G (2) g,2(U + S)(T) + G (2) g,2(S + T)(U)
 55. Threepoint function Three marked vertices with pairwise distances d12, d13 and d23. Reparametrize d12 = s + t, d13 = s + u, d23 = u + t. Even case: s, t, u Z. Odd case: s, t, u Z + 1 2 . Explicit expressions Geven z,x (s, t, u), Godd z,x (s, t, u) known. [Fusy, Guitter, 14] Scaling as before: G (1) g,3(S, T, U) = Geven g,3 (S, T, U) + Godd g,3 (S, T, U). Unless cycle vanishes, i.e. completely conuent, even and odd occur with equal probability. Hence Gconf g,3 = Geven g,3 Godd g,3 Bivalent marked vertices G (2) g,3 =1 8 (1 + S )(1 + T )(1 + U )G (1) g,3(S, T, U) + G (2) g,2(T + U)(S) + G (2) g,2(U + S)(T) + G (2) g,2(S + T)(U)
 56. Expected hop count for random cubic map Consider the limit T 0 (but S, T, U > 0) of the completely conuent threepoint function, i.e. G (2) g,2(S, U) := lim T0 1 8 (1+S )(1+T )(1+U )[Geven g,3 (S, T, U)Godd g,3 (S, T, U)]
 57. Expected hop count for random cubic map Consider the limit T 0 (but S, T, U > 0) of the completely conuent threepoint function, i.e. G (2) g,2(S, U) := lim T0 1 8 (1+S )(1+T )(1+U )[Geven g,3 (S, T, U)Godd g,3 (S, T, U)]
 58. Expected hop count for random cubic map Consider the limit T 0 (but S, T, U > 0) of the completely conuent threepoint function, i.e. G (2) g,2(S, U) := lim T0 1 8 (1+S )(1+T )(1+U )[Geven g,3 (S, T, U)Godd g,3 (S, T, U)]
 59. Expected hop count for random cubic map Consider the limit T 0 (but S, T, U > 0) of the completely conuent threepoint function, i.e. G (2) g,2(S, U) := lim T0 1 8 (1+S )(1+T )(1+U )[Geven g,3 (S, T, U)Godd g,3 (S, T, U)] Hence, for xed number of faces F, G (2) F,2(S, T S) G (2) F,2(T) is the expected density of vertices (at distance S) along a geodesic of length/passage time T.
 60. Expected hop count for random cubic map Consider the limit T 0 (but S, T, U > 0) of the completely conuent threepoint function, i.e. G (2) g,2(S, U) := lim T0 1 8 (1+S )(1+T )(1+U )[Geven g,3 (S, T, U)Godd g,3 (S, T, U)] Hence, for xed number of faces F, G (2) F,2(S, T S) G (2) F,2(T) is the expected density of vertices (at distance S) along a geodesic of length/passage time T. Scaling limit: G (2) g,2(S, T S) (1+ 1 3 )G (2) g,2(T).
 61. Expected hop count for random cubic map Consider the limit T 0 (but S, T, U > 0) of the completely conuent threepoint function, i.e. G (2) g,2(S, U) := lim T0 1 8 (1+S )(1+T )(1+U )[Geven g,3 (S, T, U)Godd g,3 (S, T, U)] Hence, for xed number of faces F, G (2) F,2(S, T S) G (2) F,2(T) is the expected density of vertices (at distance S) along a geodesic of length/passage time T. Scaling limit: G (2) g,2(S, T S) (1+ 1 3 )G (2) g,2(T). Asymptotic hop count to passage time ratio is h /T 1 + 1 3 .
 62. Time constants for FPP on a random cubic graph We have seen h /T 1 + 1 3 . This is conrmed numerically (random triangulation 128k triangles).
 63. Time constants for FPP on a random cubic graph We have seen h /T 1 + 1 3 . This is conrmed numerically (random triangulation 128k triangles). Let us assume that h, T and the graph distance d are asymptotically (almost surely) linearly related.
 64. Time constants for FPP on a random cubic graph We have seen h /T 1 + 1 3 . This is conrmed numerically (random triangulation 128k triangles). Let us assume that h, T and the graph distance d are asymptotically (almost surely) linearly related. Then we can determine the ratio d/T simply by comparing the corresponding twopoint functions.
 65. Time constants for FPP on a random cubic graph We have seen h /T 1 + 1 3 . This is conrmed numerically (random triangulation 128k triangles). Let us assume that h, T and the graph distance d are asymptotically (almost surely) linearly related. Then we can determine the ratio d/T simply by comparing the corresponding twopoint functions. Deduce from transfer matrix approach in [Kawai et al, 93]: h/T 1 + 1 2 3 .
 66. Time constants for FPP on a random cubic graph We have seen h /T 1 + 1 3 . This is conrmed numerically (random triangulation 128k triangles). Let us assume that h, T and the graph distance d are asymptotically (almost surely) linearly related. Then we can determine the ratio d/T simply by comparing the corresponding twopoint functions. Deduce from transfer matrix approach in [Kawai et al, 93]: h/T 1 + 1 2 3 . Simple relation: d = (h + T)/2. Is it true more generally?
 67. Expected hop count for pregular maps Two vertices a distance T apart.
 68. Expected hop count for pregular maps Two vertices a distance T apart.
 69. Expected hop count for pregular maps Two vertices a distance T apart. Determine the balls of radius S and T S S. They almost touch.
 70. Expected hop count for pregular maps Two vertices a distance T apart. Determine the balls of radius S and T S S. They almost touch.
 71. Expected hop count for pregular maps Two vertices a distance T apart. Determine the balls of radius S and T S S. They almost touch.
 72. Expected hop count for pregular maps Two vertices a distance T apart. Determine the balls of radius S and T S S. They almost touch.
 73. Expected hop count for pregular maps Two vertices a distance T apart. Determine the balls of radius S and T S S. They almost touch. As S 0, conditioned on the balls, a vertex occurs with probability P S with P = (p1) WN1,d+p2 WN,d in terms of the disk function Wx (z) = N=0 d=1 zd1 xN WN,d .
 74. Expected hop count for pregular maps Two vertices a distance T apart. Determine the balls of radius S and T S S. They almost touch. As S 0, conditioned on the balls, a vertex occurs with probability P S with P = (p1) WN1,d+p2 WN,d in terms of the disk function Wx (z) = N=0 d=1 zd1 xN WN,d .
 75. Expected hop count for pregular maps Two vertices a distance T apart. Determine the balls of radius S and T S S. They almost touch. As S 0, conditioned on the balls, a vertex occurs with probability P S with P = (p1) WN1,d+p2 WN,d in terms of the disk function Wx (z) = N=0 d=1 zd1 xN WN,d .
 76. Expected hop count for pregular maps Two vertices a distance T apart. Determine the balls of radius S and T S S. They almost touch. As S 0, conditioned on the balls, a vertex occurs with probability P S with P = (p1) WN1,d+p2 WN,d N,d (p1)xc zp2 c , in terms of the disk function Wx (z) = N=0 d=1 zd1 xN WN,d .
 77. Expected hop count for pregular maps Two vertices a distance T apart. Determine the balls of radius S and T S S. They almost touch. As S 0, conditioned on the balls, a vertex occurs with probability P S with P = (p1) WN1,d+p2 WN,d N,d (p1)xc zp2 c , in terms of the disk function Wx (z) = N=0 d=1 zd1 xN WN,d . Asymptotic hop count to passage time ratio is h /T (p 1)xc zp2 c .
 78. Expected hop count for pregular maps Two vertices a distance T apart. Determine the balls of radius S and T S S. They almost touch. As S 0, conditioned on the balls, a vertex occurs with probability P S with P = (p1) WN1,d+p2 WN,d N,d (p1)xc zp2 c , in terms of the disk function Wx (z) = N=0 d=1 zd1 xN WN,d . Asymptotic hop count to passage time ratio is h /T (p 1)xc zp2 c . p = 3: xc = 1/(2 33/4 ), zc = 33/4 (1 + 1/ 3), h T = 1 + 1/ 3.
 79. Expected hop count for pregular maps Two vertices a distance T apart. Determine the balls of radius S and T S S. They almost touch. As S 0, conditioned on the balls, a vertex occurs with probability P S with P = (p1) WN1,d+p2 WN,d N,d (p1)xc zp2 c , in terms of the disk function Wx (z) = N=0 d=1 zd1 xN WN,d . Asymptotic hop count to passage time ratio is h /T (p 1)xc zp2 c . p = 3: xc = 1/(2 33/4 ), zc = 33/4 (1 + 1/ 3), h T = 1 + 1/ 3. p even: xc = p2 p p 2 4 p2 p p/2 1 , zc = 4p p2 , h T = 2p2 p2 p2 2 1 .
 80. Transfer matrix for FPP Take a pregular weighted planar map with two marked points.
 81. Transfer matrix for FPP Take a pregular weighted planar map with two marked points. Identify baby universes.
 82. Transfer matrix for FPP Take a pregular weighted planar map with two marked points. Identify baby universes. Cut into small passage time intervals.
 83. Transfer matrix for FPP Take a pregular weighted planar map with two marked points. Identify baby universes. Cut into small passage time intervals. Write Gx (z, T) := d1 zd11 G (d1,d2) x,2 (T) and the disk function Wx (z) := d zd1 G (d) x,1 (weight x per vertex). Then T Gx (z, T)= z ( z A xzp1 B 2 Wx (z) C )Gx (z, T)
 84. Transfer matrix for FPP Take a pregular weighted planar map with two marked points. Identify baby universes. Cut into small passage time intervals. Write Gx (z, T) := d1 zd11 G (d1,d2) x,2 (T) and the disk function Wx (z) := d zd1 G (d) x,1 (weight x per vertex). Then T Gx (z, T)= z ( z xzp1 2Wx (z))Gx (z, T) Precisely this formula was used in [Ambjorn, Watabiki, 95] as an approximation to derive the 2point function for triangulations. Now we know it is not just an approximation!
 85. Transfer matrix for graph distance [Kawai et al, 93] Now take all edges to have length 1. Again we can build a transfer matrix.
 86. Transfer matrix for graph distance [Kawai et al, 93] Now take all edges to have length 1. Again we can build a transfer matrix.
 87. Transfer matrix for graph distance [Kawai et al, 93] Now take all edges to have length 1. Again we can build a transfer matrix.
 88. Transfer matrix for graph distance [Kawai et al, 93] Now take all edges to have length 1. Again we can build a transfer matrix. Two of the building blocks are quite similar. . .
 89. Transfer matrix for graph distance [Kawai et al, 93] Now take all edges to have length 1. Again we can build a transfer matrix. Two of the building blocks are quite similar. . . . . . but there are p 1 extra ones.
 90. Transfer matrix for graph distance [Kawai et al, 93] Now take all edges to have length 1. Again we can build a transfer matrix. Two of the building blocks are quite similar. . . . . . but there are p 1 extra ones. Can work out transfer matrix PDE explicitly and compare, but heuristically xc zp2 c
 91. Transfer matrix for graph distance [Kawai et al, 93] Now take all edges to have length 1. Again we can build a transfer matrix. Two of the building blocks are quite similar. . . . . . but there are p 1 extra ones. Can work out transfer matrix PDE explicitly and compare, but heuristically xc zp2 c and therefore + + + + 2 1 2 (1 + (p 1)xc zp2 c ) Graph distance to passage time ratio is d/T (1 + h/T)/2.
 92. Conclusion & open questions Conclusions Both the two and threepoint functions of weighted cubic maps converge to those of the Brownian map in the scaling limit. We can compute the time constants of FPP on random pregular planar maps, and in each case they satisfy d = (h + T)/2.
 93. Conclusion & open questions Conclusions Both the two and threepoint functions of weighted cubic maps converge to those of the Brownian map in the scaling limit. We can compute the time constants of FPP on random pregular planar maps, and in each case they satisfy d = (h + T)/2. Open questions Can weighted cubic maps be shown to converge to the Brownian map?
 94. Conclusion & open questions Conclusions Both the two and threepoint functions of weighted cubic maps converge to those of the Brownian map in the scaling limit. We can compute the time constants of FPP on random pregular planar maps, and in each case they satisfy d = (h + T)/2. Open questions Can weighted cubic maps be shown to converge to the Brownian map? Is d = (h + T)/2 a coincidence or does it hold for a (much) larger class of random graphs? Does it hold in the case of . . . . . . weights on the faces, e.g. pangulations? . . . other edge length distributions? . . . random geometric graphs (in certain limits)?
 95. Conclusion & open questions Conclusions Both the two and threepoint functions of weighted cubic maps converge to those of the Brownian map in the scaling limit. We can compute the time constants of FPP on random pregular planar maps, and in each case they satisfy d = (h + T)/2. Open questions Can weighted cubic maps be shown to converge to the Brownian map? Is d = (h + T)/2 a coincidence or does it hold for a (much) larger class of random graphs? Does it hold in the case of . . . . . . weights on the faces, e.g. pangulations? . . . other edge length distributions? . . . random geometric graphs (in certain limits)? Is there a bijective explanation for d = (h + T)/2?
 96. Conclusion & open questions Conclusions Both the two and threepoint functions of weighted cubic maps converge to those of the Brownian map in the scaling limit. We can compute the time constants of FPP on random pregular planar maps, and in each case they satisfy d = (h + T)/2. Open questions Can weighted cubic maps be shown to converge to the Brownian map? Is d = (h + T)/2 a coincidence or does it hold for a (much) larger class of random graphs? Does it hold in the case of . . . . . . weights on the faces, e.g. pangulations? . . . other edge length distributions? . . . random geometric graphs (in certain limits)? Is there a bijective explanation for d = (h + T)/2? How do the relative uctuations of d, h, T scale? KardarParisiZhang scaling exponents on a random surface?
 97. Conclusion & open questions Conclusions Both the two and threepoint functions of weighted cubic maps converge to those of the Brownian map in the scaling limit. We can compute the time constants of FPP on random pregular planar maps, and in each case they satisfy d = (h + T)/2. Open questions Can weighted cubic maps be shown to converge to the Brownian map? Is d = (h + T)/2 a coincidence or does it hold for a (much) larger class of random graphs? Does it hold in the case of . . . . . . weights on the faces, e.g. pangulations? . . . other edge length distributions? . . . random geometric graphs (in certain limits)? Is there a bijective explanation for d = (h + T)/2? How do the relative uctuations of d, h, T scale? KardarParisiZhang scaling exponents on a random surface? Thanks!