First-passage percolation on random planar maps

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Probability on trees and planar graphs, Banff, Canada, 15-09-2014 First-passage percolation on random planar maps Timothy Budd Niels Bohr Institute, Copenhagen. [email protected], http://www.nbi.dk/ ~ budd/ Partially based on arXiv:1408.3040

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  1. 1. Probability on trees and planar graphs, Ban, Canada, 15-09-2014 First-passage 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. 2. First-passage 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. 3. First-passage 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. 4. First-passage 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. 5. First-passage 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. 6. First-passage 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. 7. First-passage 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. 8. First-passage 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. 9. First-passage 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. 10. First-passage 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. 11. First-passage 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. 12. First-passage 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. 13. First-passage 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. 14. First-passage 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. 15. First-passage 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. 16. First-passage 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. 17. First-passage 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. 18. First-passage 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. 19. Time constants For many plane graphs (Z2 , Poisson-Voronoi, 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. 20. Time constants For many plane graphs (Z2 , Poisson-Voronoi, 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. 21. Time constants For many plane graphs (Z2 , Poisson-Voronoi, 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. 22. Time constants For many plane graphs (Z2 , Poisson-Voronoi, 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. 23. Time constants For many plane graphs (Z2 , Poisson-Voronoi, 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. 24. Multi-point functions of general planar maps FPP on a random cubic map with exponentially distributed edge weights shows up as a by-product of the study of distances on a random general planar map.
  25. 25. Multi-point functions of general planar maps FPP on a random cubic map with exponentially distributed edge weights shows up as a by-product 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. 26. Multi-point functions of general planar maps FPP on a random cubic map with exponentially distributed edge weights shows up as a by-product 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. 27. Multi-point functions of general planar maps FPP on a random cubic map with exponentially distributed edge weights shows up as a by-product 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: Two-point function (rooted) [Ambjorn, TB, 13]
  28. 28. Multi-point functions of general planar maps FPP on a random cubic map with exponentially distributed edge weights shows up as a by-product 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: Two-point function (rooted) [Ambjorn, TB, 13] Two-point function (unrooted) [Bouttier, Fusy, Guitter, 13]
  29. 29. Multi-point functions of general planar maps FPP on a random cubic map with exponentially distributed edge weights shows up as a by-product 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: Two-point function (rooted) [Ambjorn, TB, 13] Two-point function (unrooted) [Bouttier, Fusy, Guitter, 13] Three-point function [Fusy, Guitter, 14]
  30. 30. Multi-point functions of general planar maps FPP on a random cubic map with exponentially distributed edge weights shows up as a by-product 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: Two-point function (rooted) [Ambjorn, TB, 13] Two-point function (unrooted) [Bouttier, Fusy, Guitter, 13] Three-point 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. 31. From general maps to weighted cubic maps Take a planar map with marked vertices and weight x per edge.
  32. 32. From general maps to weighted cubic maps Take a planar map with marked vertices and weight x per edge. Remove dangling edges .
  33. 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. 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. 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. 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. 37. Two-point function 2-point 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. 38. Two-point function 2-point 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. 39. Two-point function 2-point 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. 40. Two-point function 2-point 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. 41. Two-point function 2-point 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. 42. Two-point function 2-point 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 distance-prole of the Brownian map.
  43. 43. Three-point function Three marked vertices with pair-wise distances d12, d13 and d23.
  44. 44. Three-point function Three marked vertices with pair-wise distances d12, d13 and d23.
  45. 45. Three-point function Three marked vertices with pair-wise distances d12, d13 and d23.
  46. 46. Three-point function Three marked vertices with pair-wise distances d12, d13 and d23. Reparametrize d12 = s + t, d13 = s + u, d23 = u + t.
  47. 47. Three-point function Three marked vertices with pair-wise 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. 48. Three-point function Three marked vertices with pair-wise 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. 49. Three-point function Three marked vertices with pair-wise 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. 50. Three-point function Three marked vertices with pair-wise 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. 51. Three-point function Three marked vertices with pair-wise 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 3-point function.
  52. 52. Three-point function Three marked vertices with pair-wise 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. 53. Three-point function Three marked vertices with pair-wise 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. 54. Three-point function Three marked vertices with pair-wise 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. 55. Three-point function Three marked vertices with pair-wise 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. 56. Expected hop count for random cubic map Consider the limit T 0 (but S, T, U > 0) of the completely conuent three-point 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. 57. Expected hop count for random cubic map Consider the limit T 0 (but S, T, U > 0) of the completely conuent three-point 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. 58. Expected hop count for random cubic map Consider the limit T 0 (but S, T, U > 0) of the completely conuent three-point 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. 59. Expected hop count for random cubic map Consider the limit T 0 (but S, T, U > 0) of the completely conuent three-point 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. 60. Expected hop count for random cubic map Consider the limit T 0 (but S, T, U > 0) of the completely conuent three-point 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. 61. Expected hop count for random cubic map Consider the limit T 0 (but S, T, U > 0) of the completely conuent three-point 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. 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. 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. 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 two-point functions.
  65. 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 two-point functions. Deduce from transfer matrix approach in [Kawai et al, 93]: h/T 1 + 1 2 3 .
  66. 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 two-point 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. 67. Expected hop count for p-regular maps Two vertices a distance T apart.
  68. 68. Expected hop count for p-regular maps Two vertices a distance T apart.
  69. 69. Expected hop count for p-regular maps Two vertices a distance T apart. Determine the balls of radius S and T S S. They almost touch.
  70. 70. Expected hop count for p-regular maps Two vertices a distance T apart. Determine the balls of radius S and T S S. They almost touch.
  71. 71. Expected hop count for p-regular maps Two vertices a distance T apart. Determine the balls of radius S and T S S. They almost touch.
  72. 72. Expected hop count for p-regular maps Two vertices a distance T apart. Determine the balls of radius S and T S S. They almost touch.
  73. 73. Expected hop count for p-regular 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. 74. Expected hop count for p-regular 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. 75. Expected hop count for p-regular 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. 76. Expected hop count for p-regular 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. 77. Expected hop count for p-regular 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. 78. Expected hop count for p-regular 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. 79. Expected hop count for p-regular 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. 80. Transfer matrix for FPP Take a p-regular weighted planar map with two marked points.
  81. 81. Transfer matrix for FPP Take a p-regular weighted planar map with two marked points. Identify baby universes.
  82. 82. Transfer matrix for FPP Take a p-regular weighted planar map with two marked points. Identify baby universes. Cut into small passage time intervals.
  83. 83. Transfer matrix for FPP Take a p-regular 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. 84. Transfer matrix for FPP Take a p-regular 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 2-point function for triangulations. Now we know it is not just an approximation!
  85. 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. 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. 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. 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. 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. 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. 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. 92. Conclusion & open questions Conclusions Both the two- and three-point 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 p-regular planar maps, and in each case they satisfy d = (h + T)/2.
  93. 93. Conclusion & open questions Conclusions Both the two- and three-point 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 p-regular 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. 94. Conclusion & open questions Conclusions Both the two- and three-point 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 p-regular 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. p-angulations? . . . other edge length distributions? . . . random geometric graphs (in certain limits)?
  95. 95. Conclusion & open questions Conclusions Both the two- and three-point 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 p-regular 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. p-angulations? . . . other edge length distributions? . . . random geometric graphs (in certain limits)? Is there a bijective explanation for d = (h + T)/2?
  96. 96. Conclusion & open questions Conclusions Both the two- and three-point 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 p-regular 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. p-angulations? . . . 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? Kardar-Parisi-Zhang scaling exponents on a random surface?
  97. 97. Conclusion & open questions Conclusions Both the two- and three-point 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 p-regular 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. p-angulations? . . . 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? Kardar-Parisi-Zhang scaling exponents on a random surface? Thanks!