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Jason Miller (MIT) Liouville quantum gravity and the Brownian map Jason Miller and Scott Sheffield Cambridge and MIT July 15, 2015 Jason Miller (Cambridge) LQG and TBM July 15, 2015 1 / 24 Overview Part I: Picking surfaces at random

  1. The Gaussian free field ◮ The discrete Gaussian free field (DGFF) is a Gaussian random surface model. ◮ Measure on functions h : D → R for D ⊆ Z 2 and h | ∂ D = ψ with density respect to Lebesgue measure on R | D | : � � 1 − 1 � ( h ( x ) − h ( y )) 2 Z exp 2 x ∼ y ◮ Natural perturbation of a harmonic function ◮ Fine mesh limit: converges to the continuum GFF, i.e. the standard Gaussian wrt the Dirichlet inner product ( f , g ) ∇ = 1 � ∇ f ( x ) · ∇ g ( x ) dx . 2 π Jason Miller (Cambridge) LQG and TBM July 15, 2015 8 / 24

  2. The Gaussian free field ◮ The discrete Gaussian free field (DGFF) is a Gaussian random surface model. ◮ Measure on functions h : D → R for D ⊆ Z 2 and h | ∂ D = ψ with density respect to Lebesgue measure on R | D | : � � 1 − 1 � ( h ( x ) − h ( y )) 2 Z exp 2 x ∼ y ◮ Natural perturbation of a harmonic function ◮ Fine mesh limit: converges to the continuum GFF, i.e. the standard Gaussian wrt the Dirichlet inner product ( f , g ) ∇ = 1 � ∇ f ( x ) · ∇ g ( x ) dx . 2 π ◮ Continuum GFF not a function — only a generalized function Jason Miller (Cambridge) LQG and TBM July 15, 2015 8 / 24

  3. Liouville quantum gravity γ = 0 . 5 ◮ Liouville quantum gravity: e γ h ( z ) dz where h is a GFF and γ ∈ [0 , 2) (Number of subdivisions) Jason Miller (Cambridge) LQG and TBM July 15, 2015 9 / 24

  4. Liouville quantum gravity γ = 0 . 5 ◮ Liouville quantum gravity: e γ h ( z ) dz where h is a GFF and γ ∈ [0 , 2) ◮ Introduced by Polyakov in the 1980s (Number of subdivisions) Jason Miller (Cambridge) LQG and TBM July 15, 2015 9 / 24

  5. Liouville quantum gravity γ = 0 . 5 ◮ Liouville quantum gravity: e γ h ( z ) dz where h is a GFF and γ ∈ [0 , 2) ◮ Introduced by Polyakov in the 1980s ◮ Does not make literal sense since h takes values in the space of distributions (Number of subdivisions) Jason Miller (Cambridge) LQG and TBM July 15, 2015 9 / 24

  6. Liouville quantum gravity γ = 0 . 5 ◮ Liouville quantum gravity: e γ h ( z ) dz where h is a GFF and γ ∈ [0 , 2) ◮ Introduced by Polyakov in the 1980s ◮ Does not make literal sense since h takes values in the space of distributions ◮ Has been made sense of as a random area measure using a regularization procedure ◮ Can compute areas of regions and lengths of curves ◮ Does not come with an obvious notion of “distance” (Number of subdivisions) Jason Miller (Cambridge) LQG and TBM July 15, 2015 9 / 24

  7. Liouville quantum gravity γ = 1 . 0 ◮ Liouville quantum gravity: e γ h ( z ) dz where h is a GFF and γ ∈ [0 , 2) ◮ Introduced by Polyakov in the 1980s ◮ Does not make literal sense since h takes values in the space of distributions ◮ Has been made sense of as a random area measure using a regularization procedure ◮ Can compute areas of regions and lengths of curves ◮ Does not come with an obvious notion of “distance” (Number of subdivisions) Jason Miller (Cambridge) LQG and TBM July 15, 2015 9 / 24

  8. Liouville quantum gravity γ = 1 . 5 ◮ Liouville quantum gravity: e γ h ( z ) dz where h is a GFF and γ ∈ [0 , 2) ◮ Introduced by Polyakov in the 1980s ◮ Does not make literal sense since h takes values in the space of distributions ◮ Has been made sense of as a random area measure using a regularization procedure ◮ Can compute areas of regions and lengths of curves ◮ Does not come with an obvious notion of “distance” (Number of subdivisions) Jason Miller (Cambridge) LQG and TBM July 15, 2015 9 / 24

  9. Liouville quantum gravity γ = 2 . 0 ◮ Liouville quantum gravity: e γ h ( z ) dz where h is a GFF and γ ∈ [0 , 2) ◮ Introduced by Polyakov in the 1980s ◮ Does not make literal sense since h takes values in the space of distributions ◮ Has been made sense of as a random area measure using a regularization procedure ◮ Can compute areas of regions and lengths of curves ◮ Does not come with an obvious notion of “distance” (Number of subdivisions) Jason Miller (Cambridge) LQG and TBM July 15, 2015 9 / 24

  10. LQG and TBM ◮ Two “canonical” (but very different) constructions of random surfaces: Liouville quantum gravity (LQG) and the Brownian map (TBM) Jason Miller (Cambridge) LQG and TBM July 15, 2015 10 / 24

  11. LQG and TBM ◮ Two “canonical” (but very different) constructions of random surfaces: Liouville quantum gravity (LQG) and the Brownian map (TBM) ◮ For γ ∈ [0 , 2), Liouville quantum gravity (LQG) is the “random surface” with “Riemannian metric” e γ h ( z ) ( dx 2 + dy 2 ) Jason Miller (Cambridge) LQG and TBM July 15, 2015 10 / 24

  12. LQG and TBM ◮ Two “canonical” (but very different) constructions of random surfaces: Liouville quantum gravity (LQG) and the Brownian map (TBM) ◮ For γ ∈ [0 , 2), Liouville quantum gravity (LQG) is the “random surface” with “Riemannian metric” e γ h ( z ) ( dx 2 + dy 2 ) ◮ So far, only made sense of as an area measure using a regularization procedure Jason Miller (Cambridge) LQG and TBM July 15, 2015 10 / 24

  13. LQG and TBM ◮ Two “canonical” (but very different) constructions of random surfaces: Liouville quantum gravity (LQG) and the Brownian map (TBM) ◮ For γ ∈ [0 , 2), Liouville quantum gravity (LQG) is the “random surface” with “Riemannian metric” e γ h ( z ) ( dx 2 + dy 2 ) ◮ So far, only made sense of as an area measure using a regularization procedure ◮ LQG has a conformal structure (compute angles, etc...) and an area measure Jason Miller (Cambridge) LQG and TBM July 15, 2015 10 / 24

  14. LQG and TBM ◮ Two “canonical” (but very different) constructions of random surfaces: Liouville quantum gravity (LQG) and the Brownian map (TBM) ◮ For γ ∈ [0 , 2), Liouville quantum gravity (LQG) is the “random surface” with “Riemannian metric” e γ h ( z ) ( dx 2 + dy 2 ) ◮ So far, only made sense of as an area measure using a regularization procedure ◮ LQG has a conformal structure (compute angles, etc...) and an area measure ◮ In contrast, TBM has a metric structure and an area measure Jason Miller (Cambridge) LQG and TBM July 15, 2015 10 / 24

  15. LQG and TBM ◮ Two “canonical” (but very different) constructions of random surfaces: Liouville quantum gravity (LQG) and the Brownian map (TBM) ◮ For γ ∈ [0 , 2), Liouville quantum gravity (LQG) is the “random surface” with “Riemannian metric” e γ h ( z ) ( dx 2 + dy 2 ) ◮ So far, only made sense of as an area measure using a regularization procedure ◮ LQG has a conformal structure (compute angles, etc...) and an area measure ◮ In contrast, TBM has a metric structure and an area measure This talk is about endowing each of these objects with the other’s structure and showing they are equivalent. Jason Miller (Cambridge) LQG and TBM July 15, 2015 10 / 24

  16. Canonical embedding of TBM into S 2 ◮ TBM is an abstract metric measure space homeomorphic to S 2 , but it does not obviously come with a canonical embedding into S 2 Jason Miller (Cambridge) LQG and TBM July 15, 2015 11 / 24

  17. Canonical embedding of TBM into S 2 ◮ TBM is an abstract metric measure space homeomorphic to S 2 , but it does not obviously come with a canonical embedding into S 2 ◮ It is believed that there should be a “natural embedding” of TBM into S 2 and that the embedded surface is described by a form of Liouville quantum gravity (LQG) � with γ = 8 / 3 Jason Miller (Cambridge) LQG and TBM July 15, 2015 11 / 24

  18. Canonical embedding of TBM into S 2 ◮ TBM is an abstract metric measure space homeomorphic to S 2 , but it does not obviously come with a canonical embedding into S 2 ◮ It is believed that there should be a “natural embedding” of TBM into S 2 and that the embedded surface is described by a form of Liouville quantum gravity (LQG) � with γ = 8 / 3 ψ ◮ Discrete approach: take a uniformly random planar map and embed it conformally into S 2 (circle packing, uniformization, etc...), then in the n → ∞ limit it converges � to a form of 8 / 3-LQG. Jason Miller (Cambridge) LQG and TBM July 15, 2015 11 / 24

  19. Canonical embedding of TBM into S 2 ◮ TBM is an abstract metric measure space homeomorphic to S 2 , but it does not obviously come with a canonical embedding into S 2 ◮ It is believed that there should be a “natural embedding” of TBM into S 2 and that the embedded surface is described by a form of Liouville quantum gravity (LQG) � with γ = 8 / 3 ψ ◮ Discrete approach: take a uniformly random planar map and embed it conformally into S 2 (circle packing, uniformization, etc...), then in the n → ∞ limit it converges � to a form of 8 / 3-LQG. Not the approach we will describe today ... Jason Miller (Cambridge) LQG and TBM July 15, 2015 11 / 24

  20. Main result Theorem (M., Sheffield) Suppose that ( M , d , µ ) is an instance of TBM. Then there exists a H¨ older homeomorphism ϕ : ( M , d ) → S 2 such that the pushforward of µ by ϕ has the law of a � 8 / 3 -LQG sphere ( S 2 , h ) . Jason Miller (Cambridge) LQG and TBM July 15, 2015 12 / 24

  21. Main result Theorem (M., Sheffield) Suppose that ( M , d , µ ) is an instance of TBM. Then there exists a H¨ older homeomorphism ϕ : ( M , d ) → S 2 such that the pushforward of µ by ϕ has the law of a � 8 / 3 -LQG sphere ( S 2 , h ) . Moreover, Jason Miller (Cambridge) LQG and TBM July 15, 2015 12 / 24

  22. Main result Theorem (M., Sheffield) Suppose that ( M , d , µ ) is an instance of TBM. Then there exists a H¨ older homeomorphism ϕ : ( M , d ) → S 2 such that the pushforward of µ by ϕ has the law of a � 8 / 3 -LQG sphere ( S 2 , h ) . Moreover, ◮ ϕ is determined by ( M , d , µ ) Jason Miller (Cambridge) LQG and TBM July 15, 2015 12 / 24

  23. Main result Theorem (M., Sheffield) Suppose that ( M , d , µ ) is an instance of TBM. Then there exists a H¨ older homeomorphism ϕ : ( M , d ) → S 2 such that the pushforward of µ by ϕ has the law of a � 8 / 3 -LQG sphere ( S 2 , h ) . Moreover, ◮ ϕ is determined by ( M , d , µ ) (TBM determines its conformal structure) Jason Miller (Cambridge) LQG and TBM July 15, 2015 12 / 24

  24. Main result Theorem (M., Sheffield) Suppose that ( M , d , µ ) is an instance of TBM. Then there exists a H¨ older homeomorphism ϕ : ( M , d ) → S 2 such that the pushforward of µ by ϕ has the law of a � 8 / 3 -LQG sphere ( S 2 , h ) . Moreover, ◮ ϕ is determined by ( M , d , µ ) (TBM determines its conformal structure) ◮ ( M , d , µ ) and ϕ are determined by ( S 2 , h ) Jason Miller (Cambridge) LQG and TBM July 15, 2015 12 / 24

  25. Main result Theorem (M., Sheffield) Suppose that ( M , d , µ ) is an instance of TBM. Then there exists a H¨ older homeomorphism ϕ : ( M , d ) → S 2 such that the pushforward of µ by ϕ has the law of a � 8 / 3 -LQG sphere ( S 2 , h ) . Moreover, ◮ ϕ is determined by ( M , d , µ ) (TBM determines its conformal structure) ◮ ( M , d , µ ) and ϕ are determined by ( S 2 , h ) (LQG determines its metric structure) Jason Miller (Cambridge) LQG and TBM July 15, 2015 12 / 24

  26. Main result Theorem (M., Sheffield) Suppose that ( M , d , µ ) is an instance of TBM. Then there exists a H¨ older homeomorphism ϕ : ( M , d ) → S 2 such that the pushforward of µ by ϕ has the law of a � 8 / 3 -LQG sphere ( S 2 , h ) . Moreover, ◮ ϕ is determined by ( M , d , µ ) (TBM determines its conformal structure) ◮ ( M , d , µ ) and ϕ are determined by ( S 2 , h ) (LQG determines its metric structure) That is, ( M , d , µ ) and ( S 2 , h ) are equivalent. Jason Miller (Cambridge) LQG and TBM July 15, 2015 12 / 24

  27. Main result Theorem (M., Sheffield) Suppose that ( M , d , µ ) is an instance of TBM. Then there exists a H¨ older homeomorphism ϕ : ( M , d ) → S 2 such that the pushforward of µ by ϕ has the law of a � 8 / 3 -LQG sphere ( S 2 , h ) . Moreover, ◮ ϕ is determined by ( M , d , µ ) (TBM determines its conformal structure) ◮ ( M , d , µ ) and ϕ are determined by ( S 2 , h ) (LQG determines its metric structure) That is, ( M , d , µ ) and ( S 2 , h ) are equivalent. Comments Jason Miller (Cambridge) LQG and TBM July 15, 2015 12 / 24

  28. Main result Theorem (M., Sheffield) Suppose that ( M , d , µ ) is an instance of TBM. Then there exists a H¨ older homeomorphism ϕ : ( M , d ) → S 2 such that the pushforward of µ by ϕ has the law of a � 8 / 3 -LQG sphere ( S 2 , h ) . Moreover, ◮ ϕ is determined by ( M , d , µ ) (TBM determines its conformal structure) ◮ ( M , d , µ ) and ϕ are determined by ( S 2 , h ) (LQG determines its metric structure) That is, ( M , d , µ ) and ( S 2 , h ) are equivalent. Comments 1. Construction is purely in the continuum Jason Miller (Cambridge) LQG and TBM July 15, 2015 12 / 24

  29. Main result Theorem (M., Sheffield) Suppose that ( M , d , µ ) is an instance of TBM. Then there exists a H¨ older homeomorphism ϕ : ( M , d ) → S 2 such that the pushforward of µ by ϕ has the law of a � 8 / 3 -LQG sphere ( S 2 , h ) . Moreover, ◮ ϕ is determined by ( M , d , µ ) (TBM determines its conformal structure) ◮ ( M , d , µ ) and ϕ are determined by ( S 2 , h ) (LQG determines its metric structure) That is, ( M , d , µ ) and ( S 2 , h ) are equivalent. Comments 1. Construction is purely in the continuum � 2. Proof by endowing a metric space structure directly on 8 / 3-LQG using the growth process QLE (8 / 3 , 0) Jason Miller (Cambridge) LQG and TBM July 15, 2015 12 / 24

  30. Main result Theorem (M., Sheffield) Suppose that ( M , d , µ ) is an instance of TBM. Then there exists a H¨ older homeomorphism ϕ : ( M , d ) → S 2 such that the pushforward of µ by ϕ has the law of a � 8 / 3 -LQG sphere ( S 2 , h ) . Moreover, ◮ ϕ is determined by ( M , d , µ ) (TBM determines its conformal structure) ◮ ( M , d , µ ) and ϕ are determined by ( S 2 , h ) (LQG determines its metric structure) That is, ( M , d , µ ) and ( S 2 , h ) are equivalent. Comments 1. Construction is purely in the continuum � 2. Proof by endowing a metric space structure directly on 8 / 3-LQG using the growth process QLE (8 / 3 , 0) 3. Resulting metric space structure is shown to satisfy axioms which characterize TBM Jason Miller (Cambridge) LQG and TBM July 15, 2015 12 / 24

  31. Main result Theorem (M., Sheffield) Suppose that ( M , d , µ ) is an instance of TBM. Then there exists a H¨ older homeomorphism ϕ : ( M , d ) → S 2 such that the pushforward of µ by ϕ has the law of a � 8 / 3 -LQG sphere ( S 2 , h ) . Moreover, ◮ ϕ is determined by ( M , d , µ ) (TBM determines its conformal structure) ◮ ( M , d , µ ) and ϕ are determined by ( S 2 , h ) (LQG determines its metric structure) That is, ( M , d , µ ) and ( S 2 , h ) are equivalent. Comments 1. Construction is purely in the continuum � 2. Proof by endowing a metric space structure directly on 8 / 3-LQG using the growth process QLE (8 / 3 , 0) 3. Resulting metric space structure is shown to satisfy axioms which characterize TBM � 4. Separate argument shows the embedding of TBM into 8 / 3-LQG is determined by TBM Jason Miller (Cambridge) LQG and TBM July 15, 2015 12 / 24

  32. Main result Theorem (M., Sheffield) Suppose that ( M , d , µ ) is an instance of TBM. Then there exists a H¨ older homeomorphism ϕ : ( M , d ) → S 2 such that the pushforward of µ by ϕ has the law of a � 8 / 3 -LQG sphere ( S 2 , h ) . Moreover, ◮ ϕ is determined by ( M , d , µ ) (TBM determines its conformal structure) ◮ ( M , d , µ ) and ϕ are determined by ( S 2 , h ) (LQG determines its metric structure) That is, ( M , d , µ ) and ( S 2 , h ) are equivalent. Comments 1. Construction is purely in the continuum � 2. Proof by endowing a metric space structure directly on 8 / 3-LQG using the growth process QLE (8 / 3 , 0) 3. Resulting metric space structure is shown to satisfy axioms which characterize TBM � 4. Separate argument shows the embedding of TBM into 8 / 3-LQG is determined by TBM � 5. Metric construction is for the 8 / 3-LQG sphere. By absolute continuity, can construct a � metric on any 8 / 3-LQG surface. Jason Miller (Cambridge) LQG and TBM July 15, 2015 12 / 24

  33. Part II: � 8 / 3-LQG Construction of the metric on Jason Miller (Cambridge) LQG and TBM July 15, 2015 13 / 24

  34. Detour: first passage percolation (FPP) ◮ Associate with a graph ( V , E ) i.i.d. exp(1) edge weights Jason Miller (Cambridge) LQG and TBM July 15, 2015 14 / 24

  35. Detour: first passage percolation (FPP) ◮ Associate with a graph ( V , E ) i.i.d. exp(1) edge weights 1 . 36 4 . 61 0 . 42 0 . 32 0 . 75 0 . 16 1 . 84 1 . 27 0 . 47 Jason Miller (Cambridge) LQG and TBM July 15, 2015 14 / 24

  36. Detour: first passage percolation (FPP) ◮ Associate with a graph ( V , E ) i.i.d. exp(1) edge weights ◮ Introduced by Eden (1961) and Hammersley and Welsh (1965) 1 . 36 4 . 61 0 . 42 0 . 32 0 . 75 0 . 16 1 . 84 1 . 27 0 . 47 Jason Miller (Cambridge) LQG and TBM July 15, 2015 14 / 24

  37. Detour: first passage percolation (FPP) ◮ Associate with a graph ( V , E ) i.i.d. exp(1) edge weights ◮ Introduced by Eden (1961) and Hammersley and Welsh (1965) ◮ On Z 2 ? Jason Miller (Cambridge) LQG and TBM July 15, 2015 14 / 24

  38. Detour: first passage percolation (FPP) ◮ Associate with a graph ( V , E ) i.i.d. exp(1) edge weights ◮ Introduced by Eden (1961) and Hammersley and Welsh (1965) ◮ On Z 2 ? ◮ Question: Large scale behavior of shape of ball wrt perturbed metric? Jason Miller (Cambridge) LQG and TBM July 15, 2015 14 / 24

  39. Detour: first passage percolation (FPP) ◮ Associate with a graph ( V , E ) i.i.d. exp(1) edge weights ◮ Introduced by Eden (1961) and Hammersley and Welsh (1965) ◮ On Z 2 ? ◮ Question: Large scale behavior of shape of ball wrt perturbed metric? Jason Miller (Cambridge) LQG and TBM July 15, 2015 14 / 24

  40. Detour: first passage percolation (FPP) ◮ Associate with a graph ( V , E ) i.i.d. exp(1) edge weights ◮ Introduced by Eden (1961) and Hammersley and Welsh (1965) ◮ On Z 2 ? ◮ Question: Large scale behavior of shape of ball wrt perturbed metric? ◮ Cox and Durrett (1981) showed that the macroscopic shape is convex Jason Miller (Cambridge) LQG and TBM July 15, 2015 14 / 24

  41. Detour: first passage percolation (FPP) ◮ Associate with a graph ( V , E ) i.i.d. exp(1) edge weights ◮ Introduced by Eden (1961) and Hammersley and Welsh (1965) ◮ On Z 2 ? ◮ Question: Large scale behavior of shape of ball wrt perturbed metric? ◮ Cox and Durrett (1981) showed that the macroscopic shape is convex ◮ Computer simulations show that it is not a Euclidean disk Jason Miller (Cambridge) LQG and TBM July 15, 2015 14 / 24

  42. Detour: first passage percolation (FPP) ◮ Associate with a graph ( V , E ) i.i.d. exp(1) edge weights ◮ Introduced by Eden (1961) and Hammersley and Welsh (1965) ◮ On Z 2 ? ◮ Question: Large scale behavior of shape of ball wrt perturbed metric? ◮ Cox and Durrett (1981) showed that the macroscopic shape is convex ◮ Computer simulations show that it is not a Euclidean disk ◮ Z 2 is not isotropic enough Jason Miller (Cambridge) LQG and TBM July 15, 2015 14 / 24

  43. Detour: first passage percolation (FPP) ◮ Associate with a graph ( V , E ) i.i.d. exp(1) edge weights ◮ Introduced by Eden (1961) and Hammersley and Welsh (1965) ◮ On Z 2 ? ◮ Question: Large scale behavior of shape of ball wrt perturbed metric? ◮ Cox and Durrett (1981) showed that the macroscopic shape is convex ◮ Computer simulations show that it is not a Euclidean disk ◮ Z 2 is not isotropic enough ◮ Vahidi-Asl and Weirmann (1990) showed that the rescaled ball converges to a disk if Z 2 is replaced by the Voronoi tesselation associated with a Poisson process Jason Miller (Cambridge) LQG and TBM July 15, 2015 14 / 24

  44. FPP on random planar maps I ◮ RPM, random vertex x . Perform FPP from x (Angel’s peeling process). Jason Miller (Cambridge) LQG and TBM July 15, 2015 15 / 24

  45. FPP on random planar maps I ◮ RPM, random vertex x . Perform FPP from x (Angel’s peeling process). Jason Miller (Cambridge) LQG and TBM July 15, 2015 15 / 24

  46. FPP on random planar maps I ◮ RPM, random vertex x . Perform FPP from x (Angel’s peeling process). Jason Miller (Cambridge) LQG and TBM July 15, 2015 15 / 24

  47. FPP on random planar maps I ◮ RPM, random vertex x . Perform FPP from x (Angel’s peeling process). Jason Miller (Cambridge) LQG and TBM July 15, 2015 15 / 24

  48. FPP on random planar maps I ◮ RPM, random vertex x . Perform FPP from x (Angel’s peeling process). Jason Miller (Cambridge) LQG and TBM July 15, 2015 15 / 24

  49. FPP on random planar maps I ◮ RPM, random vertex x . Perform FPP from x (Angel’s peeling process). Jason Miller (Cambridge) LQG and TBM July 15, 2015 15 / 24

  50. FPP on random planar maps I ◮ RPM, random vertex x . Perform FPP from x (Angel’s peeling process). Jason Miller (Cambridge) LQG and TBM July 15, 2015 15 / 24

  51. FPP on random planar maps I ◮ RPM, random vertex x . Perform FPP from x (Angel’s peeling process). Jason Miller (Cambridge) LQG and TBM July 15, 2015 15 / 24

  52. FPP on random planar maps I ◮ RPM, random vertex x . Perform FPP from x (Angel’s peeling process). Jason Miller (Cambridge) LQG and TBM July 15, 2015 15 / 24

  53. FPP on random planar maps I ◮ RPM, random vertex x . Perform FPP from x (Angel’s peeling process). Jason Miller (Cambridge) LQG and TBM July 15, 2015 15 / 24

  54. FPP on random planar maps I ◮ RPM, random vertex x . Perform FPP from x (Angel’s peeling process). Jason Miller (Cambridge) LQG and TBM July 15, 2015 15 / 24

  55. FPP on random planar maps I ◮ RPM, random vertex x . Perform FPP from x (Angel’s peeling process). Important observations: ◮ Conditional law of map given growth at time n only depends on the boundary lengths of the outside components. Jason Miller (Cambridge) LQG and TBM July 15, 2015 15 / 24

  56. FPP on random planar maps I ◮ RPM, random vertex x . Perform FPP from x (Angel’s peeling process). Important observations: ◮ Conditional law of map given growth at time n only depends on the boundary lengths of the outside components. Exploration respects the Markovian structure of the map. Jason Miller (Cambridge) LQG and TBM July 15, 2015 15 / 24

  57. FPP on random planar maps I ◮ RPM, random vertex x . Perform FPP from x (Angel’s peeling process). Important observations: ◮ Conditional law of map given growth at time n only depends on the boundary lengths of the outside components. Exploration respects the Markovian structure of the map. Belief: Isotropic enough so that at large scales this is close to a ball in the graph metric (now proved by Curien and Le Gall) Jason Miller (Cambridge) LQG and TBM July 15, 2015 15 / 24

  58. First passage percolation on random planar maps II Goal: Make sense of FPP in the continuum on top of a LQG surface ◮ We do not know how to take a continuum limit of FPP on a random planar map and couple it directly with LQG ◮ Explain a discrete variant of FPP that involves two operations that we do know how to perform in the continuum: ◮ Sample random points according to boundary length ◮ Draw (scaling limits of) critical percolation interfaces ( SLE 6 ) Jason Miller (Cambridge) LQG and TBM July 15, 2015 16 / 24

  59. FPP on random planar maps II Variant: ◮ Pick two edges on outer boundary of cluster Jason Miller (Cambridge) LQG and TBM July 15, 2015 17 / 24

  60. FPP on random planar maps II Variant: ◮ Pick two edges on outer boundary of cluster ◮ Color vertices between edges blue and yellow Jason Miller (Cambridge) LQG and TBM July 15, 2015 17 / 24

  61. FPP on random planar maps II Variant: ◮ Pick two edges on outer boundary of cluster ◮ Color vertices between edges blue and yellow ◮ Color vertices on rest of map blue or yellow with prob. 1 2 Jason Miller (Cambridge) LQG and TBM July 15, 2015 17 / 24

  62. FPP on random planar maps II Variant: ◮ Pick two edges on outer boundary of cluster ◮ Color vertices between edges blue and yellow ◮ Color vertices on rest of map blue or yellow with prob. 1 2 ◮ Explore percolation (blue/yellow) interface Jason Miller (Cambridge) LQG and TBM July 15, 2015 17 / 24

  63. FPP on random planar maps II Variant: ◮ Pick two edges on outer boundary of cluster ◮ Color vertices between edges blue and yellow ◮ Color vertices on rest of map blue or yellow with prob. 1 2 ◮ Explore percolation (blue/yellow) interface ◮ Forget colors Jason Miller (Cambridge) LQG and TBM July 15, 2015 17 / 24

  64. FPP on random planar maps II Variant: ◮ Pick two edges on outer boundary of cluster ◮ Color vertices between edges blue and yellow ◮ Color vertices on rest of map blue or yellow with prob. 1 2 ◮ Explore percolation (blue/yellow) interface ◮ Forget colors ◮ Repeat Jason Miller (Cambridge) LQG and TBM July 15, 2015 17 / 24

  65. FPP on random planar maps II Variant: ◮ Pick two edges on outer boundary of cluster ◮ Color vertices between edges blue and yellow ◮ Color vertices on rest of map blue or yellow with prob. 1 2 ◮ Explore percolation (blue/yellow) interface ◮ Forget colors ◮ Repeat Jason Miller (Cambridge) LQG and TBM July 15, 2015 17 / 24

  66. FPP on random planar maps II Variant: ◮ Pick two edges on outer boundary of cluster ◮ Color vertices between edges blue and yellow ◮ Color vertices on rest of map blue or yellow with prob. 1 2 ◮ Explore percolation (blue/yellow) interface ◮ Forget colors ◮ Repeat Jason Miller (Cambridge) LQG and TBM July 15, 2015 17 / 24

  67. FPP on random planar maps II Variant: ◮ Pick two edges on outer boundary of cluster ◮ Color vertices between edges blue and yellow ◮ Color vertices on rest of map blue or yellow with prob. 1 2 ◮ Explore percolation (blue/yellow) interface ◮ Forget colors ◮ Repeat Jason Miller (Cambridge) LQG and TBM July 15, 2015 17 / 24

  68. FPP on random planar maps II Variant: ◮ Pick two edges on outer boundary of cluster ◮ Color vertices between edges blue and yellow ◮ Color vertices on rest of map blue or yellow with prob. 1 2 ◮ Explore percolation (blue/yellow) interface ◮ Forget colors ◮ Repeat Jason Miller (Cambridge) LQG and TBM July 15, 2015 17 / 24

  69. FPP on random planar maps II Variant: ◮ Pick two edges on outer boundary of cluster ◮ Color vertices between edges blue and yellow ◮ Color vertices on rest of map blue or yellow with prob. 1 2 ◮ Explore percolation (blue/yellow) interface ◮ Forget colors ◮ Repeat Jason Miller (Cambridge) LQG and TBM July 15, 2015 17 / 24

  70. FPP on random planar maps II Variant: ◮ Pick two edges on outer boundary of cluster ◮ Color vertices between edges blue and yellow ◮ Color vertices on rest of map blue or yellow with prob. 1 2 ◮ Explore percolation (blue/yellow) interface ◮ Forget colors ◮ Repeat Jason Miller (Cambridge) LQG and TBM July 15, 2015 17 / 24

  71. FPP on random planar maps II Variant: ◮ Pick two edges on outer boundary of cluster ◮ Color vertices between edges blue and yellow ◮ Color vertices on rest of map blue or yellow with prob. 1 2 ◮ Explore percolation (blue/yellow) interface ◮ Forget colors ◮ Repeat Jason Miller (Cambridge) LQG and TBM July 15, 2015 17 / 24

  72. FPP on random planar maps II Variant: ◮ Pick two edges on outer boundary of cluster ◮ Color vertices between edges blue and yellow ◮ Color vertices on rest of map blue or yellow with prob. 1 2 ◮ Explore percolation (blue/yellow) interface ◮ Forget colors ◮ Repeat Jason Miller (Cambridge) LQG and TBM July 15, 2015 17 / 24

  73. FPP on random planar maps II Variant: ◮ Pick two edges on outer boundary of cluster ◮ Color vertices between edges blue and yellow ◮ Color vertices on rest of map blue or yellow with prob. 1 2 ◮ Explore percolation (blue/yellow) interface ◮ Forget colors ◮ Repeat Jason Miller (Cambridge) LQG and TBM July 15, 2015 17 / 24

  74. FPP on random planar maps II Variant: ◮ Pick two edges on outer boundary of cluster ◮ Color vertices between edges blue and yellow ◮ Color vertices on rest of map blue or yellow with prob. 1 2 ◮ Explore percolation (blue/yellow) interface ◮ Forget colors ◮ Repeat ◮ This exploration also respects the Markovian structure of the map. Jason Miller (Cambridge) LQG and TBM July 15, 2015 17 / 24

  75. FPP on random planar maps II Variant: ◮ Pick two edges on outer boundary of cluster ◮ Color vertices between edges blue and yellow ◮ Color vertices on rest of map blue or yellow with prob. 1 2 ◮ Explore percolation (blue/yellow) interface ◮ Forget colors ◮ Repeat ◮ This exploration also respects the Markovian structure of the map. ◮ Expect that at large scales this growth process looks the same as FPP, hence the same as the graph metric ball Jason Miller (Cambridge) LQG and TBM July 15, 2015 17 / 24

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