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Liouville Quantum Gravity as a Mating of Trees Bertrand Duplantier, Jason Miller, and Scott Sheffield CEA/Saclay and Massachusetts Institute of Technology September 30, 2014 Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of

  1. Constructing a sphere from a pair of trees ◮ X , Y ind. Brownian excursions on [0 , 1] ◮ Red / green lines give an ∼ =-relation on S 2 H = horizontal, V = vertical ◮ Types of equivalence classes: 1. Outer boundary of rectangle C − Y t 2. V line which does not share an endpoint with a H line 3. H line below X or above C − Y with two V lines with common endpoint 4. H line below X or above C − Y with two V lines with common endpoint and a third V line hitting in the middle ◮ ∼ = is topologically closed and does not separate S 2 into two or more components, X t thus S 2 / ∼ = is homeomorphic to S 2 t ◮ Following the V lines from left to right gives a space-filling path on S 2 / ∼ = The sphere/space-filling path pair is a peanoshere Q: What is the canonical embedding of this peanoshere into the Euclidean sphere S 2 ? Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 6 / 30

  2. Part II: Scaling limits of random planar maps and Liouville quantum gravity Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 7 / 30

  3. Random planar maps ◮ A planar map is a finite graph together with an embedding in the plane so that no edges cross Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 8 / 30

  4. Random planar maps ◮ A planar map is a finite graph together with an embedding in the plane so that no edges cross ◮ Its faces are the connected components of the complement of edges Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 8 / 30

  5. Random planar maps ◮ A planar map is a finite graph together with an embedding in the plane so that no edges cross ◮ Its faces are the connected components of the complement of edges ◮ A map is a quadrangulation if each face has 4 adjacent edges Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 8 / 30

  6. Random planar maps ◮ A planar map is a finite graph together with an embedding in the plane so that no edges cross ◮ Its faces are the connected components of the complement of edges ◮ A map is a quadrangulation if each face has 4 adjacent edges ◮ Interested in random quadrangulations with n faces — random planar map (RPM). Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 8 / 30

  7. Random planar maps ◮ A planar map is a finite graph together with an embedding in the plane so that no edges cross ◮ Its faces are the connected components of the complement of edges ◮ A map is a quadrangulation if each face has 4 adjacent edges ◮ Interested in random quadrangulations with n faces — random planar map (RPM). ◮ First studied by Tutte in 1960s while working on the four color theorem ◮ Combinatorics : enumeration formulas ◮ Physics : statistical physics models: percolation, Ising, UST ... ◮ Probability : “uniformly random surface,” Brownian surface Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 8 / 30

  8. Random quadrangulation with 25,000 faces (Simulation due to J.F. Marckert) Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 9 / 30

  9. Laws on quadrangulations ◮ Natural laws on quadrangulations with n faces. Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 10 / 30

  10. Laws on quadrangulations ◮ Natural laws on quadrangulations with n faces. ◮ Uniform measure Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 10 / 30

  11. Laws on quadrangulations ◮ Natural laws on quadrangulations with n faces. ◮ Uniform measure ◮ Weighted by the partition function of the FK model with q ∈ (0 , 4): ◮ For a fixed quadrangulation M , the probability of picking it is proportional to L q # L / 2 where the sum is over loop configurations L and # L is the Z M = � number of loops in L Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 10 / 30

  12. Laws on quadrangulations ◮ Natural laws on quadrangulations with n faces. ◮ Uniform measure ◮ Weighted by the partition function of the FK model with q ∈ (0 , 4): ◮ For a fixed quadrangulation M , the probability of picking it is proportional to L q # L / 2 where the sum is over loop configurations L and # L is the Z M = � number of loops in L ◮ Natural to pick a map/loop-configuration pair ( M , L ) in the FK weighted case Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 10 / 30

  13. Laws on quadrangulations ◮ Natural laws on quadrangulations with n faces. ◮ Uniform measure ◮ Weighted by the partition function of the FK model with q ∈ (0 , 4): ◮ For a fixed quadrangulation M , the probability of picking it is proportional to L q # L / 2 where the sum is over loop configurations L and # L is the Z M = � number of loops in L ◮ Natural to pick a map/loop-configuration pair ( M , L ) in the FK weighted case ◮ Can encode the loops in terms of a tree/dual tree pair ◮ Generate the tree by first picking a root ◮ Generate the branch from the root to any vertex by following the boundaries of the loop configuration until the vertex is cut off from the root, at which point you branch towards the vertex and continue Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 10 / 30

  14. Laws on quadrangulations ◮ Natural laws on quadrangulations with n faces. ◮ Uniform measure ◮ Weighted by the partition function of the FK model with q ∈ (0 , 4): ◮ For a fixed quadrangulation M , the probability of picking it is proportional to L q # L / 2 where the sum is over loop configurations L and # L is the Z M = � number of loops in L ◮ Natural to pick a map/loop-configuration pair ( M , L ) in the FK weighted case ◮ Can encode the loops in terms of a tree/dual tree pair ◮ Generate the tree by first picking a root ◮ Generate the branch from the root to any vertex by following the boundaries of the loop configuration until the vertex is cut off from the root, at which point you branch towards the vertex and continue Sheffield’s Hamburger-Cheeseburger (H-C) bijection encodes an FK-weighted planar map by describing the pair of contour functions which correspond to the tree/dual tree pair Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 10 / 30

  15. Random quadrangulation Sampled using H-C bijection. Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 11 / 30

  16. Red tree Sampled using H-C bijection. Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 11 / 30

  17. Red and blue trees Sampled using H-C bijection. Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 11 / 30

  18. Path snaking between the trees. Encodes the trees and how they are glued together. Sampled using H-C bijection. Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 11 / 30

  19. How was the graph embedded into R 2 ? Sampled using H-C bijection. Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 11 / 30

  20. Can subivide each quadrilateral to obtain a triangulation without multiple edges. Sampled using H-C bijection. Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 11 / 30

  21. Circle pack the resulting triangulation. Sampled using H-C bijection. Packed with Stephenson’s CirclePack. Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 11 / 30

  22. Circle pack the resulting triangulation. Sampled using H-C bijection. Packed with Stephenson’s CirclePack. Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 11 / 30

  23. Circle pack the resulting triangulation. Sampled using H-C bijection. Packed with Stephenson’s CirclePack. Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 11 / 30

  24. What is the “limit” of this embedding? Circle packings are related to conformal maps. Sampled using H-C bijection. Packed with Stephenson’s CirclePack. Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 11 / 30

  25. Random planar map convergence results Uniformly random ◮ Diameter is ≍ n 1 / 4 , profile of distances from random point (Chaissang-Schaefer) Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 12 / 30

  26. Random planar map convergence results Uniformly random ◮ Diameter is ≍ n 1 / 4 , profile of distances from random point (Chaissang-Schaefer) ◮ Existence of subsequential limits after rescaling distances by n − 1 / 4 (Le Gall) Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 12 / 30

  27. Random planar map convergence results Uniformly random ◮ Diameter is ≍ n 1 / 4 , profile of distances from random point (Chaissang-Schaefer) ◮ Existence of subsequential limits after rescaling distances by n − 1 / 4 (Le Gall) ◮ Existence of limit to the Brownian map (Le Gall, Miermont) Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 12 / 30

  28. Random planar map convergence results Uniformly random ◮ Diameter is ≍ n 1 / 4 , profile of distances from random point (Chaissang-Schaefer) ◮ Existence of subsequential limits after rescaling distances by n − 1 / 4 (Le Gall) ◮ Existence of limit to the Brownian map (Le Gall, Miermont) FK-weighted Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 12 / 30

  29. Random planar map convergence results Uniformly random ◮ Diameter is ≍ n 1 / 4 , profile of distances from random point (Chaissang-Schaefer) ◮ Existence of subsequential limits after rescaling distances by n − 1 / 4 (Le Gall) ◮ Existence of limit to the Brownian map (Le Gall, Miermont) FK-weighted ◮ H-C bijection encodes an FK weighted random planar map in terms of a pair of random discrete trees glued together along a space-filling path Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 12 / 30

  30. Random planar map convergence results Uniformly random ◮ Diameter is ≍ n 1 / 4 , profile of distances from random point (Chaissang-Schaefer) ◮ Existence of subsequential limits after rescaling distances by n − 1 / 4 (Le Gall) ◮ Existence of limit to the Brownian map (Le Gall, Miermont) FK-weighted ◮ H-C bijection encodes an FK weighted random planar map in terms of a pair of random discrete trees glued together along a space-filling path ◮ Sheffield proved that the contour functions of these two discrete trees properly rescaled converge to a pair of Brownian excursions Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 12 / 30

  31. Random planar map convergence results Uniformly random ◮ Diameter is ≍ n 1 / 4 , profile of distances from random point (Chaissang-Schaefer) ◮ Existence of subsequential limits after rescaling distances by n − 1 / 4 (Le Gall) ◮ Existence of limit to the Brownian map (Le Gall, Miermont) FK-weighted ◮ H-C bijection encodes an FK weighted random planar map in terms of a pair of random discrete trees glued together along a space-filling path ◮ Sheffield proved that the contour functions of these two discrete trees properly rescaled converge to a pair of Brownian excursions ◮ For UST weighted random planar maps ( q = 0), the CRTs are independent. For general q ∈ (0 , 4), the CRTs are correlated Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 12 / 30

  32. Random planar map convergence results Uniformly random ◮ Diameter is ≍ n 1 / 4 , profile of distances from random point (Chaissang-Schaefer) ◮ Existence of subsequential limits after rescaling distances by n − 1 / 4 (Le Gall) ◮ Existence of limit to the Brownian map (Le Gall, Miermont) FK-weighted ◮ H-C bijection encodes an FK weighted random planar map in terms of a pair of random discrete trees glued together along a space-filling path ◮ Sheffield proved that the contour functions of these two discrete trees properly rescaled converge to a pair of Brownian excursions ◮ For UST weighted random planar maps ( q = 0), the CRTs are independent. For general q ∈ (0 , 4), the CRTs are correlated ◮ Canonical embedding of peanospheres that come from gluing correlated CRTs is thus related to the problem of describing the scaling limits of FK weighted random planar maps embedded into C ∪ {∞} Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 12 / 30

  33. 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) Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 13 / 30

  34. 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 as a generalization of the path integral to the setting of surfaces (Number of subdivisions) Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 13 / 30

  35. 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 as a generalization of the path integral to the setting of surfaces ◮ Does not make literal sense since h takes values in the space of distributions (Number of subdivisions) Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 13 / 30

  36. 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 as a generalization of the path integral to the setting of surfaces ◮ Does not make literal sense since h takes values in the space of distributions ◮ Can be made sense of as a random area measure using a regularization procedure ◮ Can compute areas of regions and lengths of curves (Number of subdivisions) Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 13 / 30

  37. 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 as a generalization of the path integral to the setting of surfaces ◮ Does not make literal sense since h takes values in the space of distributions ◮ Can be made sense of as a random area measure using a regularization procedure ◮ Can compute areas of regions and lengths of curves ◮ Conjectured to describe the limit of conformally embedded FK-weighted random planar maps (Number of subdivisions) Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 13 / 30

  38. 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 as a generalization of the path integral to the setting of surfaces ◮ Does not make literal sense since h takes values in the space of distributions ◮ Can be made sense of as a random area measure using a regularization procedure ◮ Can compute areas of regions and lengths of curves ◮ Conjectured to describe the limit of conformally embedded FK-weighted random planar maps (Number of subdivisions) Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 13 / 30

  39. 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 as a generalization of the path integral to the setting of surfaces ◮ Does not make literal sense since h takes values in the space of distributions ◮ Can be made sense of as a random area measure using a regularization procedure ◮ Can compute areas of regions and lengths of curves ◮ Conjectured to describe the limit of conformally embedded FK-weighted random planar maps (Number of subdivisions) Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 13 / 30

  40. 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 as a generalization of the path integral to the setting of surfaces ◮ Does not make literal sense since h takes values in the space of distributions ◮ Can be made sense of as a random area measure using a regularization procedure ◮ Can compute areas of regions and lengths of curves ◮ Conjectured to describe the limit of conformally embedded FK-weighted random planar maps (Number of subdivisions) Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 13 / 30

  41. Scaling limit conjectures ψ (Simulation due to J.-F. Marckert) ◮ Uniform RPM conformally embedded into S 2 converges to � 8 / 3-LQG as n → ∞ Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 14 / 30

  42. Scaling limit conjectures ψ (Simulation due to J.-F. Marckert) ◮ Uniform RPM conformally embedded into S 2 converges to � 8 / 3-LQG as n → ∞ ◮ For q ∈ [0 , 4), FK weighted RPM together with loop configuration conformally embedded into S 2 converges to γ -LQG as n → ∞ decorated by an independent CLE κ ′ where √ q = 2 + 2 cos 8 π κ ′ ∈ (4 , 8] . 16 /κ ′ ∈ [ � κ ′ , γ = 2 , 2) , Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 14 / 30

  43. Part III: Results Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 15 / 30

  44. Main result Theorem (Duplantier, M., Sheffield) For each γ ∈ (0 , 2) there is a type of γ -LQG surface such that the following are true: Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 16 / 30

  45. Main result Theorem (Duplantier, M., Sheffield) For each γ ∈ (0 , 2) there is a type of γ -LQG surface such that the following are true: ◮ If we explore with an independent space-filling SLE κ ′ process, κ ′ = 16 γ 2 , then the LQG lengths of its left and right sides evolve as a 2 D Brownian motion ( L , R ) Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 16 / 30

  46. Main result Theorem (Duplantier, M., Sheffield) For each γ ∈ (0 , 2) there is a type of γ -LQG surface such that the following are true: ◮ If we explore with an independent space-filling SLE κ ′ process, κ ′ = 16 γ 2 , then the LQG lengths of its left and right sides evolve as a 2 D Brownian motion ( L , R ) ◮ ( L , R ) almost surely determine both the γ -LQG surface and the SLE κ ′ Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 16 / 30

  47. Main result Theorem (Duplantier, M., Sheffield) For each γ ∈ (0 , 2) there is a type of γ -LQG surface such that the following are true: ◮ If we explore with an independent space-filling SLE κ ′ process, κ ′ = 16 γ 2 , then the LQG lengths of its left and right sides evolve as a 2 D Brownian motion ( L , R ) ◮ ( L , R ) almost surely determine both the γ -LQG surface and the SLE κ ′ Comments Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 16 / 30

  48. Main result Theorem (Duplantier, M., Sheffield) For each γ ∈ (0 , 2) there is a type of γ -LQG surface such that the following are true: ◮ If we explore with an independent space-filling SLE κ ′ process, κ ′ = 16 γ 2 , then the LQG lengths of its left and right sides evolve as a 2 D Brownian motion ( L , R ) ◮ ( L , R ) almost surely determine both the γ -LQG surface and the SLE κ ′ Comments ◮ Space-filling SLE κ ′ is the peano curve associated with the continuum tree/dual tree pair which encodes CLE κ ′ Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 16 / 30

  49. Main result Theorem (Duplantier, M., Sheffield) For each γ ∈ (0 , 2) there is a type of γ -LQG surface such that the following are true: ◮ If we explore with an independent space-filling SLE κ ′ process, κ ′ = 16 γ 2 , then the LQG lengths of its left and right sides evolve as a 2 D Brownian motion ( L , R ) ◮ ( L , R ) almost surely determine both the γ -LQG surface and the SLE κ ′ Comments ◮ Space-filling SLE κ ′ is the peano curve associated with the continuum tree/dual tree pair which encodes CLE κ ′ ◮ Combined with the convergence for the H-C bijection, this says that FK weighted RPM converge to CLE -decorated LQG with respect to the topology where two loop-decorated surfaces are close if the contour functions of their tree/dual tree pair are close Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 16 / 30

  50. Main result Theorem (Duplantier, M., Sheffield) For each γ ∈ (0 , 2) there is a type of γ -LQG surface such that the following are true: ◮ If we explore with an independent space-filling SLE κ ′ process, κ ′ = 16 γ 2 , then the LQG lengths of its left and right sides evolve as a 2 D Brownian motion ( L , R ) ◮ ( L , R ) almost surely determine both the γ -LQG surface and the SLE κ ′ Comments ◮ Space-filling SLE κ ′ is the peano curve associated with the continuum tree/dual tree pair which encodes CLE κ ′ ◮ Combined with the convergence for the H-C bijection, this says that FK weighted RPM converge to CLE -decorated LQG with respect to the topology where two loop-decorated surfaces are close if the contour functions of their tree/dual tree pair are close ◮ For planar lattices, the FK models which have been shown to converge to SLE are the UST ( q = 0), percolation ( q = 1), FK-Ising model ( q = 2) (Lawler-Schramm-Werner, Smirnov). Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 16 / 30

  51. Main result Theorem (Duplantier, M., Sheffield) For each γ ∈ (0 , 2) there is a type of γ -LQG surface such that the following are true: ◮ If we explore with an independent space-filling SLE κ ′ process, κ ′ = 16 γ 2 , then the LQG lengths of its left and right sides evolve as a 2 D Brownian motion ( L , R ) ◮ ( L , R ) almost surely determine both the γ -LQG surface and the SLE κ ′ Comments ◮ Space-filling SLE κ ′ is the peano curve associated with the continuum tree/dual tree pair which encodes CLE κ ′ ◮ Combined with the convergence for the H-C bijection, this says that FK weighted RPM converge to CLE -decorated LQG with respect to the topology where two loop-decorated surfaces are close if the contour functions of their tree/dual tree pair are close ◮ For planar lattices, the FK models which have been shown to converge to SLE are the UST ( q = 0), percolation ( q = 1), FK-Ising model ( q = 2) (Lawler-Schramm-Werner, Smirnov). ◮ The above result implies the convergence for all q ∈ [0 , 4) on RPM to SLE κ ′ with √ q = 2 + 2 cos 8 π κ ′ ∈ (4 , 8] . � 16 /κ ′ ∈ [ κ ′ , γ = 2 , 2) , Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 16 / 30

  52. Main result Theorem (Duplantier, M., Sheffield) For each γ ∈ (0 , 2) there is a type of γ -LQG surface such that the following are true: ◮ If we explore with an independent space-filling SLE κ ′ process, κ ′ = 16 γ 2 , then the LQG lengths of its left and right sides evolve as a 2 D Brownian motion ( L , R ) ◮ ( L , R ) almost surely determine both the γ -LQG surface and the SLE κ ′ Comments ◮ Space-filling SLE κ ′ is the peano curve associated with the continuum tree/dual tree pair which encodes CLE κ ′ ◮ Combined with the convergence for the H-C bijection, this says that FK weighted RPM converge to CLE -decorated LQG with respect to the topology where two loop-decorated surfaces are close if the contour functions of their tree/dual tree pair are close ◮ For planar lattices, the FK models which have been shown to converge to SLE are the UST ( q = 0), percolation ( q = 1), FK-Ising model ( q = 2) (Lawler-Schramm-Werner, Smirnov). ◮ The above result implies the convergence for all q ∈ [0 , 4) on RPM to SLE κ ′ with √ q = 2 + 2 cos 8 π κ ′ ∈ (4 , 8] . � 16 /κ ′ ∈ [ κ ′ , γ = 2 , 2) , ◮ As in the discrete setting, the contour functions of the continuum tree/dual tree pair determine everything Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 16 / 30

  53. Random quadrangulation as a gluing of trees Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 17 / 30

  54. Continuum space-filling path Space-filling SLE 6 on a LQG surface. Random path which encodes the limit of a RPM. Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 18 / 30

  55. A calculus of random surfaces ◮ Types of surfaces: quantum wedges, cones, disks, and spheres ◮ Operations: welding and cutting ◮ Interfaces between welded surfaces are variants of SLE which can be described as GFF flow lines ◮ Conversely, natural to cut these surfaces with SLE -type paths Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 19 / 30

  56. External inputs Imaginary geometry: calculus of flow lines of e ih /χ where h is a GFF. Paths are types of SLE curves. Regions between paths are independent wedges. Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 20 / 30

  57. External inputs Imaginary geometry: calculus of flow lines of e ih /χ where h is a GFF. Paths are types of SLE curves. Regions between paths are independent wedges. Conformal welding: Certain special case of “quantum wedge welding” due to Sheffield. Interface almost surely determined by welding, lengths on left and right sides of interface almost surely agree. Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 20 / 30

  58. Types of random surfaces Quantum wedges W θ ◮ Start with a free boundary GFF h on a Euclidean wedge W θ with angle θ h Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 21 / 30

  59. Types of random surfaces Quantum wedges W θ ◮ Start with a free boundary GFF h on a Euclidean wedge W θ with angle θ ◮ Change coordinates to H with z θ/π . Yields free h boundary GFF plus Q ( θ π − 1) log | z | ψ ( z )= z θ/π H h ◦ ψ + Q log | ψ ′ | Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 21 / 30

  60. Types of random surfaces Quantum wedges W θ ◮ Start with a free boundary GFF h on a Euclidean wedge W θ with angle θ ◮ Change coordinates to H with z θ/π . Yields free h boundary GFF plus Q ( θ π − 1) log | z | ◮ Defined modulo global additive constant; fix additive constant in canonical way ψ ( z )= z θ/π H h ◦ ψ + Q log | ψ ′ | Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 21 / 30

  61. Types of random surfaces Quantum wedges W θ ◮ Start with a free boundary GFF h on a Euclidean wedge W θ with angle θ ◮ Change coordinates to H with z θ/π . Yields free h boundary GFF plus Q ( θ π − 1) log | z | ◮ Defined modulo global additive constant; fix additive constant in canonical way ◮ Parameterize space of wedges by multiple α of ψ ( z )= z θ/π − log | z | or by weight W = γ ( γ + 2 γ − α ) H h ◦ ψ + Q log | ψ ′ | Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 21 / 30

  62. Types of random surfaces Quantum wedges W θ ◮ Start with a free boundary GFF h on a Euclidean wedge W θ with angle θ ◮ Change coordinates to H with z θ/π . Yields free h boundary GFF plus Q ( θ π − 1) log | z | ◮ Defined modulo global additive constant; fix additive constant in canonical way ◮ Parameterize space of wedges by multiple α of ψ ( z )= z θ/π − log | z | or by weight W = γ ( γ + 2 γ − α ) H Quantum cones ◮ Similar to a wedge except start with a GFF on a h ◦ ψ + Q log | ψ ′ | Euclidean cone with angle θ ◮ Parameterize space of cones with multiple α of − log | z | or by weight W = 2 γ ( Q − α ) Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 21 / 30

  63. Types of random surfaces Quantum wedges W θ ◮ Start with a free boundary GFF h on a Euclidean wedge W θ with angle θ ◮ Change coordinates to H with z θ/π . Yields free h boundary GFF plus Q ( θ π − 1) log | z | ◮ Defined modulo global additive constant; fix additive constant in canonical way ◮ Parameterize space of wedges by multiple α of ψ ( z )= z θ/π − log | z | or by weight W = γ ( γ + 2 γ − α ) H Quantum cones ◮ Similar to a wedge except start with a GFF on a h ◦ ψ + Q log | ψ ′ | Euclidean cone with angle θ ◮ Parameterize space of cones with multiple α of − log | z | or by weight W = 2 γ ( Q − α ) Quantum disks and spheres (finite volume surfaces) ◮ Constructed with free boundary GFF and Bessel excursion measures Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 21 / 30

  64. Welding and slicing independent wedges Can “weld” and “slice” quantum wedges to obtain larger/smaller wedges. ◮ Weight parameter W = γ ( γ + 2 γ − α ) is additive under the welding operation. ◮ Interface between welding of independent wedges W 1 , W 2 of weight W 1 and W 2 is an SLE κ ( W 1 − 2; W 2 − 2). ◮ Interface is a deterministic function of W 1 , W 2 . Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 22 / 30

  65. Welding many wedges Can also weld together many wedges W 1 , . . . , W n of weight W 1 , . . . , W n to obtain a wedge W with weight W 1 + · · · + W n . Interfaces are SLE κ ( ρ 1 ; ρ 2 ) type processes coupled together as flow lines of a GFF and are a deterministic function of W 1 , . . . , W n . Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 23 / 30

  66. Welding a wedge to itself Can “weld” left and right sides of a wedge to obtain a cone. Conversely, can slice a cone with an independent SLE to obtain a wedge. ◮ Weight parameter W = 2 γ ( Q − α ) ◮ Welding left and right sides of weight W wedge yields a weight W cone; the interface is an independent whole-plane SLE κ ( W − 2) ◮ Interface is simple if the wedge is “thick” as on the left (homeomorphic to H ); it is self-intersecting if the wedge is thin as on the right (not homeomorphic to H ) Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 24 / 30

  67. Exploring an LQG surface with an SLE κ ′ with κ ′ ∈ (4 , 8) η ′ √ ◮ Draw an independent SLE κ ′ on top of a 3 γ 2 2 − 2 wedge, γ = 4 / κ ′ Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 25 / 30

  68. Exploring an LQG surface with an SLE κ ′ with κ ′ ∈ (4 , 8) η ′ √ ◮ Draw an independent SLE κ ′ on top of a 3 γ 2 2 − 2 wedge, γ = 4 / κ ′ ◮ Quantum disks cut out by the path have a Poissonian structure Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 25 / 30

  69. Exploring an LQG surface with an SLE κ ′ with κ ′ ∈ (4 , 8) η ′ √ ◮ Draw an independent SLE κ ′ on top of a 3 γ 2 2 − 2 wedge, γ = 4 / κ ′ ◮ Quantum disks cut out by the path have a Poissonian structure ◮ Conditionally independent given their boundary lengths Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 25 / 30

  70. Exploring an LQG surface with an SLE κ ′ with κ ′ ∈ (4 , 8) η ′ √ ◮ Draw an independent SLE κ ′ on top of a 3 γ 2 2 − 2 wedge, γ = 4 / κ ′ ◮ Quantum disks cut out by the path have a Poissonian structure ◮ Conditionally independent given their boundary lengths ◮ Change in the left/right γ -LQG boundary lengths given by independent κ ′ 4 -stable L´ evy processes Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 25 / 30

  71. Exploring an LQG surface with an SLE κ ′ with κ ′ ∈ (4 , 8) η ′ √ ◮ Draw an independent SLE κ ′ on top of a 3 γ 2 2 − 2 wedge, γ = 4 / κ ′ ◮ Quantum disks cut out by the path have a Poissonian structure ◮ Conditionally independent given their boundary lengths ◮ Change in the left/right γ -LQG boundary lengths given by independent κ ′ 4 -stable L´ evy processes Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 25 / 30

  72. Exploring an LQG surface with an SLE κ ′ with κ ′ ∈ (4 , 8) η ′ √ ◮ Draw an independent SLE κ ′ on top of a 3 γ 2 2 − 2 wedge, γ = 4 / κ ′ ◮ Quantum disks cut out by the path have a Poissonian structure ◮ Conditionally independent given their boundary lengths ◮ Change in the left/right γ -LQG boundary lengths given by independent κ ′ 4 -stable L´ evy processes Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 25 / 30

  73. Exploring an LQG surface with an SLE κ ′ with κ ′ ∈ (4 , 8) η ′ √ ◮ Draw an independent SLE κ ′ on top of a 3 γ 2 2 − 2 wedge, γ = 4 / κ ′ ◮ Quantum disks cut out by the path have a Poissonian structure ◮ Conditionally independent given their boundary lengths ◮ Change in the left/right γ -LQG boundary lengths given by independent κ ′ 4 -stable L´ evy processes Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 25 / 30

  74. Exploring an LQG surface with an SLE κ ′ with κ ′ ∈ (4 , 8) η ′ √ ◮ Draw an independent SLE κ ′ on top of a 3 γ 2 2 − 2 wedge, γ = 4 / κ ′ ◮ Quantum disks cut out by the path have a Poissonian structure ◮ Conditionally independent given their boundary lengths ◮ Change in the left/right γ -LQG boundary lengths given by independent κ ′ 4 -stable L´ evy processes Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 25 / 30

  75. Gluing independent L´ evy trees Can view SLE κ ′ process, κ ′ ∈ (4 , 8) as a gluing of two κ ′ 4 -stable L´ evy trees. C − Y t X t t Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 26 / 30

  76. Gluing independent L´ evy trees Can view SLE κ ′ process, κ ′ ∈ (4 , 8) as a gluing of two κ ′ 4 -stable L´ evy trees. C − Y t X t t Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 26 / 30

  77. Gluing independent L´ evy trees Can view SLE κ ′ process, κ ′ ∈ (4 , 8) as a gluing of two κ ′ 4 -stable L´ evy trees. C − Y t X t t Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 26 / 30

  78. Gluing independent L´ evy trees Can view SLE κ ′ process, κ ′ ∈ (4 , 8) as a gluing of two κ ′ 4 -stable L´ evy trees. C − Y t X t t ◮ The two trees of quantum disks almost surely determine both the SLE κ ′ and the LQG surface on which it is drawn Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 26 / 30

  79. Gluing independent L´ evy trees Can view SLE κ ′ process, κ ′ ∈ (4 , 8) as a gluing of two κ ′ 4 -stable L´ evy trees. C − Y t X t t ◮ The two trees of quantum disks almost surely determine both the SLE κ ′ and the LQG surface on which it is drawn ◮ Can convert questions about SLE κ ′ into questions about κ ′ 4 -stable processes Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 26 / 30

  80. Gluing independent L´ evy trees Can view SLE κ ′ process, κ ′ ∈ (4 , 8) as a gluing of two κ ′ 4 -stable L´ evy trees. C − Y t X t t ◮ The two trees of quantum disks almost surely determine both the SLE κ ′ and the LQG surface on which it is drawn ◮ Can convert questions about SLE κ ′ into questions about κ ′ 4 -stable processes ◮ Question: Is the graph of components of an SLE κ ′ process connected? Duplantier, Miller, Sheffield Liouville Quantum Gravity as a Mating of Trees September 30, 2014 26 / 30

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