Scaling limit of uniform spanning tree in three dimensions Daisuke - PowerPoint PPT Presentation

Scaling limit of uniform spanning tree in three dimensions Daisuke Shiraishi, Kyoto University ongoing work with Omer Angel (UBC), David Croydon (Kyoto University) and Sarai Hernandez Torres (UBC) August 2019, Kyushu University 1 / 22

Uniform Spanning Tree (UST) ◮ For a graph G = ( V , E ), a spanning tree T of G is a subgraph of G that is a tree with (the vertex set of T ) = V . 2 / 22

Uniform Spanning Tree (UST) ◮ For a graph G = ( V , E ), a spanning tree T of G is a subgraph of G that is a tree with (the vertex set of T ) = V . ◮ A uniform spanning tree (UST) in G is a random spanning tree chosen uniformly from a set of all spanning trees. 2 / 22

Uniform Spanning Tree (UST) ◮ For a graph G = ( V , E ), a spanning tree T of G is a subgraph of G that is a tree with (the vertex set of T ) = V . ◮ A uniform spanning tree (UST) in G is a random spanning tree chosen uniformly from a set of all spanning trees. ◮ UST has important connections to several areas: 2 / 22

Uniform Spanning Tree (UST) ◮ For a graph G = ( V , E ), a spanning tree T of G is a subgraph of G that is a tree with (the vertex set of T ) = V . ◮ A uniform spanning tree (UST) in G is a random spanning tree chosen uniformly from a set of all spanning trees. ◮ UST has important connections to several areas: ◮ Loop-erased random walk (LERW) ◮ Loop soup ◮ Conformally invariant scaling limits ◮ The Abelian sandpile model ◮ Gaussian free field ◮ Domino tiling ◮ Random cluster model ◮ Random interlacements ◮ Potential theory ◮ Amenability · · · 2 / 22

Uniform Spanning Tree (UST) 2D UST in a fine grid. Picture credit: Adrien Kassel. 3 / 22

Uniform Spanning Tree (UST) ◮ Today’s talk : Scaling limit of UST in δ Z 3 as δ → 0 w.r.t. the spatial Gromov-Hausdorff topology. 4 / 22

Uniform Spanning Tree (UST) ◮ Today’s talk : Scaling limit of UST in δ Z 3 as δ → 0 w.r.t. the spatial Gromov-Hausdorff topology. In particular, we want to define a random metric χ in R 3 which is the limit of the rescaled graph distance in UST. Namely, χ satisfies that for all x , y ∈ R 3 , the rescaled graph distance between x and y in UST in δ Z 3 converges weakly to χ ( x , y ). 4 / 22

The Gromov-Hausdorff convergence ◮ A pointed metric space ( X , ρ ) is a pair of a metric space X and a distinguished point ρ of X . 5 / 22

The Gromov-Hausdorff convergence ◮ A pointed metric space ( X , ρ ) is a pair of a metric space X and a distinguished point ρ of X . ◮ For two metric spaces ( X 1 , d 1 ) and ( X 2 , d 2 ), a correspondence between X 1 and X 2 is a subset R of X 1 × X 2 s.t. ∀ x 1 ∈ X 1 , ∃ x 2 ∈ X 2 s.t. ( x 1 , x 2 ) ∈ R and conversely ∀ y 2 ∈ X 2 , ∃ y 1 ∈ X 1 s.t. ( y 1 , y 2 ) ∈ R . 5 / 22

The Gromov-Hausdorff convergence ◮ A pointed metric space ( X , ρ ) is a pair of a metric space X and a distinguished point ρ of X . ◮ For two metric spaces ( X 1 , d 1 ) and ( X 2 , d 2 ), a correspondence between X 1 and X 2 is a subset R of X 1 × X 2 s.t. ∀ x 1 ∈ X 1 , ∃ x 2 ∈ X 2 s.t. ( x 1 , x 2 ) ∈ R and conversely ∀ y 2 ∈ X 2 , ∃ y 1 ∈ X 1 s.t. ( y 1 , y 2 ) ∈ R . ◮ The distortion of the correspondence R is defined by � : ( x 1 , x 2 ) , ( y 1 , y 2 ) ∈ R �� � � dis( R ) = sup � d 1 ( x 1 , y 1 ) − d 2 ( x 2 , y 2 ) . 5 / 22

The Gromov-Hausdorff convergence x 2 x 1 y 1 y 2 X 1 X 2 Correspondence between X 1 and X 2 . Picture credit: Daisuke Shiraishi. 6 / 22

The Gromov-Hausdorff convergence ◮ A pointed metric space ( X , ρ ) is a pair of a metric space X and a distinguished point ρ of X . ◮ For two metric spaces ( X 1 , d 1 ) and ( X 2 , d 2 ), a correspondence between X 1 and X 2 is a subset R of X 1 × X 2 s.t. ∀ x 1 ∈ X 1 , ∃ x 2 ∈ X 2 s.t. ( x 1 , x 2 ) ∈ R and conversely ∀ y 2 ∈ X 2 , ∃ y 1 ∈ X 1 s.t. ( y 1 , y 2 ) ∈ R . ◮ The distortion of the correspondence R is defined by � : ( x 1 , x 2 ) , ( y 1 , y 2 ) ∈ R �� � � dis( R ) = sup � d 1 ( x 1 , y 1 ) − d 2 ( x 2 , y 2 ) . 7 / 22

The Gromov-Hausdorff convergence ◮ A pointed metric space ( X , ρ ) is a pair of a metric space X and a distinguished point ρ of X . ◮ For two metric spaces ( X 1 , d 1 ) and ( X 2 , d 2 ), a correspondence between X 1 and X 2 is a subset R of X 1 × X 2 s.t. ∀ x 1 ∈ X 1 , ∃ x 2 ∈ X 2 s.t. ( x 1 , x 2 ) ∈ R and conversely ∀ y 2 ∈ X 2 , ∃ y 1 ∈ X 1 s.t. ( y 1 , y 2 ) ∈ R . ◮ The distortion of the correspondence R is defined by � : ( x 1 , x 2 ) , ( y 1 , y 2 ) ∈ R �� � � dis( R ) = sup � d 1 ( x 1 , y 1 ) − d 2 ( x 2 , y 2 ) . ◮ For two pointed compact metric spaces ( X 1 , ρ 1 ) and ( X 2 , ρ 2 ), define the distance d GH ( X 1 , X 2 ) by d GH ( X 1 , X 2 ) = inf dis( R ) , where the infimum is over all correspondences R between X 1 and X 2 with ( ρ 1 , ρ 2 ) ∈ R . 7 / 22

The Gromov-Hausdorff convergence isometry Two equivalent trees in the Gromov-Hausdorff topology. 8 / 22

The spatial Gromov-Hausdorff convergence ◮ A quadruplet X = ( X , d X , ρ X , φ X ) is called a pointed spatial compact metric space if ( X , d X , ρ X ) is a pointed compact metric space and φ X is a continuous map from ( X , d X ) to R 3 . 9 / 22

The spatial Gromov-Hausdorff convergence ◮ A quadruplet X = ( X , d X , ρ X , φ X ) is called a pointed spatial compact metric space if ( X , d X , ρ X ) is a pointed compact metric space and φ X is a continuous map from ( X , d X ) to R 3 . ◮ For two pointed spatial compact metric spaces X i = ( X i , d i , ρ i , φ i ) ( i = 1 , 2), define d sp GH ( X 1 , X 2 ) by � �� d sp � GH ( X 1 , X 2 ) = inf dis( R ) ∨ sup d Euclid φ 1 ( x 1 ) , φ 2 ( x 2 ) , ( x 1 , x 2 ) ∈R where the infimum is over all correspondences R between X 1 and X 2 with ( ρ 1 , ρ 2 ) ∈ R . 9 / 22

The spatial Gromov-Hausdorff convergence isometry These two trees are distinguished in the spatial Gromov-Hausdorff topology. 10 / 22

Main Result ◮ Let U be the UST in Z 3 endowed with the graph distance d U . 11 / 22

Main Result ◮ Let U be the UST in Z 3 endowed with the graph distance d U . ◮ Suppose that ( U , d U ) is pointed at the origin. 11 / 22

Main Result ◮ Let U be the UST in Z 3 endowed with the graph distance d U . ◮ Suppose that ( U , d U ) is pointed at the origin. ◮ φ U : U → R 3 : the identity on vertices, with linear interpolation along edges of U . 11 / 22

Main Result ◮ Let U be the UST in Z 3 endowed with the graph distance d U . ◮ Suppose that ( U , d U ) is pointed at the origin. ◮ φ U : U → R 3 : the identity on vertices, with linear interpolation along edges of U . ◮ Let LERW n be the loop-erased random walk from 0 to ∂ B (2 n ) in Z 3 . Denote the number of steps of LERW n by � � � LERW n � . 11 / 22

SRW and LERW 2 n Erase Loops O O SRW (left) and Loop-erased random walk (right) in Z 3 . 12 / 22

Main Result ◮ Let U be the UST in Z 3 endowed with the graph distance d U . ◮ Suppose that ( U , d U ) is pointed at the origin. ◮ φ U : U → R 3 : the identity on vertices, with linear interpolation along edges of U . ◮ Let LERW n be the loop-erased random walk from 0 to ∂ B (2 n ). Denote the number of steps of LERW n by � � � LERW n � . 13 / 22

Main Result ◮ Let U be the UST in Z 3 endowed with the graph distance d U . ◮ Suppose that ( U , d U ) is pointed at the origin. ◮ φ U : U → R 3 : the identity on vertices, with linear interpolation along edges of U . ◮ Let LERW n be the loop-erased random walk from 0 to ∂ B (2 n ). Denote the number of steps of LERW n by � � � LERW n � . ◮ (S. ’14, Li-S. ’18) It is proved that ∃ a constant β ∈ (1 , 5 3 ] s.t. �� � n →∞ 2 − β n E � lim � LERW n ∈ (0 , ∞ ) . � 13 / 22

Main Result ◮ Let U be the UST in Z 3 endowed with the graph distance d U . ◮ Suppose that ( U , d U ) is pointed at the origin. ◮ φ U : U → R 3 : the identity on vertices, with linear interpolation along edges of U . ◮ Let LERW n be the loop-erased random walk from 0 to ∂ B (2 n ). Denote the number of steps of LERW n by � � � LERW n � . ◮ (S. ’14, Li-S. ’18) It is proved that ∃ a constant β ∈ (1 , 5 3 ] s.t. �� � n →∞ 2 − β n E � lim � LERW n ∈ (0 , ∞ ) . � ◮ (Wilson ’10) Numerical simulation: β = 1 . 624 · · · . 13 / 22

Main Result ◮ Let U be the UST in Z 3 endowed with the graph distance d U . ◮ Suppose that ( U , d U ) is pointed at the origin. ◮ φ U : U → R 3 : the identity on vertices, with linear interpolation along edges of U . ◮ Let LERW n be the loop-erased random walk from 0 to ∂ B (2 n ). Denote the number of steps of LERW n by � � � LERW n � . ◮ (S. ’14, Li-S. ’18) It is proved that ∃ a constant β ∈ (1 , 5 3 ] s.t. �� � n →∞ 2 − β n E � lim � LERW n ∈ (0 , ∞ ) . � ◮ (Wilson ’10) Numerical simulation: β = 1 . 624 · · · . Theorem (Angel-Croydon-S.-Hernandez Torres. ’19+) As n → ∞ , the pointed spatial tree ( U , 2 − β n d U , 0 , 2 − n φ U ) converges weakly w.r.t. the metric d sp GH . 13 / 22

Remarks ◮ Remark 1 : This is the first result to prove the existence of the scaling limit of 3D UST! 14 / 22