Simple random walk on the two-dimensional uniform spanning tree - PowerPoint PPT Presentation

Simple random walk on the two-dimensional uniform spanning tree STATISTICAL MECHANICS SEMINAR, UNIVERSITY OF WARWICK, 28 MAY 2015 David Croydon (University of Warwick) joint with Martin Barlow (University of British Columbia) Takashi Kumagai (Kyoto University)

UNIFORM SPANNING TREE IN TWO DIMENSIONS Let Λ n := [ − n, n ] 2 ∩ Z 2 . A subgraph of the lattice is a spanning tree of Λ n if it connects all vertices and has no cycles. Let U ( n ) be a spanning tree of Λ n se- lected uniformly at random from all pos- sibilities. The UST on Z 2 , U , is then the local limit of U ( n ) . NB. Wired/free boundary conditions unimportant. Almost-surely, U is a spanning tree of Z 2 . (Forest for d > 4.) [Aldous, Benjamini, Broder, H¨ aggstr¨ om, Kirchoff, Lyons, Pe- mantle, Peres, Schramm, Wilson,. . . ]

UNIFORM SPANNING TREE IN TWO DIMENSIONS Let Λ n := [ − n, n ] 2 ∩ Z 2 . A subgraph of the lattice is a spanning tree of Λ n if it connects all vertices and has no cycles. Let U ( n ) be a spanning tree of Λ n se- lected uniformly at random from all pos- sibilities. The UST on Z 2 , U , is then the local limit of U ( n ) . NB. Wired/free boundary conditions unimportant. Almost-surely, U is a spanning tree of Z 2 . (Forest for d > 4.) [Aldous, Benjamini, Broder, H¨ aggstr¨ om, Kirchoff, Lyons, Pe- mantle, Peres, Schramm, Wilson,. . . ]

UNIFORM SPANNING TREE IN TWO DIMENSIONS The distances in the tree to the path between opposite corners in a uniform spanning tree in a 200 × 200 grid. Picture: Lyons/Peres: Probability on trees and networks

WILSON’S ALGORITHM ON Z 2 Let x 0 = 0 , x 1 , x 2 , . . . be an enumeration of Z 2 . Let U (0) be the graph tree consisting of the single vertex x 0 . Given U ( k − 1) for some k ≥ 1, define U ( k ) to be the union of U ( k − 1) and the loop-erased random walk (LERW) path run from x k to U ( k − 1). The UST U is then the local limit of U ( k ). x 1 x 1 x 1 x 2 x 0 x 0 x 0 x 0

LERW SCALING IN Z d Consider LERW as a process ( L n ) n ≥ 0 (assume original random walk is transient). In Z d , d ≥ 5, L rescales diffusively to Brownian motion [Lawler]. In Z 4 , with logarithmic corrections rescales to Brownian motion [Lawler]. In Z 3 , { L n : n ∈ [0 , τ ] } has a scaling limit [Kozma, Shiraishi]. In Z 2 , { L n : n ∈ [0 , τ ] } has SLE(2) scaling limit [Lawler/Schramm/Werner]. Growth exponent is 5 / 4 [Kenyon, Masson, Lawler]. Picture: Ariel Yadin

VOLUME AND RESISTANCE ESTIMATES [BARLOW/MASSON] With high probability, B E ( x, λ − 1 R ) ⊆ B U ( x, R 5 / 4 ) ⊆ B E ( x, λR ) , as R → ∞ then λ → ∞ . In particular, ≤ c 1 e − c 2 λ 1 / 9 . R − 8 / 5 µ U ( B U ( x, R )) �∈ [ λ − 1 , λ ] � � P Also, ≤ c 1 e − c 2 λ 2 / 9 . Resistance( x, B U ( x, R ) c ) /R �∈ [ λ − 1 , 1] � � P Implies exit time for intrinsic ball radius R is R 13 / 5 , also heat ker- nel bounds p U 2 n (0 , 0) ≍ n − 8 / 13 . cf. [Barlow/Jarai/Kumagai/Slade] How about scaling limit for UST? And for SRW on the UST?

VOLUME AND RESISTANCE ESTIMATES [BARLOW/MASSON] With high probability, B E ( x, λ − 1 R ) ⊆ B U ( x, R 5 / 4 ) ⊆ B E ( x, λR ) , as R → ∞ then λ → ∞ . In particular, ≤ c 1 e − c 2 λ 1 / 9 . R − 8 / 5 µ U ( B U ( x, R )) �∈ [ λ − 1 , λ ] � � P Also, ≤ c 1 e − c 2 λ 2 / 9 . Resistance( x, B U ( x, R ) c ) /R �∈ [ λ − 1 , 1] � � P Implies exit time for intrinsic ball radius R is R 13 / 5 , also heat ker- nel bounds p U 2 n (0 , 0) ≍ n − 8 / 13 . cf. [Barlow/Jarai/Kumagai/Slade] How about scaling limit for UST? And for SRW on the UST?

UST SCALING [SCHRAMM] Consider U as an ensemble of paths: ( a, b, π ab ) : a, b ∈ Z 2 � � U = , where π ab is the unique arc connecting a and b in U , as an element R 2 × ˙ R 2 × H ( ˙ R 2 )), of the compact space H ( ˙ cf. [Aizenman/Burchard/Newman/Wilson]. The trunk of the scaling limit = T { ( a, b, π ab ) : a, b ∈ R 2 } , ∪ T π ab \{ a, b } , is a dense topological tree with degree at most 3, almost-surely. Picture: Oded Schramm ISSUE: This topology does not carry information about intrinsic distance, volume, or resistance.

GENERALISED GROMOV-HAUSDORFF TOPOLOGY (cf. [GROMOV, LE GALL/DUQUESNE]) Define T to be the collection of measured, rooted, spatial trees, i.e. ( T , d T , µ T , φ T , ρ T ) , where: • ( T , d T ) is a complete and locally compact real tree; • µ T is a locally finite Borel measure on ( T , d T ); • φ T is a continuous map from ( T , d T ) into R 2 ; • ρ T is a distinguished vertex in T . On T c (compact trees only), define a distance ∆ c by    � �   d Z P ( µ T ◦ ψ − 1 , µ ′ T ◦ ψ ′− 1 )+ � d Z ( ψ ( x ) , ψ ′ ( x ′ )) + � φ T ( x ) − φ ′ T ( x ′ ) � inf sup  . � � � Z,ψ,ψ ′ , C : ( x,x ′ ) ∈C ( ρ T ,ρ ′ T ) ∈C Can be extended to locally compact case.

A CRITERION FOR TIGHTNESS Let A be a subset of T such that, for every r > 0: (i) for every ε > 0, there exists a finite integer N ( r, ε ) such that for any element T of A there is an ε -cover of B T ( ρ T , r ) of cardinality less than N ( r, ε ); (ii) it holds that sup µ T ( B T ( ρ T , r )) < ∞ ; T ∈A (iii) { φ T ( ρ T ) : T ∈ A} is a bounded subset of R 2 , and for every ε > 0, there exists a δ = δ ( r, ε ) > 0 such that sup sup | φ T ( x ) − φ T ( y ) | < ε. T ∈A x,y ∈ B T ( ρ T ,r ): d T ( x,y ) ≤ δ Then A is relatively compact.

TIGHTNESS OF UST Theorem. If P δ is the law of the measured, rooted spatial tree � U , δ 5 / 4 d U , δ 2 µ U ( · ) , δφ U , 0 � under P , then the collection ( P δ ) δ ∈ (0 , 1) is tight in M 1 ( T ). Key estimate is an exponential bound for: � � R − 8 / 5 µ U ( B U ( x, R )) �∈ [ λ − 1 , λ ] , ∀ x ∈ B E (0 , n ) , n − 5 / 4 R ∈ [ e − λ 1 / 40 , 1] . P Proof involves: strengthening estimates of [Bar- • low/Masson], • comparison of Euclidean and intrinsic distance along paths.

TIGHTNESS OF UST Theorem. If P δ is the law of the measured, rooted spatial tree � U , δ 5 / 4 d U , δ 2 µ U ( · ) , δφ U , 0 � under P , then the collection ( P δ ) δ ∈ (0 , 1) is tight in M 1 ( T ). Key estimate is an exponential bound for: � � R − 8 / 5 µ U ( B U ( x, R )) �∈ [ λ − 1 , λ ] , ∀ x ∈ B E (0 , n ) , n − 5 / 4 R ∈ [ e − λ 1 / 40 , 1] . P Proof involves: strengthening estimates of [Bar- • low/Masson], • comparison of Euclidean and intrinsic distance along paths.

TIGHTNESS OF UST Theorem. If P δ is the law of the measured, rooted spatial tree � U , δ 5 / 4 d U , δ 2 µ U ( · ) , δφ U , 0 � under P , then the collection ( P δ ) δ ∈ (0 , 1) is tight in M 1 ( T ). Key estimate is an exponential bound for: � � R − 8 / 5 µ U ( B U ( x, R )) �∈ [ λ − 1 , λ ] , ∀ x ∈ B E (0 , n ) , n − 5 / 4 R ∈ [ e − λ 1 / 40 , 1] . P Proof involves: strengthening estimates of [Bar- • low/Masson], • comparison of Euclidean and intrinsic distance along paths.

TIGHTNESS OF UST Theorem. If P δ is the law of the measured, rooted spatial tree � U , δ 5 / 4 d U , δ 2 µ U ( · ) , δφ U , 0 � under P , then the collection ( P δ ) δ ∈ (0 , 1) is tight in M 1 ( T ). Key estimate is an exponential bound for: � � R − 8 / 5 µ U ( B U ( x, R )) �∈ [ λ − 1 , λ ] , ∀ x ∈ B E (0 , n ) , n − 5 / 4 R ∈ [ e − λ 1 / 40 , 1] . P Proof involves: strengthening estimates of [Bar- • low/Masson], • comparison of Euclidean and intrinsic distance along paths.

UST LIMIT PROPERTIES If ˜ P is a subsequential limit of ( P δ ) δ ∈ (0 , 1) , then for ˜ P -a.e. ( T , d T , µ T , φ T , ρ T ) it holds that: (a) (i) the Hausdorff dimension of ( T , d T ) is given by d f := 8 5; (ii) ( T , d T ) has precisely one end at infinity; (b) (i) µ T is non-atomic and supported on the leaves of T ; (ii) given R > 0, uniformly for x ∈ B T ( ρ T , R ) and r ∈ (0 , r 0 ), c 1 r d f (log r − 1 ) − 80 ≤ µ T ( B T ( x, r )) ≤ c 2 r d f (log r − 1 ) 80 ; (iii) uniformly for r ∈ (0 , r 0 ), c 1 r d f (log log r − 1 ) − 9 ≤ µ T ( B T ( ρ, r )) ≤ c 2 r d f (log log r − 1 ) 3 ;

(c) (i) the restriction of the continuous map φ T : T → R 2 to T o is a homeomorphism between this set and its image φ T ( T o ), which is dense in R 2 ; (ii) max x ∈T deg T ( x ) = 3 = max x ∈ R 2 | φ − 1 T ( x ) | ; (iii) µ T = L ◦ φ T .

SIMPLE RANDOM WALK ON UST After 5,000 and 50,000 steps: Picture: Thanks to Sunil Chhita

CONVERGENCE OF SRW (cf. [C., ATHREYA/L ¨ OHR/WINTER]) Let ( T n ) n ≥ 1 be a sequence of finite graph trees, and X T n the SRW on T n . Suppose that there exist null sequences ( a n ) n ≥ 1 , ( b n ) n ≥ 1 , ( c n ) n ≥ 1 with b n = o ( a n ) such that � � → ( T , d T , µ T , φ T , ρ T ) T n , a n d T n , b n µ T n , c n φ T n , ρ T n in ( T c , ∆ c ), where ( T , d T , µ T , φ T , ρ T ) is an element of T ∗ c . Let X T be Brownian motion on T , then � � �� X T n � � X T �� c n φ T n → φ T t t/a n b n t ≥ 0 t ≥ 0 in distribution in C ( R + , R 2 ), where we assume X T n = ρ T n for 0 each n , and also X T 0 = ρ T . Can also extend to locally compact case.