Asymptotic Shape of Equivariant Random Metrics on Nilpotent Groups

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1 Asymptotic Shape of Equivariant Random Metrics on Nilpotent Groups Michael Cantrell Alex Furman University of Illinois Chicago NCGOA 2015, Vanderbilt University

2 First Passage Percolation Cayley Graph Cay(Γ, S) with Edges = E X = [a, b] E, 0 < a < b < the space of assignments of a length to each edge µ Prob([a, b]) For each x X define m = µ E Prob(X) d x (γ 1, γ 2 ) = inf x(e i ) where the inf is taken over paths (e 1,..., e k ) from γ 1 to γ 2. What is the asymptotic shape of the ball of radius R in the metric d x?

3 Asymptotic Shape Theorems for FPP What is the asymptotic shape of the ball of radius R in the metric d x? Γ = Z Law of Large Numbers (1713)

4 Asymptotic Shape Theorems for FPP What is the asymptotic shape of the ball of radius R in the metric d x? Γ = Z Law of Large Numbers (1713) Γ = Z d Boivin (1990)

5 Asymptotic Shape Theorems for FPP What is the asymptotic shape of the ball of radius R in the metric d x? Γ = Z Law of Large Numbers (1713) Γ = Z d Boivin (1990) Γ =Nilpotent Benjamini-Tessera (2014) C-Furman (2015) e.g. Γ = 1 x z 0 1 y : x, y, z Z Benjamini-Tessera require weaker integrability, use statistical methods, use that FPP is i.i.d.s We consider more general random metrics

6 Ergodic Equivariant Random Metrics Definition An ergodic equivariant random metric on a finitely generated group Γ is a probability measure space (X, m) and a measurable family of metrics d x on Γ satisfying d γ x (γ 1, γ 2 ) = d x (γγ 1, γγ 2 ) γ, γ 1, γ 2 Γ a.e. x such that Γ (X, m) is measure-preserving and ergodic.

7 Ergodic Equivariant Random Metrics Definition An ergodic equivariant random metric on a finitely generated group Γ is a probability measure space (X, m) and a measurable family of metrics d x on Γ satisfying d γ x (γ 1, γ 2 ) = d x (γγ 1, γγ 2 ) γ, γ 1, γ 2 Γ a.e. x such that Γ (X, m) is measure-preserving and ergodic. First Passage Percolation is a special case of EERM with i.i.d.s.

8 The Asymptotic Cone Of A Nilpotent Group Theorem (Pansu s Theorem) If Γ is a finitely generated nilpotent group equipped with a left-invariant inner metric d (e.g. word metric) then there exists a unique connected, simply-connected, graded nilpotent Lie group G and a homogeneous Carnot-Caratheodory metric d such that in Gromov-Hausdorff convergence (Γ, 1 n d, id) (G, d, id). Example Γ = G = 1 x z 0 1 y x z 0 1 y : x, y, z Z : x, y, z R

9 The Asymptotic Cone of an Ergodic Equivariant Random Metric on a Nilpotent Group Theorem (C-Furman 2015) Suppose (X, d x ) is an ergodic equivariant random metric on a finitely generated nilpotent group Γ such that 1. d x is inner a.e. x X 2. 0 < a < b < such that a < d x /d < b for a.e. x X for some (any) word metric d

10 The Asymptotic Cone of an Ergodic Equivariant Random Metric on a Nilpotent Group Theorem (C-Furman 2015) Suppose (X, d x ) is an ergodic equivariant random metric on a finitely generated nilpotent group Γ such that 1. d x is inner a.e. x X 2. 0 < a < b < such that a < d x /d < b for a.e. x X for some (any) word metric d Then there exists a homogeneous Carnot-Caratheodory metric d on G such that in Gromov-Hausdorff convergence for a.e. x (Γ, 1 n d x, e) (G, d, e).

11 From ERM to Subadditive Cocycles Definition Given Γ (X, m) a probability measure-preserving action a subadditive cocycle is c : Γ X R + such that c(γ 1 γ 2, x) c(γ 1, γ 2 x) + c(γ 2, x).

12 From ERM to Subadditive Cocycles Definition Given Γ (X, m) a probability measure-preserving action a subadditive cocycle is c : Γ X R + such that c(γ 1 γ 2, x) c(γ 1, γ 2 x) + c(γ 2, x). {subadditive cocycles} {equivariant random metrics} Given c : Γ X R + define d x (γ 1, γ 2 ) = c(γ 1 γ 1 2, γ 2 x). Given {d x } define c(γ, x) = d x (e, γ). Cocycle inequality triangle inequality.

13 Previous Subadditive Ergodic Theorems Γ = Z - Kingmann (1973): For a.e. x as n 1 1 c(n, x) inf c(n, x)dm(x). n n 1 n X implies Birkhoff s Ergodic Theorem implies Asymptotic Shape Theorem for FPP on Z many other applications...e.g. LLN for matrices

14 Previous Subadditive Ergodic Theorems Γ = Z - Kingmann (1973): For a.e. x as n 1 1 c(n, x) inf c(n, x)dm(x). n n 1 n X implies Birkhoff s Ergodic Theorem implies Asymptotic Shape Theorem for FPP on Z many other applications...e.g. LLN for matrices Γ = Z d - Björklund (2010): There exists a norm L on R d s.t. for a.e. x as Z d n c( n, x) L( n) n 0. implies Boivin s Asymptotic Shape Theorem for FPP on Z d

15 Previous Subadditive Ergodic Theorems Γ = Z - Kingmann (1973): For a.e. x as n 1 1 c(n, x) inf c(n, x)dm(x). n n 1 n X implies Birkhoff s Ergodic Theorem implies Asymptotic Shape Theorem for FPP on Z many other applications...e.g. LLN for matrices Γ = Z d - Björklund (2010): There exists a norm L on R d s.t. for a.e. x as Z d n c( n, x) L( n) n 0. implies Boivin s Asymptotic Shape Theorem for FPP on Z d Γ = Nilpotent?

16 Subadditive Ergodic Theorem for Nilpotent Groups Theorem (C-Furman 2015) Suppose Γ is a finitely generated nilpotent group, Γ (X, m) is an ergodic probability measure preserving and c : Γ X R + is a subadditive cocycle such that 1. 0 < a < b < such that a < c(γ, x)/ γ < b for a.e. x γ for some (any) word norm 2. d x is inner for a.e. x; i.e. for a.e. x ɛ > 0 finite set F Γ s.t. every γ Γ may be written γ = γ n γ 1, γ i F and c(γ 1, x)+c(γ 2, γ 1 x)+ +c(γ n, γ n 1 γ 1 x) (1+ɛ)c(γ, x)

17 Subadditive Ergodic Theorem for Nilpotent Groups Theorem (C-Furman 2015) Suppose Γ is a finitely generated nilpotent group, Γ (X, m) is an ergodic probability measure preserving and c : Γ X R + is a subadditive cocycle such that 1. 0 < a < b < such that a < c(γ, x)/ γ < b for a.e. x γ for some (any) word norm 2. d x is inner for a.e. x; i.e. for a.e. x ɛ > 0 finite set F Γ s.t. every γ Γ may be written γ = γ n γ 1, γ i F and c(γ 1, x)+c(γ 2, γ 1 x)+ +c(γ n, γ n 1 γ 1 x) (1+ɛ)c(γ, x) Then there exists a unique Carnot-Caratheodory norm φ on G s.t. for a.e. x whenever N Γ (t i, γ i ) g G 1 t i c(γ i, x) φ(g).

18 Where Does φ Come From? Idea: integrate and apply Pansu s construction. Carnot-Caratheodory construction

19 I. Benjamini, R. Tessera. First passage percolation on nilpotent cayley graphs and beyond. M. Björklund. The asymptotic shape theorem for generalized first passage percolation. Ann. Probab. 38 (2010), J. F. C. Kingman. Subadditive ergodic theory. Ann. Probab P. Pansu. Croissance des boules et des géodésiques fermées dan les nilvariétés. Ergodic Dyn. Syst. 3(1983),