Random Walks on Hyperbolic Groups IV

Transcription

1 Random Walks on Hyperbolic Groups IV Steve Lalley University of Chicago January 2014

2 Automatic Structure Thermodynamic Formalism Local Limit Theorem Final Challenge: Excursions

3 I. Automatic Structure of a Hyperbolic Group Definition: A discrete group Γ is hyperbolic (also called word-hyperbolic) if its Cayley graph has the thin triangle property: δ > 0 such that every geodesic triangle in the Cayley graph is δ thin, i.e., every vertex on side A is within distance δ of side B or side C (or both). Theorem (Bonk-Schramm): If Γ is hyperbolic then for any finite symmetric generating set A the Cayley graph G(Γ; A) is quasi-isometric to a convex subset of H n.

4 Geometric Boundary and Gromov Compactification Two geodesic rays (x n ) n 0 and (y n ) n 0 are equivalent if there exists k Z such that for all large n, d(x n, y n+k ) 2δ. Geometric Boundary Γ: Set of equivalence classes of geodesic rays. Topology on Γ Γ: Basic open sets: (a) singletons {x} with x Γ; and (b) sets B m (ξ) = with m 1 and ξ Γ where B m (ξ) = set of x Γ and ζ Γ such that there exists geodesic rays (x n ) and (y n ) with initial points x 0 = y 0 = 1, endpoints ξ and ζ (or ξ and x), and such that x j = y j j m Non-elementary Hyperbolic Group: Γ =.

5 Visual Metric on Γ Visual Metric: A metric d a on Γ such that for any ξ, ζ Γ, any bi-infinite geodesic γ from ξ to ζ, and any vertex y on γ minimizing distance to 1, C 1 a d(1,y) d a (ξ, ζ) C 2 a d(1,y) Proposition: For some a > 1 a visual metric exists.

6 Automatic Structure Automaton: Finite digraph D = (V, E) with distinguished vertex s =start and labeling α : E A of the edge set E by letters of finite alphabet A. Vertex v is recurrent if nontrivial path beginning and ending at v; otherwise transient. Regular Language Σ accepted by D: Set of all finite words a 1 a 2 a n with a i A such that path in D starting at s with successive edges labeled a i. Closure of regular language Σ : Define Σ = Σ Σ where Σ = set of infinite words all of whose finite prefixes are in Σ. Associated Shift σ : Σ Σ mapping σ(a 1 a 2 a 3 ) = a 2 a 3

7 Automatic Structure Definition: Automaton D = (V, E) with labeling α : E A is a strongly Markov automatic structure for finitely generated group Γ with generating set A if no edge ends at s ; every vertex v V is accessible from s ; for every path γ in D with initial point s, the projection α(γ) to the Cayley graph C(Γ, A) is a geodesic starting at s ; and the endpoint map α : {paths} Γ is bijective. Theorem: (J. Cannon) Every hyperbolic group has a strongly Markov automatic structure (D, α). Moreover, every infinite path in D starting at s maps to a geodesic ray starting at 1.

8 Cone Types and Neighborhood Types Γ: hyperbolic group with symmetric generating set A. C(Γ, A): Cayley graph. For x, y Γ write x y if geodesic segment in C(Γ, A) from 1 to y that passes through x. Cone [x, ]: Labeled subgraph of C(Γ, A) with vertices y x. Two cones [x, ] and [y, ] are isomorphic if (yx) 1 [x, ] = [y, ] as labeled graphs. If [x, ] and [y, ] are isomorphic then they have same cone type. m Neighborhood N m (x): Labeled subgraph with vertices y at distance m from x. Each such vertex is given label d(1, y) d(1, x). If zx 1 N m (x) = N m (z) then x, z have same m neighborhood type. Theorem: (Cannon) If x, y have same (2δ + 1) neighborhood type then they have the same cone type.

9 Proof of Cannon Theorem (Sketch) Fix total order on generating set A. Lex-order geodesics. For each g Γ let u g be lex-smallest geodesic from 1 to g. Claim: Set {u g } g Γ is a regular language (i.e., is generated by a finite-state automaton). Construction: g say that h is a competitor if g = h, u h u g, and d(h, g) 2δ. Mark each g by (τ(g), κ(g)) where τ(g) is the (2δ + 1) neighborhood type and κ(g) = list of competitors. The set of marks is finite these are the states of the automaton.

10 Regenerative Structure Proposition: If Γ is a surface group then for any two recurrent cone types τ, τ and any cone [x, ] of type τ there is a cone of type τ contained in [x, ]. Theorem: (Haissinsky, Mathieu, Muller) Assume Γ is a surface group and X n is a FRRW on Γ. Fix a recurrent cone type τ. Then the random walk path can be partitioned into independent blocks B 0 B 1 B 2 B 3 such that each block B i (except B 0 ) consists of a finite path in a cone [x i, ] of type τ ending at a vertex x i+1 in the interior of [x i, ]. Moreover, the block lengths have finite exponential moments, and B 1, B 2,... are identically distributed. Corollary 1: d(x n, 1) obeys a central limit theorem. Corollary 2: Speed l depends analytically on the step distribution.

11 Thermodynamic Formalism Notation: (D, α) = strongly Markov automatic structure Σ = {semi-infinite paths in D} Σ = {finite paths in D} Σ = Σ Σ σ : Σ Σ the shift (x y) = min{n : x n y n } d a (x, y) = a (x y) Induced Map: α : Σ Γ bijectively Boundary Map: α : Σ Γ Fact: The boundary map is Hölder continuous.

12 Thermodynamic Formalism Ruelle Operators: Let H be the set of Hölder continuous functions ϕ : Σ R. For each ϕ H, define operator L ϕ f (ω) = exp{ϕ(ω )}f (ω ) = L n ϕf (ω) = ω σ(ω )=ω ω σ n (ω )=ω n 1 S n ϕ = ϕ σ j j=0 exp{s n ϕ(ω )}f (ω ) where Ruelle s PF Theorem: If the shift σ : Σ Σ is topologically mixing then for every ϕ H there exist constant Pr(ϕ), function h ϕ H, and probability measure ν ϕ such that f H L n ϕf = exp{npr(ϕ)} f, ν ϕ h ϕ + O(e n(pr(ϕ) ε) ).

13 Thermodynamic Formalism Gibbs State: The measure µ ϕ := h ϕ ν ϕ is called the Gibbs state associated with the potential function ϕ. Fact: µ ϕ is an ergodic, mixing, σ invariant probability measure on Σ. Fact: µ ϕ is the unique σ invariant probability measure with the following property: 0 < C < such that for every finite path ω Σ of length m µ ϕ {ω Σ : ω = ωω } exp{s m ϕ(ω) mpr(ϕ)} 1 C (1)

14 Green s Function Key to Local Limit Theorem: The Green s function of a symmetric FRRW can be lifted to a Hölder continuous function on Σ. Define ϕ r (ω) = log G r (1, α (ω)) G r (1, α (σω)) = log G r (1, α (ω)) G r (ω 0, α (ω)) ϕ r (ω) = log K r (1, α (ω)) for ω Σ K r (ω 0, α (ω)) for ω Σ where K r (x, ζ) is the Martin kernel. Then for any ω Σ of length n, G r (1, α (ω)) = G r (1, 1) exp{s n ϕ r (ω)}. Proposition: ϕ r is Hölder continuous. Proof: Yesterday!

15 Convergence to the Martin Kernel Shadowing: A geodesic segment [x y ] shadows a geodesic segment [xy] if every vertex on [xy] lies within distance 2δ of [x y ]. If geodesic segments [x y ] and [x y ] both shadow [xy] then they are fellow-traveling along [xy]. Proposition: 0 < α < 1 and C < such that if [xy] and [x y ] are fellow-traveling along a geodesic segment [x 0 y 0 ] of length m then G r (x, y)/g r (x, y) G r (x, y )/G r (x, y ) 1 Cαm

16 Application of Ruelle Hypotheses of Ruelle require that the shift σ : Σ Σ be topologically mixing. This need not be true for every hyperbolic group, however, Fact: For a co-compact Fuchsian group Γ there is a strongly Markov automatic structure or which the associated shift σ : Σ Σ is topologically mixing. Proposition: Let G r (x, y) be the Green s function of a symmetric FRRW on co-compact Fuchsian Γ. Then for r R there exist constants 0 < C r < such that as m, G r (1, x) 2 C r exp{mpr(2ϕ r )}. d(1,x)=m Note: The result remains true for nonelementary hyperbolic groups, though.

17 Properties of the Pressure Pr(ϕ r ) Proposition: The map r Pr(ϕ r ) is continuous in r. Proof: Regular perturbation theory. Proposition: If R > 1 is the inverse spectral radius then Pr(2ϕ r ) < 0 Pr(2ϕ R ) = 0 for r < R Proof: Repeat of the branching random walk construction used in the case of RW on a free group. Recall: d dr G r (1, 1) = r 1 G r (1, x) 2 r 1 G r (1, 1) x Γ

18 Local Limit Theorem Objective: Sketch proof of Theorem: (Gouezel-Lalley) For any symmetric FRRW on a co-compact Fuchsian group, P 1 {X 2n = 1} CR 2n (2n) 3/2. Theorem: (Gouezel) This also holds for any nonelementary hyperbolic group. Moreover, for Fuchsian groups the hypothesis of symmetry is unnecessary. Note: Same local limit theorem also holds for virtually free Fuchsian groups Γ, for instance SL(2, Z) and congruence subgroups.

19 Local Limit Theorem Recall: Corollary: Let G r (x, y) be the Green s function of an aperiodic and symmetric random walk on a countable group Γ. If for every x, y Γ there are constants C x,y > 0 such that d dr G r (x, y) C x,y /(R r) β as r R then for suitable C x,y > 0 the transition probabilities satisfy p n (x, y) C x,yr n n β 2 as n.

20 Differential Equations for Green s Function Strategy : Exploit the differential equations d dr G r (x, y) = r 1 G r (x, z)g r (z, y) r 1 G r (x, y) z Γ d 2 dr 2 G r (x, y) = r 2 z 1,z 2 Γ G r (x, z 1 )G r (z 1, z 2 )G r (z 2, y)+

21 Differential Equations for Green s Function Strategy : Exploit the differential equations Show that d dr G r (x, y) = r 1 G r (x, z)g r (z, y) r 1 G r (x, y) z Γ d 2 dr 2 G r (x, y) = r 2 z 1,z 2 Γ G r (x, z 1 )G r (z 1, z 2 )G r (z 2, y)+ d 2 ( ) 3 d dr 2 G r (x, y) C x,y dr G r (x, y).

22 Geometric Approach Strategy : Define H 2 (r) : = G r (1, z)g r (z, 1) and z Γ H 3 (r) : = G r (1, z 1 )G r (z 1, z 2 )G r (z 2, 1) z 1,z 2 Γ Show that as r R H 3 (r) CH 2 (r) 3

23 Geometric Approach Strategy : Define H 2 (r) : = z Γ G r (1, z)g r (z, 1) and H 3 (r) : = Show that as r R Recall: As r R, d(1,x)=m z 1,z 2 Γ G r (1, z 1 )G r (z 1, z 2 )G r (z 2, 1) H 3 (r) CH 2 (r) 3 G r (1, x) 2 C r exp{mpr(2ϕ r )}.

24 Thin Triangle Reduction Recall α : Σ Γ is a bijection. Thus, for any pair z 1, z 2 Γ at distances m, n from 1 there are paths ω 1, ω 2 Σ of lengths m, n that map to geodesic segments from 1 to z 1, z 2. Let z = z 1 z 2 = divergence point. Now relabel the points 1, z 1, z 2 in the sum by z 1 shift so that z becomes the center of the triangle, then use the automatic representation of the 3 spokes to write =. z 1,z 2 ω 1,ω 2,ω 3 Σ

25 Application of Ruelle Thus, H 3 (r) : = G r (1, z 1 )G r (z 1, z 2 )G r (z 2, 1) = z 1,z 2 Γ ω 1,ω 2,ω 3 Σ exp { 2 } 3 S mi ϕ r (ω i ) f r (ω 1, ω 2, ω 3 ) i=1 where m i = length of ω i and f r is a nonnegative, jointly Hölder continuous function. Now use Ruelle s theorem 3 times in succession to get ( H 3 (r) C r exp{mpr(2ϕ r )} m 0 C r H 2 (r) 3 as r R. ) 3

26 Brownian Excursion Excursion: Simple random walk path (x n ) n 2m on Z + of length 2m that begins and ends at 0 and remains positive for 0 < n < 2m. Let D 2m = set ofexcursions of length 2m. Renormalization: For excursion γ = (x n ) n 2m of length 2m let πγ be the continuous path (y t ) t [0,1] such that y j/2m = x j / 2m. Fact: Let µ 2m be uniform distribution on D 2m and ν 2m = µ 2m π 1 be the induced measure on C[0, 1]. Then ν 2m = ν The probability measure ν is the distribution of Brownian excursion

27 Random Walk Excursions Theorem: Let {Zn m } 0 n m be simple nearest neighbor random walk on the d regular tree conditioned to return to its starting point 1 at time m. Let Yn m = d(zn m, 1) be the distance to the root. Then as m Y[mt] m D = Brownian Excursion m

28 Random Walk Excursions Theorem: Let {Zn m } 0 n m be simple nearest neighbor random walk on the d regular tree conditioned to return to its starting point 1 at time m. Let Yn m = d(zn m, 1) be the distance to the root. Then as m Y[mt] m D = Brownian Excursion m Theorem: (Bougerol-Jeulin) For T > 0 let {Wt T } 0 t T be Brownian motion in H conditioned to return to it starting point O at time T, and let Y T t = d(w T t, O). Then Y T [Tt] T D = Brownian Excursion

29 Random Walk Excursions Conjecture: Let {Zn m } 0 n m be symmetric, FRRW on any nonelementary hyperbolic group conditioned to return to its starting point 1 at time m. Let Yn m = d(zn m, 1) be the distance to the root. Then as m Y m [mt] m D = Brownian Excursion

30 Random Walk Excursions Conjecture: Let {Zn m } 0 n m be symmetric, FRRW on any nonelementary hyperbolic group conditioned to return to its starting point 1 at time m. Let Yn m = d(zn m, 1) be the distance to the root. Then as m Y m [mt] m D = Brownian Excursion Conjecture: An analogous theorem is true for conditioned symmetric FRRW on any lattice of a connected semisimple Lie group with finite center. The limit process is the modulus of p dimensional Brownian excursion for p = function of Lie algebra structure.