Local Limit Theorems For Random Walks OnTrees And Groups

Steve Lalley2 and Sebastien Gouezel1

1University of Rennes

2University of Chicago

June 18, 2011

Infinite HomogeneousTree of Degree d = 3

Random Walk on Tree T4

Transition Probabilities:

I Up/Down probability pV

I Right/Left probability pH

I 2pV + 2pH = 1

Problem: What is theprobability of return to theroot after 2n steps?

Asymptotic Behavior of Return ProbabilitiesMore generally, consider homogeneous, nearest-neighborrandom walk Xn on the infinite regular tree Td of degree d ≥ 3.

Theorem: (Gerl-Woess 1986 PTRF) There exist constantsC > 0 and R > 1 (depending on the step distribution) such thatas n →∞,

P1{X2n = 1} ∼ CRnn3/2

Asymptotic Behavior of Return ProbabilitiesMore generally, consider homogeneous, nearest-neighborrandom walk Xn on the infinite regular tree Td of degree d ≥ 3.

Theorem: (Gerl-Woess 1986 PTRF) There exist constantsC > 0 and R > 1 (depending on the step distribution) such thatas n →∞,

P1{X2n = 1} ∼ CRnn3/2

What about non-nearest neighbor random walk on Td?

Theorem: (Lalley 1993 Annals) There exist constants C > 0and R > 1 (depending on the step distribution) such that asn →∞,

P1{X2n = 1} ∼ CRnn3/2

SLLN and CLTThe local limit theorem suggests that the event of return to theroot at large time 2n is a large deviation event.

Theorem: (SLLN; CLT) There exist µ > 0 and σ2 > 0depending on the step distribution such that as n →∞,

distance(Xn, root)/n → µ a.s., and

(distance(Xn, root)− nµ)/√

n =⇒ Normal(0, σ2)

These are less interesting that the return probabilities and canbe established by sgtandard techniques, i.e., subadditiveergodic theorem and martingale CLT.

T4 as a Free Group/Regular Language

I Vertices correspond to finite,reduced words in the lettersA, A−1, B, B−1.

I The set F2 of all such words isa regular language.

I F2 is a group underconcatenation/reduction.

I Xn = ξ1ξ2 · · · ξn where ξi arei.i.d.

Random Walk on Nonamenable GroupsNonamenable Group: A finitely generated, discrete group Γ isnonamenable if its isoperimetric constant γ is positive, that is

γ = infF⊂Γ

#(∂F )

#F> 0

Examples: The integer lattices Zd are amenable. Free groupsand discrete subgroups of matrix groups GL(n, Z) arenonammenable.

Kesten’s Theorem: For every nontrivial random walk on anonammenable group there is a constant R > 1 such that

limn→∞

n−1 log P1{X2n = 1} = − log R.

Random Walk on Nonamenable GroupsNonamenable Group: A finitely generated, discrete group Γ isnonamenable if its isoperimetric constant γ is positive, that is

γ = infF⊂Γ

#(∂F )

#F> 0

Examples: The integer lattices Zd are amenable. Free groupsand discrete subgroups of matrix groups GL(n, Z) arenonammenable.

Kesten’s Theorem: For every nontrivial random walk on anonammenable group there is a constant R > 1 such that

limn→∞

n−1 log P1{X2n = 1} = − log R.

Random Walk on PSL(2, R)

PSL(2, R) is the group of all 2× 2 real matrices withdeterminant +1. It is of particular interest to mathematiciansbecause it is the isometry group of the hyperbolic plane.

Theorem: (Bougerol 1981 Ann. Sci. de l’ENS) LetXn = ξ1ξ2 · · · ξn be a symmetric, right random walk onPSL(2, R) whose step distribution has a density with respect toHaar measure and rapidly decaying tails. Then the n−steptransition probability densities pn(x , y) satisfy

pn(x , y) ∼Cx ,y

Rnn3/2

In fact Bougerol proved that there is a similar asymptoticformula for random walk on any semi-simple Lie group. Theexponent 3/2 is replaced by k/2 for some integer k dependingonly on the structural constants of the Lie group.

The Free Group Fk as a Subgroup of PSL(2, R)

AB

I Isometries of H are linearfractional transformations.

I Let A be the LFT mapping blueline to blue line and B the LFTmapping red line to red line.

I Let Γ be the group generatedby A±1, B±1.

I Then Γ = F2

The Free Group Fk as a Subgroup of PSL(2, R)

Elements of Γ = F2 are in 1− 1correspondence with the tiles ofthe tessellation. Each element of Γacts as a permutation on the tiles.The tree T4 is naturally embeddedin the tessellation, with each tilecontaining one vertex of T4.

Theory: The fact that F2 embeds into PSL(2, R) is responsiblefor the local limit theorem.

Discrete Subgroups of PSL(2, R)

Conjecture: (L-1993) The local limittheorem holds for any symmetric,finite-range random walk on anyco-compact subgroup ofPSL(2, R), and in particular forrandom walk on any surface groupof genus ≥ 2.

Theorem: (Gouezel-Lalley 2011)It’s true! For any such randomwalk,

P1{X2n = 1} ∼ CR−nn−3/2.

Random Walk on a Regular LanguageA regular language is a set L of words (finite sequences) over afinite alphabet such that membership in L can be checked by afinite-state automaton.

Example 1: The group Z2 ∗ Z2 ∗ Z2 is the set of all finite wordsfrom a 3-letter alphabet {a, b, c} in which no letter appearstwice consecutively.

Example 2: Any group that has a free group Fk as a subgroupof finite index (for instance, PSL(2, Z) or its congruencesubgroups) has a representation as a regular language.

Random Walk on a Regular LanguageA random walk on a regular language L is a Markov chain Xnon L whose one-step transitions are such that

I the length of the word can only change by 0,+1, or −1;I only the last 2 letters are altered; andI the transition probabilities depend only on the last 2 letters.

Example: LIFO queues with finitely many job types.

Random Walk on a Regular LanguageA random walk on a regular language L is a Markov chain Xnon L whose one-step transitions are such that

I the length of the word can only change by 0,+1, or −1;I only the last 2 letters are altered; andI the transition probabilities depend only on the last 2 letters.

Example: LIFO queues with finitely many job types.

Theorem: (Lalley 2003) Under suitable irreducibility andaperiodicity hypotheses, every transient RWRL satisfies a locallimit theorem

Px{Xn = y} ∼ Cx ,yR−nn−α

for some R ≥ 1 and α = 0, 12 , or 3

2 . For random walk on avirtually free group, R > 1 and only α = 3

2 is possible.

Green’s Function of a Random WalkThe Green’s function of a Markov chain is the matrix ofgenerating functions of transition probabilities: for each pair ofstates x , y ,

Gx ,y (r) =∞∑

n=0

pn(x , y)rn.

If the Markov chain is irreducible then the radius ofconvergence R ≥ 1 does not depend on x , y . For random walkon any nonamenable group, Gx ,y (R) < ∞.

The behavior of the Green’s function at the radius ofconvergence determines and is determined by the asymptoticbehavior of the coefficients pn(x ,y). In particular,

G(R)−G(r) ∼ C√

R − r + Tauberian condition

=⇒ p2n(x , x) ∼ CxR−nn−3/2

Flajolet-Odlysko Tauberian TheoremTheorem: Let G(z) =

∑∞n=0 anzn be a power series with radius

of convergence R. Suppose that G has an analytic continuationto C \ ([R,∞) ∪ (−∞,−R − ε]), and suppose that as z → R in∆ρ,φ,

G(z)− C ∼ K (R − z)α (1)

for some K 6= 0 and α 6∈ {0, 1, 2, . . . }. Then as n →∞,

an ∼K

Γ(−α)Rnnα+1 . (2)

Basic Fact: The Green’s function G1,1(r) of an aperiodic,symmetric random walk on a discrete group admits an analyticcontinuation to C \ ([R,∞) ∪ (−∞,−R − ε]).

Green’s Function and Branching Random WalkBranching Random Walk: Particles alternately reproduceaccording to the law of a Galton-Watson process and moveaccording to the law of a symmetric random walk. The defaultinitial configuration has a single particle located at the groupidentity 1.

Proposition: Let Nn(x) be the number of particles at location xin the nth generation. Let r be the mean of the offspringdistribution. Then

ENn(x) = rnpn(1, x).

Consequently, the expected total occupation time of vertex xthrough the entire history of the BRW is

G1,x(r).

Weak SurvivalBranching random walk on a homogeneous tree has a weaksurvival phase. If the mean offspring number r > 1 then thetotal number of particles in generation n is a supercriticalGalton-Watson process. If 1 < r < R then the expectednumber of particles at the root decays exponentially. So withpositive probability, the number of particles grows exponentially,but the particles eventually vacate any finite set of vertices.

Limit Set: Define

Λ = {ends in which the BRW survives}

Branching Brownian Motion in the Hyperbolic Plane

Branching Brownian Motion:Individual particles executeindependent Brownian motions inH beginning at theirbirthplaces/times, andindependently fission at rate λ > 0.

Theorem: (Lalley-Sellke 1996) (Lalley-Sellke) If λ < 1/8 thenthe process survives weakly. The limit set Λ is wp1 a Cantor setof Hausdorff dimension

δ(λ) = (1−√

1− 8λ)/2

Branching Brownian Motion in the Hyperbolic Plane

Branching Brownian Motion:Individual particles executeindependent Brownian motions inH beginning at theirbirthplaces/times, andindependently fission at rate λ > 0.

Theorem: (Lalley-Sellke 1996) (Lalley-Sellke) If λ < 1/8 thenthe process survives weakly. The limit set Λ is wp1 a Cantor setof Hausdorff dimension

δ(λ) = (1−√

1− 8λ)/2

The square-root singularity is caused by a correspondingsquare-root singularity in the Green’s function of hyperbolicBrownian motion.

Branching Brownian Motion in the Hyperbolic Plane

Branching Brownian Motion:Individual particles executeindependent Brownian motions inH beginning at theirbirthplaces/times, andindependently fission at rate λ > 0.

Theorem: (Lalley-Sellke 1996) (Lalley-Sellke) If λ < 1/8 thenthe process survives weakly. The limit set Λ is wp1 a Cantor setof Hausdorff dimension

δ(λ) = (1−√

1− 8λ)/2

There is a similar exact formula for BRW on the homogeneoustree Td , also with square-root singularity at the phase transitionpoint.

Asymptotic Behavior of the Green’s Function at r = R

Special Case: RW on T3.

Color all edges Red, Blue, orGreen in such a way thatevery vertex has one incidentedge of each color. Let Xn bethe random walk that makesRed, Blue, or Green stepswith probabilities pR, pB, pG.

Let τ(i) = first time that Xnvisits vertex i (wherei = R, B, G).

Functional Equations for the Green’s FunctionsDefine Fi(r) = E1r τ(i). Then

G1,1(r) = 1 +∑

i=R,B,G

piFi(r)G1,1(r)

=

1−∑

i=R,B,G

piFi(r)

−1

andFi(r) = pi r +

∑j 6=i

pj rFj(r)Fi(r)

These equations determine an algebraic variety in the variablesFR, FG, FB and r . Hence, the functions Fi and G1,1 arealgebraic functions, and thus their only singularities are isolatedbranch points.

Functional Equations for the Green’s FunctionsThe algebraic system in vector notation has the form

F = rP + rQ(F)

where Q is a vector of quadratic polynomials with positivecoefficients. By the Implicit Function Theorem, analyticcontinuation of the vector-valued function F is possible at everypoint where the linearized system is solvable for dF in terms ofdr :

dF = (dr)P + (dr)Q(F) + r(

∂Q∂F

)(dF) ⇐⇒(

I− r∂Q∂F

)(dF) = (dr) (P + rQ(F))

Functional Equations for the Green’s FunctionsThus, the smallest positive singularity of the function F(r)occurs at the value r = R where the spectral radius(=Perron-Frobenius eigenvalue) of r(∂Q/∂F ) reaches 1. If vT isthe left Perron-Frobenius eigenvector then near r = R,

vT((

I− r∂Q∂F

)dF

)=(dr)(vT P + rvT Q(F))

+ r(quadratic form in dF)

The quadratic terms have positive coefficients, hence thesingularity at r = R is a square-root singularity (branch point oforder 2).

Life Without Functional EquationsUnfortunately, this analysis depends in an essential way on therecursive structure of a regular language or regular tree. Ingeneral, for discrete groups there seems to be no finite systemof algebraic equations that determine the Green’s functions.

Conjecture: (P. Sarnak) The Green’s functions of a nearestneighbor random walk on a discrete subgroup Γ of PSL(2, R)are algebraic if and only if Γ has a free subgoup of finite index.

The Keys to the KingdomThe Green’s functions of any Markov chain satisfy a system ofquadratic ordinary differential equations:

ddr

Gx ,y (r) = r−1∑

z

Gx ,z(r)Gz,y (r)− r−1Gx ,y (r).

The Keys to the KingdomThe Green’s functions of any Markov chain satisfy a system ofquadratic ordinary differential equations:

ddr

Gx ,y (r) = r−1∑

z

Gx ,z(r)Gz,y (r)− r−1Gx ,y (r).

For random walk on a discrete subgroup of PSL(2, R) the sumblows up as r → R. The rate of blow-up is determined by thebehavior of Gx ,z(r) as z converges to the Martin boundary.

The Bottom Line: As r → R,

Gx ,y ∼ Cx ,y√

R − r .

What We Still Don’t Know

(1) Is there a local limit theorem for random walks on Tdwhose step distributions have infinite support?

(2) Is the square-root singularity at the weak/strong survivaltransition point for branching random walk on Td inheritedby similar particle systems, e.g., the contact process?

(3) What about PSL(3, Z) and PSL(4, Z) and · · ·?