The Local Limit Theorem for Random Walks on Free Groups

Steve Lalley

Purdue University

Algorithms Seminar

May 31, 1999

[summary by Cyril Banderier]

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Abstract

Local limit theorems and saddlepoint approximations are given for random walks on a free group whose step distributions have finite support. These are derived by exploiting a set of algebraic relations among certain generating functions that arise naturally in connection with the transition probabilities of the random walks. Basic tools involved in the analysis are the elementary theory of algebraic functions, the Perron-Frobenius theory of nonnegative matrices, and standard techniques of singularity analysis.

## 1   Walks on Groups Figure 1: A representation (Cayley graph) of the infinite free group Z * Z

Let G be a the free group with generators a1,..., aL. For example, the free group Z * Z has two generators a and b. The word a b a a_ b_ a_ a_ b_ a a corresponds to the point B, the associated reduced word is a_ b_ a a.

A finite-range random walk {Zn}n³ 0 is a Markov chain on G with Z0:=e (the identity of the group, the ``origin'', the starting point) and transition probabilities
Pr{Zn+1=yx|Zn=y}=px      " x,y Î G, n³ 0,
where px for xÎ G is a probability distribution with finite support (in other words, px is the probability of the ``jump'' x).

Note p*n(x) the probability of being in x after n steps. It is assumed that the random walk is irreducible and aperiodic, that is, that
" x Î G
 å n³ 1
p*n(x) >0      (irreducibility);
GCD{n; p*n(e)>0}=1      (aperiodicity).
Another important condition is the following:
Positivity:      pe>0 and pg>0 for all generators g of G (and their inverses).
Similarly to the random walk in the Euclidian case Zd, a local limit theorem is given by the asymptotics
p*n(x)~
Bx R-n
(2 p R)1/2n3/2
.

Similar results were already known when all the steps are of size one (nearest neighbour random walk ) or when all the words at a same distance from the origin are equiprobable (the so-called isotropic random walk [2, 8, 9]).

## 2   Singularity Analysis

This section is devoted to the analysis of some probability generating functions (PGF) related to the walk. For xÎ G and zÎ C (|z|<1), define
• the random variable coding where one is after n steps: Zn,
• the PGF to reach x in n steps: Gx(z):=ån p*n(x)zn,
• the PGF of the excursions (Green's function): G(z):=Ge(z),
• the first time x is reached: Tx:=inf{n³ 0: Zn=x},
• the PGF to reach x for the first time in n steps: Fx(z):=ån Pr{Tx=n} zn.
Note that aperiodicity and irreducibility imply that for all sufficiently large n³ 1, p*n(e)>0 and p*n(y)>0, for any (inverse of a) generator y.

The following (combinatorially trivial) relations
Gx(z)=Fx(z)G(x),
G(z)=1+z æ
ç
ç
è
pe+
 å x¹ e
px F
 x-1
(z) ö
÷
÷
ø
G(z)= æ
ç
ç
è
1-zpe-z
 å x¹ e
px F
 x-1
(z) ö
÷
÷
ø
 -1
allow to prove that all of the functions Fx and Gx have the same radius of convergence R, 1<R<¥ (the less obvious is that R is strictly greater than 1).

Let B be the set of points at distance £ K, where K is such that there is no smaller ball in which the support of the step distribution {px} is contained. Define now
• the first time that x is exceeded: tx
• the PGF to go from a to xb while x as never been exceeded before:
Hxab=  å n
Pr {Zn=xb|Z0=a and Zi<nÏxB } zn
• the PGF to go from x to the origin: Fx-1(z)
The formula
" x¹ e      Fx(z)=
 å bÎ B-{e}
Hxeb (z) F
 b-1
(z)
leads to an expression of Fx=uHx(z)v=u Hx1(z)··· Hxm(z)v where u is the projection on e and v a vector whose entries are the Fb-1(z) for bÎ B and where the product of matrices is over x1··· xm, the reduced word associated to x.

It is then proven by the Markov property that all the non-constant Hxiab satisfy polynomial relations (Hxi=Qi(Hx1, Hx2, ...), see  for exact relations) and that they have the same radius of convergence.

By elimination (Gröbner basis or resultants), the functions Fx and Gx are algebraic. Their Puiseux expansion leads to an algebraic singularity with exponent 1/2.

Here is a sketch of the proof that the exponent is indeed a=1/2. Define Jz the Jacobian matrix ( Qi / Hxj), the polynomials Qi have nonnegative coefficients, thus there exists n such that Jzn is an aperiodic and irreducible matrix with strictly positive coefficients. By the Perron-Frobenius theorem, Jz has a positive eigenvalue lz of multiplicity 1. The function lz is increasing and real-analytic and lR=1/R. Considering a left eigenvector of JR and using the shape of the Qi yields to the relations (R-z)(C+···)=C'(R-z)2a+···, thus 2a=1.

As z=R is the dominant singularity of Fx and Gx, one has the two following theorems:

Theorem 1  [Local limit theorem, access]   Assuming irreducibility and aperiodicity, one has, for a positive constant Bx:
p*n(x)~
Bx
(2p R)1/2Rn n3/2
.

Theorem 2  [Local limit theorem, first access]   Assuming positivity, one has, for a positive constant Ax:
Pr {x is reached for the first time after n steps} ~
Ax
(2p R)1/2Rn n3/2
.

The probability to reach a point x at a distance m of the origin in n steps is
p*n(x)~
exp(nb(m/n))
(
 .. y
(m/n))1/2
C(m/n)
for appropriate functions b, C and y.. (see the correct definitions / notations in ).

This uniform asymptotics in x and n corresponds to the classical saddlepoint approximation (sharp large deviations theorems) for sums of iid random vectors in Rd.

The saddlepoint approximations are of interest for another reason. For large n, nearly all the mass in the probability distribution p*n(x) is concentrated in the region |x|³ e n, where the local limit approximations are not accurate. This contrasts with the situation for finite range random walk in Euclidean space. In fact, Guivarch  has shown that for random walks in G, the distance from the origin grows linearly in n. Sawyer and Steger  have further shown that (|Zn|-nb)/(n)1/2 converges in law to a normal distribution.

Finally, S. Lalley, using a special matrix product and results on Ruelle's Perron-Frobenius operators, derives a saddlepoint approximation, uniformly for m/n in a given compact
Pr{|Zn|=m}~
exp(nB(m/n))
(2 p m D(m/n))1/2
C(m/n).

## References


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Lalley (Steven P.) and Hueter (Irene). -- Anisotropic branching random walks on homogeneous trees. Preprint, 1999.


Picardello (Massimo A.). -- Spherical functions and local limit theorems on free groups. Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 133, 1983, pp. 177--191.


Sawyer (Stanley). -- Isotropic random walks in a tree. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 42, n°4, 1978, pp. 279--292.


Sawyer (Stanley) and Steger (Tim). -- The rate of escape for anisotropic random walks in a tree. Probability Theory and Related Fields, vol. 76, n°2, 1987, pp. 207--230.


Seneta (E.). -- Nonnegative matrices and Markov chains. -- Springer-Verlag, New York, 1981, second edition, xiii+279p.

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