ANISOTROPIC HARMONIC ANALYSIS ON TREES 7 Note that if x G G, then d(x, e) (with respect to the Cayley graph) is equal to the length of x as a reduced word in the generators. In general we have d(x, y) = \x~1y\1 where | | denotes the length of a reduced word. We introduce here some notation associated with G. We shall be interested in the function spaces 0(G) = {/: G - C ||/||p = ( £ |/(x)|") ' oo}. . x€G The group G acts on these spaces according to (x f)(y) = f(x~ly) and this representation is called the {left) regular representation. When £P(G) = £2(G), this representation is unitary. If 6X G ^X(G), is given by then x / = 6X* f. Sometimes we write 6xy with the same meaning as Sx(y). Finally (,l(G) is a convolution algebra with 6e as the identity. All the above applies to any discrete group. A random walk on G is defined by giving an infinite matrix R = (R(y, x))y,xeG with nonnegative coefficients, satisfying ^ZyR(y^x) = 1. The random walk is called symmetric if R(x, y) = R(y, x) for all x and y. The number R(y, x) represents the probability of taking a step from x to y. The matrix R = (R(y,x)) is called the transition matrix of the random walk. Observe that the conditions R(y, x) 0 and ]Ty R(y, x) = 1 imply that we can define the powers of R as infinite matrices Rn = (Rn(y,x)), where Rn(y,x) = ^2tRn~1(y,t)R(t,x). Induction shows that ]"(*,, x) = J ^ J T * - 1 (»,*)*(*,*) = I . y y t The number Rn(y, x) represents the probability of going from x to y in n steps. The conditions R(y, x) 0 and ^ R(y, x) 1 imply that the matrix R defines a bounded linear operator on ^{G) (Rf)(y) = ^2X R{y'»x) jT(x), and P/III^EE^^I/^IHI/III- y£Gx£G We say that the random walk R is G-invariant if R{ty,tx) = R(y,x) for all x,y,t G G. A G-invariant random walk may also be invariant for other isometries of the tree. An important special case is that of a random walk which is invariant with respect to any isometry. In the latter case R(y, x) will depend only on ^he distance between x and y. Such a random walk is called isotropic. From now on we consider only G-invariant random walks. Define a probability measure \i on G by \±{x) R(x,e). Since R is G-invariant, R(y,x) = ji(x~ly), and Rf(y) = ^2X fi(x~1y)f(x) = (/ fi)(y). Moreover if /in denotes the n-th convolution power of /x, then the coefficients of Rn are jin{x~ly). A G-invariant
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