Random walks on the torus with several generators

Given n vectors {~ αi}i=1 ∈ [0, 1), consider a random walk on the ddimensional torus T = R/Z generated by these vectors by successive addition and subtraction. For certain sets of vectors, this walk converges to Haar (uniform) measure on the torus. We show that the discrepancy distance D(Q) between the k-th step distribution of the walk and Haar measure is bounded below byD(Q) ≥ C1k, where C1 = C(n, d) is a constant. If the vectors are badly approximated by rationals (in a sense we will define) then D(Q) ≤ C2k for C2 = C(n, d, ~ αj) a constant. Let T = R/Z denote the d-dimensional torus. As a quotient group of R it is an additive group, so the group elements may be viewed as elements of [0, 1), with the group operation defined as coordinate-wise addition mod 1. Let ~ α1, ~ α2, . . . , ~ αn be vectors in T d = [0, 1), and consider the random walk on the d-dimensional torus T that proceeds as follows. Start at ~0. At each step, choose one the vectors ~ αi with probability 1/n and add or subtract that vector (with probability 1/2) to the current position to get to the next position in the walk. As a random walk on a group, the k-th step distribution of the walk converges to a limiting distribution [6], and in many cases this will be Haar measure, the unique translation-invariant measure on the group. For the torus T, Haar measure may be thought of as the uniform distribution on the “flat” cube [0, 1), since addition corresponds to translation on R/Z. We shall prove bounds for how quickly this random walk approaches Haar measure on the torus. ∗Department of Mathematics, University of California, Los Angeles, CA 90095, [email protected]. ∗∗Research partially supported by NSF Grant DMS-0301129. Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, [email protected] (corresponding author). 1 2 TIMOTHY PRESCOTT AND FRANCIS EDWARD SU We first note that for certain sets of vectors, this walk may not converge to Haar measure. For instance, if all the entries of each ~ αi are rational, then the random walk will not converge to Haar measure, but will converge to a limiting distribution supported on a discrete subgroup of T. As another example, if there is only one generator ~ α1 = (x, x, ..., x) for some irrational x, then the walk will be supported on a circle along the “diagonal” of the torus. (However, a single vector can generate a walk that does converge to Haar measure, provided it is chosen well.) LetQ denote the generating measure for this random walk, i.e., if S = ∪i=1{+~ αi,−~ αi} is the set of generators of the random walk, then for a set B ⊆ T, let Q(B) = |B ∩ S|/|S| where | · | denotes the size of a finite set. The k-th step probability distribution is then given by the k-th convolution power of Q, which we denote by Q. Let U denote Haar measure. As a measure of distance between the probability distributionsQ and U , we will use the discrepancy metric, which is defined to be the supremum of the difference of two probability measures over all “boxes” in T = R/Z with sides parallel to the axes in R, i.e., of the form [a1, b1)× [a2, b2)× ...× [ad, bd). Let D(Q) denote the discrepancy of Q from Haar measure U : D(Q) := sup boxB⊆Td |Q(B)− U(B)|. The discrepancy metric has been used by number theorists to study the uniform distribution of sequences mod 1, e.g., see [2, 7]. Diaconis [1] suggested its use for the study of rates of convergence for random walks on groups. It admits Fourier bounds [4] and has many other nice properties and connections with other probability metrics [3]. Although the total variation metric is more commonly used to study the convergence of random walks, we do not use it here because this random walk does not converge in total variation (in fact, the total variation distance between Q and U is always 1, since at any step Q is supported on a finite set). The possibility of using Fourier analysis to bound the discrepancy distance makes it a more desirable RANDOM WALKS ON THE TORUS 3 choice than other common metrics on probabilities, such as the Prohorov metric, and has allowed many recent results for the study of discrete random walks on continuous state spaces (e.g., [12, 14]). Most of the literature for rates of convergence of random walks have been limited to walks on finite groups or state spaces, and those that have focused on infinite compact groups (e.g., [8], [9], [13]) have studied walks generated by continuous measures. By contrast, the walk we study is generated by a discrete set of generators on an infinite group. We prove: Theorem 1. Let Q denote the generating measure of the the random walk on the d-torus generated by n vectors ~ α1, ..., ~ αn. Then the k-th step probability distribution Q satisfies: D(Q) ≥ 1 πd5n+1dn/2 k. This result holds for any set of vector generators. On the other hand, for certain sets of badly approximable generators (to be defined later), we can establish the following upper bound. Theorem 2. Let An×d be a badly approximable matrix, with rows ~ α1, ..., ~ αn, and approximation constant CA. If Q is the generating measure of the random walk on the d-torus generated by the ~ αi, then the k-th step probability distribution Q ∗k satisfies: D(Q) ≤ ( 3 2 )d 20 ( n CA √ 2 )n/d k. We note that the case d = 1 corresponds to a random walk on the circle, which has been studied for a single generator [12] and for several generators [4]. 1. Lower Bound The following notation will be used throughout this paper: ‖x‖: the Euclidean (L) norm of a vector x ‖x‖∞: the supremum norm of a vector x {x}: the Euclidean (L) distance from x to the nearest integral point 4 TIMOTHY PRESCOTT AND FRANCIS EDWARD SU {x}∞: the supremum distance from x to the nearest integral point To establish a lower bound for the discrepancy, we use a lemma due to Dirichlet: Lemma 3 (Dirichlet 1842). Given any real n × d matrix A and q ≥ 1, there is some h ∈ Z such that 0 < ‖h‖∞ ≤ q and {Ah}∞ < 1/q. A simple proof using a pigeonhole argument may be found in [11]. We now prove Theorem 1. Proof. Su [13] has shown that for any probability distribution P on T: (1) D(P ) ≥ sup r∈(0,.5]d   ∑ 0 6=h∈Zd |P̂ (h)| d ∏ i=1 { sin(2πhiri) πh2i if hi 6= 0 4r i if hi = 0 }  1/2 where P̂ (h) is the Fourier transform of P , i.e., P̂ (h) = ∫ Td eQ(dx). We will use this formula to boundD(Q) where Q is the generating measure of our random walk. Note that: