On mixing behavior of a family of random walks determined by a linear recurrence
Introduction
Let be a positive increasing integer sequence given by the linear recurrence with constant coefficients and . The sequence determines a family of random walks: for fixed n, consider the Markov chain whose state space is . The initial state is and from the current state , the next state is where is chosen from the set uniformly at random. So for each n we have an associated Markov chain, more specifically, a random walk on the finite abelian group . By the assumption , the set generates the group and hence the random walk is irreducible. Further as , the walk is aperiodic. The stationary distribution , to which the random walk converges, is uniform over .
This paper examines the number of steps required for the distribution of to be close to its stationary distribution. It is well-known that this number, or mixing time, is governed by the second largest eigenvalue modulus (SLEM) and, in general, related to the collection of nontrivial eigenvalues of the transition matrix. In the next section, we formalize the notion of mixing time and make the relationship between that and eigenvalues concrete. We also introduce notations and established results that will be used throughout this work. Section 3 details explicit formulas for the eigenvalues and we prove that for a random walk arising from subject to certain conditions, at most steps will suffice where κ is some constant that depends on . Section 4 focuses on random walks arising from first order recurrences. In that case we show that steps will suffice, where γ is also some constant that depends on .
Our results on the eigenvalues of these Markov chains also allow us to derive lower bounds on the mixing times in the case that grows like an exponential function. For general linear recurrences of exponential growth, we have the lower bound of the form and in the first order case we get a lower bound of the form κn.
Though we have proven these upper and lower bounds on the mixing times we suspect from simulations that the mixing time has growth somewhere between n and . The table below displays the mixing times (see Definition 2.3) for random walks arising from three integer sequences.
Random walks on the integers modulo an integer have been studied frequently, as they are a prototypical example of a Markov chain on a group, and are amenable to techniques based on discrete Fourier analysis. In his review article [9], Saloff-Coste considers random walks on given by where for some choice of . Hildebrand [5] considers walks on given by where is uniform on a set of k random elements of . He shows that if n is prime then it suffices to take steps to be close to uniformly distributed for almost all choices of k elements. Hildebrand also considers the case where the size of the random step set grows with n, and the situation studied in this paper provides an interesting deterministic boundary case between Theorems 3 and 4 of [5]. Diaconis [1] discusses various random walks on given by , where and are subject to various restrictions. More recent related work includes work of Hermon and Olesker-Taylor [3] which extends results of Hildebrand to many other abelian groups with random generating sets. The work of Hough [6] considers both random generating sets, and the case of a fixed deterministic generating set in , p a prime. In particular it achieves tight results of order in for mixing on with generating steps . While this result does not precisely fit the context of our paper, it is a close analogue of the results in Section 4 on first order recurrences.
Our attention to this particular problem arose from a project in which we set out to generalize the abelian sandpile Markov chain introduced in [7]. For a more complete description of this connection see Chapter 2 of [10].
Section snippets
Preliminary results
This section collects relevant definitions, notations, and theorems on linear recurrences, mixing times, and Markov chains on groups that we will use. A more detailed study of probability and mixing time related items can be found in [8], and the importance of group structure for analyzing eigenvalues of Markov chains appears in [1].
A standard theorem of elementary combinatorics characterizes the solutions of linear recurrence relations (see, e.g. [11, Chapter 4]):
Theorem 2.1 The sequence satisfies
General linear recurrences
In this section, we prove bounds on nontrivial eigenvalue moduli for linear recurrence relations of arbitrary order. From this we are able to deduce lower and upper bounds on the mixing time of the Markov chain. In the next section, we specialize to the case of first order linear recurrences, where we are able to prove stronger upper and lower bounds. The upcoming results often make use of the sum of the positive coefficients in the defining linear recurrence, so for
First order recurrences
This section considers sequences generated by first order recurrences , that is, geometric series of the form , where is a positive integer. For these sequences, we show that the order of the mixing time of associated family of random walks is between n and . The main result of this section is the following upper bound on mixing time:
Theorem 4.1 For the random walk determined by the sequence , where is an integer,
Conclusion
We have shown that the order of the mixing time of random walks determined by a general linear recurrence exhibiting exponential growth is between and . A situation that requires further study is the special case where the integer sequence defined by the linear recurrence exhibits polynomial growth instead. This occurs when the characteristic equation of the recurrence is for some . For this case, the result and proof of Theorem 3.1 still holds and the corresponding upper
Declaration of Competing Interest
There are no conflicts of interest to report.
Acknowledgements
Caprice Stanley was partially supported by the US National Science Foundation (DMS 1615660). Seth Sullivant was partially supported by the US National Science Foundation (DMS 1615660) and by the David and Lucile Packard Foundation.
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