Mixing time of fractional random walk on finite fields

Abstract

We study a random walk on Fp defined by Xn+1 = 1/Xn + εn+1 if Xn = 0, and Xn+1 = εn+1 if Xn = 0, where εn+1 are independent and identically distributed. This can be seen as a non-linear analogue of the Chung-Diaconis-Graham process. We show that the mixing time is of order log p, answering a question of Chatterjee and Diaconis [12].

Accelerating Abelian Random Walks with Hyperbolic Dynamics

“…They proved for a = 2 that after t = O(log n log log n) steps the distribution of X t is close to uniform, thus showing a dramatic speed-up over the simple random walk, which needs Ω(n 2 ) steps. Recently, attention has been brought back to the potential speed-up obtained by applying deterministic functions to Markov chains [10,15,16,5]. In this work, we revisit the Chung-Diaconis-Graham process in the multi-dimensional case.…”

“…Since expansion is related to the fast-mixing of simple random walks, the same tools can be used to prove fast mixing of affine random walks. Incidentally, such results have already been used in [16] to prove speed-up for the inverse mapping on Z pZ. Let us give a few examples.…”

“…On the cycle Z pZ with p prime, He [15] proves an almost linear mixing time in p when the bijection considered is a rational function. This bound has been improved in [16] in the specific case of the inverse function f ∶ x ↦ 1 x if x ≠ 0, f (0) ∶= 0, where He, Pham and Wei shows that O(log p) steps suffice to reach stationarity.…”

Fast mixing of a randomized shift-register Markov chain

“…sequences of random variables supported on Z according to some fixed probability measures. Further variations on this chain are studied in [10].…”

Cutoff for permuted Markov chains

“…He, Pham, and Xu [18] recently proved that, for f (x) = x −1 when x = 0, and f (0) = 0, this chain takes only order log n steps to mix. For f (x) = 2x, it was shown by Eberhard and Varjú [16] that the chain has cutoff at time c log n, for some absolute constant c > 0.…”