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Markov chains on finite fields with deterministic jumps

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... Hence the Chung-Diaconis-Graham process gives an example of a speedup phenomenon, that is, a phenomenon of the decreasing time of convergence. In [8] a more general nonlinear version of the Chung-Diaconis-Graham process was studied, defined by ...
... where f is a bijection of F p . In particular, it was proved that for rational functions of bounded degree (defined appropriately at the poles; see [8]) the mixing time is ...
... As a byproduct, we show that in the case when f (x) = x 2 and p ≡ 3 (mod 4) the mixing time of (1.2) is actually O(p log p): see Remark 2. We underline again that the expected order of t mix in all these problems is probably O(log p), but this can be a hard question (especially, in view of some special constructions in the affine group which show that there are families of so-called 'rich' transformations having exactly subexponential lower bounds for the number of incidences; see [17], Theorem 15). Our approach is not analytic as in [8], but it uses some methods from additive combinatorics and incidence geometry. In particular, we apply some results on growth in the affine group Aff(F p ). ...
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We study the lazy Markov chain on Fp\mathbb{F}_p defined as follows: Xn+1=XnX_{n+1}=X_n with probability 1/2 and Xn+1=f(Xn)εn+1X_{n+1}=f(X_n) \cdot \varepsilon_{n+1}, where the εn\varepsilon_n are random variables distributed uniformly on the set {γ,γ1}\{\gamma, \gamma^{-1}\}, γ\gamma is a primitive root and f(x)=x/(x1)f(x)=x/(x-1) or f(x)=ind(x)f(x)=\mathrm{ind}(x). Then we show that the mixing time of XnX_n is exp(O(logplogloglogp/loglogp))\exp(O(\log p \cdot \log \log \log p/ \log \log p)). Also, we obtain an application to an additive-combinatorial question concerning a certain Sidon-type family of sets. Bibliography: 34 titles.
... 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 [11,16,17,5]. In this work, we study an analog of the Chung-Diaconis-Graham process in the multi-dimensional case. ...
... Finally, let us mention two other examples of explicit deterministic bijections implying speed-ups that have been studied. On the cycle Z pZ with p prime, He [16] proves an almost linear mixing time in p when the bijection considered is a rational function. This bound has been improved in [17] in the specific case of the inverse function ...
... By definition, rectangles of R 1 are closed sets which contain no point of W , and R 1 , W are both finite sets, thus η ∶= min R∈R1 d(R, W ) must be positive. By equation (16) and the continuity of the function f (13), there exists γ = γ(η, α) ∈ (0, 1) such that f is bounded by ...
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Given integers d2,n1d \ge 2, n \ge 1, we consider affine random walks on torii (Z/nZ)d(\mathbb {Z}/ n \mathbb {Z})^{d} defined as Xt+1=AXt+BtmodnX_{t+1} = A X_{t} + B_{t} \mod n, where AGLd(Z)A \in \mathrm {GL}_{d}(\mathbb {Z}) is a invertible matrix with integer entries and (Bt)t0(B_{t})_{t \ge 0} is a sequence of iid random increments on Zd\mathbb {Z}^{d}. We show that when A has no eigenvalues of modulus 1, this random walk mixes in O(lognloglogn)O(\log n \log \log n) steps as nn \rightarrow \infty , and mixes actually in O(logn)O(\log n) steps only for almost all n. These results are similar to those of Chung et al. (Ann Probab 15(3):1148–1165, 1987) on the so-called Chung–Diaconis–Graham process, which corresponds to the case d=1. Our proof is based on the initial arguments of Chung, Diaconis and Graham, and relies extensively on the properties of the dynamical system xAxx \mapsto A^{\top } x on the continuous torus Rd/Zd\mathbb {R}^{d} / \mathbb {Z}^{d}. Having no eigenvalue of modulus one makes this dynamical system a hyperbolic toral automorphism, a typical example of a chaotic system known to have a rich behaviour. As such our proof sheds new light on the speed-up gained by applying a deterministic map to a Markov chain.
... In fact, it can even be shown that appropriate homogeneous reversible random walks are optimal in some sense [12]. A different recent technique is based on alternating between stepping with the original Markov chain and moving by a permutation fixed apriori, see [3,8,13] for the initial theory, refined cut-off analysis for random permutations, and some specialized case study. But this does clearly involve very significant modification of the Markov chain structure. ...
... which implies (13). The last part of the result follows from the definitions of ∆ k , δ k and ϵ k . ...
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We analyze the absolute spectral gap of Markov chains on graphs obtained from a cycle of n vertices and perturbed only at approximately n1/ρn^{1/\rho} random locations with an appropriate, possibly sparse, interconnection structure. Together with a strong asymmetry along the cycle, the gap of the resulting chain can be bounded inversely proportionally by the longest arc length (up to logarithmic factors) with high probability, providing a significant mixing speedup compared to the reversible version.
... 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. ...
... Finally, let us mention two other examples of explicit deterministic bijections implying speed-ups that have been studied. 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 ...
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Given integers d2,n1d \geq 2, n \geq 1, we consider affine random walks on torii (Z/nZ)d(\mathbb{Z} / n \mathbb{Z})^{d} defined as Xt+1=AXt+BtmodnX_{t+1} = A X_{t} + B_{t} \mod n, where AGLd(Z)A \in \mathrm{GL}_{d}(\mathbb{Z}) is an invertible matrix with integer entries and (Bt)t0(B_{t})_{t \geq 0} is a sequence of iid random increments on Zd\mathbb{Z}^{d}. We show that when A has no eigenvalues of modulus 1, this random walk mixes in O(lognloglogn)O(\log n \log \log n) steps as nn \rightarrow \infty, and mixes actually in O(logn)O(\log n) steps only for almost all n. These results generalize those on the so-called Chung-Diaconis-Graham process, which corresponds to the case d=1. Our proof is based on the initial arguments of Chung, Diaconis and Graham, and relies extensively on the properties of the dynamical system xAxx \mapsto A^{\top} x on the continuous torus Rd/Zd\mathbb{R}^{d} / \mathbb{Z}^{d}. Having no eigenvalue of modulus one makes this dynamical system a hyperbolic toral automorphism, a typical example of a chaotic system known to have a rich behaviour. As such our proof sheds new light on the speed-up gained by applying a deterministic map to a Markov chain.
... In [22], the first author studied a more general non-linear version of the Chung-Diaconis-Graham process, defined by ...
... In [22], the first author studied a more general non-linear version of the Chung-Diaconis-Graham process, defined by ...
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We study a random walk on Fp\mathbb{F}_p defined by Xn+1=1/Xn+εn+1X_{n+1}=1/X_n+\varepsilon_{n+1} if Xn0X_n\neq 0, and Xn+1=εn+1X_{n+1}=\varepsilon_{n+1} if Xn=0X_n=0, where εn+1\varepsilon_{n+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 logp\log p, answering a question of Chatterjee and Diaconis.
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Chung, Diaconis, and Graham considered random processes of the form X n+1 = 2Xn + bn (mod p) where X0 = 0, p is odd, and bn for n = 0, 1, 2, . . . are i.i.d. random variables on {-1, 0, 1}. If Pr(bn = -1) = Pr(bn = 1) = β and Pr(bn = 0) = 1 - 2β, they asked which value of β makes Xn get close to uniformly distributed on the integers mod p the slowest. In this paper, we extend the results of Chung, Diaconis, and Graham in the case p = 2t - 1 to show that for 0 < β ≤ 1/2, there is no such value of β.
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Chung, Diaconis, and Graham considered random processes of the form X n + 1 = a n X n + b n ( mod p ) X_{n+1}=a_nX_n+b_n\pmod p where p p is odd, X 0 = 0 X_0=0 , a n = 2 a_n=2 always, and b n b_n are i.i.d. for n = 0 , 1 , 2 , … n=0,1,2,\dots . In this paper, we show that if P ( b n = − 1 ) = P ( b n = 0 ) = P ( b n = 1 ) = 1 / 3 P(b_n=-1)=P(b_n=0)=P(b_n=1)=1/3 , then there exists a constant c > 1 c>1 such that c log 2 ⁡ p c\log _2p steps are not enough to make X n X_n get close to being uniformly distributed on the integers mod p p .
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