Questions tagged [mixing]

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Does Bernoulli imply exponential mixing?

This question comes from this paper where the authors proved that exponential mixing implies Bernoulli. They also mentioned in the introduction that Bernoulli is the strongest ergodic property and ...
Kousaka_Reina's user avatar
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0 answers
33 views

Existence of a minimal, weakly mixing and Lipschitz selfmap?

I am looking for an example of a dynamical system $(M,f)$ such that: $M$ is a metric space; $f:M \to M$ is Lipschitz; $f$ is weakly mixing (that is $f \times f$ is topologically transitive) $f$ is ...
Baguette's user avatar
5 votes
1 answer
289 views

Weak mixing and entering time

Let $X$ be a compact metric space and $f$ a continuous map from $X$ to $X$. Is it true, that if $f$ is weakly mixing, then the entering time $$N(U,V) = \{n \in \mathbb{N}\mid f^n(U) \cap V \neq \...
Matěj Moravík's user avatar
0 votes
0 answers
71 views

Numerical method for mixed system of equations and nonlinear inequalities

I am currently encountering challenges in determining the solution method for the following system of equations and inequalities: $$ \begin{aligned} &F(x) = 0\\ &G(x) < 0\\ \end{aligned} $$ ...
AnNam's user avatar
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1 vote
0 answers
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Reference for the asymptotic mixing time of the random walk on the cycle

In Diaconis's book Group Representations in Probability and Statistics, Chapter 3C, there are explicit computations for the mixing time of the random walk on the cycle graph $\mathbb{Z}_{p}$, with $p$ ...
Austin80's user avatar
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2 votes
1 answer
253 views

Equivalence of the definitions of exactness and mixing

Let $f:X \to X$ be a continuous map, where $X$ is a compact metric space. We say that $f$ is (locally) expanding if there are constants $\lambda >1$ and $\delta_0 > 0$ such that, for all $x, y\...
Mrcrg's user avatar
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1 vote
0 answers
53 views

Bounding difference of characteristic functions with mixing coefficients

Setting Let $X = \{ X(t), t \in \mathbb{R} \}$ be a stationary, $\alpha$-mixing stochastic process and define $$ \begin{align} I_{n, l} &= \{ (l - 1)b_{n} + 1 + r_{n}, \ldots, lb_{n} \}, \quad l = ...
AlbertRapp's user avatar
9 votes
1 answer
325 views

Equivalent definitions of topological weak mixing

A dynamical system $f:X\to X$ is said to be topologically transitive if for any two nonempty open sets $U,V$ there exists $n \in \mathbb{Z}$ such that $f^{\circ n}(U) \cap V \neq \emptyset$. The ...
Wrt's user avatar
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2 votes
0 answers
75 views

Constructing weakly-dependent process with certain decay rate of dependency coefficients

Let $(X_{t})_{t \in \mathbb{N}}$ be a real-valued stationary stochastic process over probability $(\Omega,\mathcal{F},\mathbb{P})$, such that for $p\geq 2$, $X_{t} \in L_{p}(\mathbb{P})$ and it holds: ...
Oleksandr Z.'s user avatar
2 votes
1 answer
267 views

Uniform upper bound on contraction coefficient w.r.t total-variation metric, of a certain set of block-diagonal Markov kernels

Disclaimer. This is related to another question I've asked on the TCS site https://cstheory.stackexchange.com/q/46097/44644. I'm new to information theory (and other relevant fields). It's even ...
dohmatob's user avatar
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2 votes
1 answer
142 views

Existence of topologically mixing (discrete) dynamical system on manifold

If $M$ is a connected $(d\geq 2)$-dimensional smooth closed manifold, then does there exist a class $C^1$-diffeomorphism $\phi$ from $M$ onto itself, such that $(M,\phi)$ is a topologically mixing (...
ABIM's user avatar
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0 votes
1 answer
313 views

Mixing time of random walks on graphs

Suppose that we start a lazy random walk on a connected graph. However, the starting node is picked from a distribution of $\mu$ and $||\mu-\pi||_{TV}<1/8$, where $\pi$ is the stationary ...
user86217's user avatar
1 vote
0 answers
111 views

Must this upper bound on mixing time depend on the minimum stationary probability?

It is known fact that for a finite-state, reversible and ergodic Markov chain with transition matrix $M$, the following control on the mixing time holds $$\left( \frac{1}{\gamma_\star - 1}\right)\ln{...
geo.wolfer's user avatar
2 votes
1 answer
862 views

Comparing mixing time of lazy and non-lazy Markov chains

Suppose we have a probability distribution $\pi : X \rightarrow [0,1]$ where $X$ is finite and let $Q : X \times X \rightarrow [0,1]$ be a Markov kernel that is reversible with respect to $\pi$. That ...
Josh R's user avatar
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1 vote
0 answers
41 views

definition of mixing component

definition of ergodic component: consider stationary dynamical system $(X, \mathcal{B}, \mu, T)$, each ergodic component is $m(\cdot)=\mathbb{E}_{\mu}^{\mathcal{I}}\mathbf{1}_{(\cdot)}$, $\mathcal{I} $...
jason's user avatar
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3 votes
0 answers
214 views

Renyi's theorem on mixing

I have been trying to understand the proof of Renyi's characterization of (strongly) mixing transformations: A measure preserving transformation $T \text{ is strongly mixing iff for every measurable }...
Anon's user avatar
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0 answers
128 views

Probability of getting through with a phone-call

Alice is quite popular. She gets called on her cell-phone in a Poisson$(\lambda)$ manner. She answers her calls when possible, and ignores them when in the middle of conversation. Since you know her ...
Matjaž Krnc's user avatar
5 votes
1 answer
163 views

On a finitary version of mixing

Let $(X_1,X_2,\ldots)$ be a stationary, mixing sequence of real random variables. Then it holds (for example) for any event $A$ that is measurable in $\sigma(X_1,X_2,\ldots)$ and any $S \subseteq \...
Vladimir's user avatar
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5 votes
1 answer
256 views

Can an amenable group have a weak mixing unitary representation without almost invariant vectors?

Does there exist a finitely generated discrete amenable group $G$ that acts on a separable Hilbert space $\mathcal{H}$ by unitary transformations, and where (1) $\mathcal{H}$ has no finite dimensional ...
Vladimir's user avatar
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2 votes
0 answers
205 views

mixing time of random walks on dense Erdos Renyi graphs

Is there anything known about the mixing time of a simple random walk on an Erdos-Renyi graph with parameter $\langle n,d \rangle$ where $d=n^a (0<a<1 )$. I know about Reed et al and Benjamini ...
shahrzad haddadan's user avatar
0 votes
1 answer
1k views

Mixing time of lazy random walk on the directed cycle $C_n$

Briefly: A hint (if this is easy), reference or derivation would be of great help. The question Let $C_n$ be the directed cycle with loops in each of its $n$ vertices, and consider the random walk ...
Tássio's user avatar
  • 188
3 votes
1 answer
488 views

positive Harris recurrent, aperiodic, stationary Markov chain

How to proof that every positive Harris recurrent, aperiodic, stationary Markov chain is alpha-mixing (strong-mixing)?
matematyk's user avatar
0 votes
1 answer
531 views

Ergodic and mixing processes [closed]

I am working with an article, where it says: "that the discrete time stationary sequence $\{Y_j\}_{j\in Z}$ is mixing and hence ergodic." where $Y_t$ is defined as $Y_t = \int_{-\infty}^{t} h_k(...
Mathy's user avatar
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7 votes
1 answer
772 views

Who introduced the concept of topological mixing?

I am writing an introduction and I want to know who introduced the concept of topological mixing?
user39115's user avatar
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2 votes
1 answer
827 views

Mixing time of a continuous time Markov chain with arbitrary rate matrix

I would like to calculate the mixing time of a continuous time starting from the rate matrix and not necessarily assuming that the time in between jumps have rate 1 - all I have is the (finite ...
Danny W.'s user avatar
  • 229
4 votes
0 answers
145 views

Mixing time for dimers on the square-octagon graph

Consider the "fortress graph" of order $n$ (see Figure 9 of http://faculty.uml.edu/jpropp/tiling/www/mdblum/arctic.html). It's been known empirically for twenty years that if one turns the set of ...
James Propp's user avatar
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7 votes
2 answers
1k views

What good is (strong) mixing in dynamical systems?

For measure-preserving dynamical systems, there exist several notions of mixing. The most basic ones are strong mixing, weak mixing and ergodicity (see the wikipedia page, for instance), asserting ...
Henry Wegener's user avatar
6 votes
2 answers
614 views

Mixing property of first return map

Let $(X,\mathcal{X},\mu)$ be a probability measure system, $T:X\to X$ be a $\mu$-preserving isomorphism on $X$. Let $A\in \mathcal{X}$ such that $\mu(\bigcup_{n\ge 0}T^nA)=1$, and $\mu_A$ be the ...
Pengfei's user avatar
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