# Questions tagged [mixing]

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19
questions

**1**

vote

**1**answer

113 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 ...

**3**

votes

**1**answer

65 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 (...

**0**

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**1**answer

122 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 ...

**1**

vote

**0**answers

39 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{...

**1**

vote

**1**answer

137 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 ...

**1**

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**0**answers

35 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} $...

**2**

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**0**answers

127 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 }...

**0**

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**0**answers

77 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 ...

**5**

votes

**1**answer

153 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 \...

**5**

votes

**1**answer

160 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 ...

**2**

votes

**0**answers

129 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 ...

**0**

votes

**1**answer

861 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 ...

**3**

votes

**1**answer

341 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)?

**0**

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**1**answer

320 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(...

**7**

votes

**1**answer

658 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?

**1**

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**1**answer

460 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 ...

**4**

votes

**0**answers

132 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 ...

**6**

votes

**2**answers

845 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 ...

**5**

votes

**2**answers

410 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 ...