# Tagged Questions

The markov-chains tag has no usage guidance.

**65**

votes

**3**answers

3k views

### Gromov's list of 7 constructions in differential topology

At the 2010 Clay Research Conference, Gromov explained that we know of only 7 different methods for constructing smooth manifolds. Working from memory, and hence not necessarily respecting the order ...

**36**

votes

**7**answers

4k views

### Deep Learning / Deep neural nets for mathematician

I am interested in finding out the math ideas behind the technologies that are under the umbrella of "Deep Learning" or "Deep neural nets".
Most of the papers/books that are often quoted in ...

**16**

votes

**9**answers

2k views

### How can I generate random permutations of [n] with k cycles, where k is much larger than log n?

I've been thinking a lot lately about random permutations. It's well-known that the mean and variance of the number of cycles of a permutation chosen uniformly at random from Sn are both ...

**15**

votes

**5**answers

3k views

### Where to publish a paper on the Mafia game?

I wrote a research paper "A mathematical model of the Mafia game" (arXiv:1009.1031 [math.PR]). However, I do not know where to publish it. As an undergraduate studying majorly physics, I have little ...

**14**

votes

**4**answers

965 views

### Eigenvectors of a particular transition matrix

I am considering a Markov chain with $n$ states with a particularly nice structure. The transition matrix is as follows:
\begin{equation}\mathbf{P}=\begin{pmatrix}
0 & 0& \dots&0 & 0 ...

**14**

votes

**1**answer

647 views

### Convexity of spectral radius of Markov operators, Random walks on non-amenable groups

Let $P_1,P_2$ denote stochastic transition matrices on a countable set $I$.
Consider $P_1,P_2$ as operators on $\ell^2(I)$ given by multiplication.
Question
Under which conditions can we show that ...

**13**

votes

**3**answers

1k views

### Markov chain on Groups

Let $G$ be a permutation group on the finite set $\Omega$. Consider the Markov chain where you start with an element $\alpha \in \Omega$ chosen from some arbitrary starting probability distribution. ...

**13**

votes

**2**answers

1k views

### Random walk is to diffusion as self-avoiding random walk is to …?

One can view a random walk as a discrete process whose continuous
analog is diffusion.
For example, discretizing the heat diffusion equation
(in both time and space) leads to random walks.
Is there a ...

**13**

votes

**0**answers

722 views

### representation theoretic interpretation of Jack polynomials

Monomial symmetric polynomials on $n$ variables $x_1, \ldots x_n$ form a natural basis of the space $\mathcal{S}_n$ of symmetric polynomials on $n$ variables and are defined by additive symmetrization ...

**13**

votes

**0**answers

725 views

### Diagonalizing some matrices arising from Fourier transform on $S_n$.

Consider the function $f$ on $S_n$ which equals $1/n$ on all adjacent transpositions $(i,i+1)$, where we let $n+1 = 1$, and $0$ otherwise, and its Fourier transform $\hat{f}(\rho)$ evaluated at the ...

**12**

votes

**3**answers

7k views

### Solving a Rubik's cube via a series of randomly selected (quarter-turn) Singmaster moves

In July of 2010, Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge demonstrated (computationally) that a $3\times3\times3$ Rubik's cube, starting in an arbitrary configuration, can ...

**12**

votes

**1**answer

511 views

### Random walk origin return monotinicity

Consider a Markov chain on $\mathbb{Z}^d$ with transition kernel $P$ for adjacent vertices (non-diagonal). Essentially this is a $d$ dimensional random walk with the probability of a transition ...

**12**

votes

**0**answers

202 views

### Ising model - phase transition vs rapid mixing

Consider a graph $G=(V,E)$ and Ising model on that graph, i.e. configuration space is $\Omega=${$-1,+1$}$^V$ and energy of a configuration $s \in \Omega$ is given by:
$H(s) = -\beta \sum_{u \sim ...

**11**

votes

**4**answers

2k views

### What is the cover time of a random walk on a cube?

I can't quite figure this problem yet. There is an ant at one vertex of a cube. The ant goes from one vertex to another by choosing one of the neighboring vertices uniformly at random. What is the ...

**10**

votes

**1**answer

871 views

### Different uses of the word “ergodic”

There appear to be two definitions of the word ergodic.
The dynamical systems definition says that a measure space $(X,\mathit B, \mu)$ and measure preserving transformation $T: X \mapsto X$ is ...

**10**

votes

**2**answers

1k views

### Markov chains: invariant measures and explosion

The following seems like such an elementary question, but I didn't get anywhere with it.
Suppose you are considering a Markov chain in continuous time which is transient and has an invariant measure ...

**9**

votes

**2**answers

1k views

### Is this a situation where triple mutual information is always non-negative?

Suppose I have three identically-distributed homogeneous continuous-time discrete state space Markov chains $X_1(t), X_2(t), X_3(t)$, $t\geq 0$. They evolve independently but share a common random ...

**8**

votes

**1**answer

578 views

### Bounds on $||P^{k+1} - P^k||$ for $n$ by $n$ stochastic matrix $P$ with trace $n-1$ and integer $k>>n$.

The problem:
We have a $n$-state Markov chain with arbitrary initial distribution and transition matrix $P$ that is arbitrary except that we know that $P$ has trace $n-1$. Of course $P$ is also a ...

**8**

votes

**1**answer

154 views

### Equalizing Geometric means of Graph Cycles

Consider a strongly connected directed graph $G$. I have been stuck on the following question: can you assign real numbers in $[0,1]$ to each edge of $G$ so that the geometric mean of all cycles are ...

**8**

votes

**1**answer

308 views

### Ising model on a cycle

The Ising model on $\mathbb{Z} / 2d\mathbb{Z}$ gives to the configuration $x=(x_0, \ldots, x_{2d-1}) \in \{-1,+1\}^{2d}$ a probability proportional to $\exp\\big(\beta \sum_i x_ix_{i+1} \\big)$. The ...

**8**

votes

**1**answer

390 views

### Potts model simulation

I was wondering what were the state-of-the-art methods to simulate low temperature configurations of Potts-like models that exhibit a discontinuous phase transition. For models with a continuous phase ...

**8**

votes

**0**answers

112 views

### Functions between Markov chains that preserve local harmonicity

Given two Markov chains with respective state-spaces $S$ and $T$, say that a function $\phi$ from $S$ to $T$ is holomorphic iff for all states $t \in T$, every real-valued function $f$ on $T$ that is ...

**7**

votes

**1**answer

250 views

### Existence of Limiting Distribution for Moving Regions in Stat. Phys. Models

As the title (hopefully) suggests, I've been trying to prove (or disprove) the existence of a limiting distribution for a certain projection in a statistical physics model. I'll give the details of ...

**7**

votes

**2**answers

346 views

### Convergence rate of the convolution of almost uniform measures on $\mathbb{Z}_p$

Statement Given a finite abelian group $G$ and two independent random variables $X,Y$ taking values in $G$ and satisfying $d_{TV}(X,U_G)\leqslant \delta$ and $d_{TV}(Y,U_G)\leqslant \delta$ (where ...

**7**

votes

**1**answer

233 views

### First Collision Time for k Random Walkers on a Torus

I consider $k$ random walkers on $\mathbb{Z}^{d}/n \mathbb{Z}^{d}$, the $d$-dimensional torus of side length $n$. More precisely, I will define a Markov chain $Z_{t} = (X_{t}[1], \ldots, X_{t}[k])$ ...

**7**

votes

**2**answers

306 views

### Examples of Slowly Mixing Chains in Statistics

This should probably be community wiki, but I don't know how to set that myself.
I'm looking for examples or Markov chains that are used in statistics or statistical physics, and which are known to ...

**6**

votes

**2**answers

873 views

### Expectation of first positive value in random walk

Let $p$ be a parameter in $]0,1[$. Let $(X_k)_{k\geq 0}$ be an independent, identically distributed sequence of random variables, such that each $X_k$ takes values only in
$\lbrace -1, \frac{1-p}{p} ...

**6**

votes

**2**answers

593 views

### Can ergodic theory help to prove ergodicity of general Markov chain?

I am a beginner in ergodic theory. I have read some lecture notes(such as this and this) about it in hope that I could find something which helps to prove the ergodicity of some Markov chain taking ...

**6**

votes

**1**answer

188 views

### A family of skew-symmetric matrices corresponding to cycles in graphs

When investigating loops in Markov chains I ran into the following observation.
A cycle in a graph $G$ with $n$ vertices may be represented by a matrix $\Gamma \in \mathbb R^{n \times n}$ having the ...

**6**

votes

**2**answers

570 views

### References for a physicist migrating to stochastic processes

I've studied "Markov Chains" - Norris and "Measure, Integral and Probability" - Capinski, Kopp. Now, I'm looking for a couple of books (or other references) that help me bridging these two topics. ...

**6**

votes

**1**answer

319 views

### Quasi-stationary distribution for a death process

In the paper, Survival in a quasi-death process by van Doorn and Pollett, the quasi-stationary distribution of a transient CTMC is discussed and QSD for a simple death process is derived.
Consider a ...

**6**

votes

**2**answers

297 views

### Deterministic finite-state automaton driven by a Markov chain

I've stumbled on some problem, and I have the feeling that this is closed to something well-studied in dynamical systems. The problem is the following. Consider a finite-state automaton with state ...

**6**

votes

**1**answer

185 views

### Finding cohesive (low exit probability) sets in a Markov process

The following is a fact about Markov chains that came up in a game theory paper. The purpose of this question is to ask if related notions or similar results are found elsewhere in probability, or are ...

**5**

votes

**3**answers

5k views

### Time-inhomogeneous Markov Chains

I'm trying to find out what is known about time-inhomogeneous ergodic Markov Chains where the transition matrix can vary over time. All textbooks and lecture notes I could find initially introduce ...

**5**

votes

**1**answer

410 views

### Elementary Markov Chain Question

Are any general conditions known on a finite transition nxn matrix that ensure that there exists at least one mth root which is also a transition matrix? It is easy to construct a 3x3 , diagonally ...

**5**

votes

**1**answer

421 views

### Probability of a set of random vectors over finite field being a spanning set

Suppose I have a set of random vectors $f(a_1, \ldots, a_\ell) := (v_1, \ldots, v_m) \subset F_p^n$, $m \ge n$, given by a matrix valued polynomial function $f$, where the $a_i$'s are independent, ...

**5**

votes

**1**answer

314 views

### Does every (generalized?) Markov chain admit transition probabilities?

To pose the question let us start by recalling the following notions:
Transition Probabilities. A transition probability matrix between two measurable spaces $(S,\mathcal{S})$ and ...

**5**

votes

**2**answers

669 views

### Is there a way to analytically compute the recurrence time of a finite Markov process?

Let $X_t$ be an ergodic (time-homogeneous) Markov process (in discrete or continuous time) on a finite state space $\{1,\dots,n\}$. Let $T(X_0)$ be the stopping time given by the infimum of times such ...

**5**

votes

**1**answer

135 views

### Relative vulnerabilities in SIS epidemic model

Consider the SIS model of epidemic spreading. There is a finite graph $G(V,E)$, link infection rates $\lambda_{ij}$ and node recovery rates $\mu_i$. There are a few initial nodes which are infected at ...

**5**

votes

**1**answer

992 views

### Efficient computation of Markov chain transition probability matrix

Consider a continuous Markov chain $X = (X_t)$ on a finite state space and let $Q$ be the (given) transition rate matrix. This matrix is very sparse, with non-zero values on 3 diagonals only (so from ...

**5**

votes

**0**answers

187 views

### Maximal inequalities for square of partial sums

Let $S_n = \sum_{i \leq n} X_i$ be the partial sums of a nice sequence of random variables $X_i$. In my application, $X_i$ is a functional of a finite-state, irreducible, aperiodic Markov chain, so ...

**4**

votes

**5**answers

431 views

### Some questions concerning a random number process

Consider the following Markov process: Start with an integer $N = N_0$. Now repeatedly choose an $N_i$ uniformly at random in the range $[1...N_{i-1}]$ until $N_i = 1$ at which point one terminates ...

**4**

votes

**1**answer

328 views

### Combinatorial descriptions of the stationary distribution of a Markov chain

When I say "Markov chain" I think of a directed positively weighted (finite) graph, such that the sum of all edges going out of a vertex equals 1. Also I assume that it is aperiodic and irreducible.
...

**4**

votes

**1**answer

732 views

### A simple problem in markov chains

I'm trying to understand a 1954 paper of Kubo intitled "Note on the stochastic theory of resonance absorption". The specific problem can be stated mathematically as follows: let $X(t)$ be a random ...

**4**

votes

**1**answer

384 views

### Uniqueness of Gibbs Measure on Ising model

If I understood this correctly, the Gibbs Specification for the Ising model on $ℤ^d$ dos not have a unique Gibbs Measure for β above the critical level. But what about the Ising model on a finite ...

**4**

votes

**3**answers

355 views

### Why does the overhand shuffle converge to the uniform distribution on $S_n$?

Pemantle 1989 proves, among other things, that the Markov chain on $S_n$ induced by repeatedly and independently performing an overhand shuffle on a deck of $n$ cards is ergodic and has limiting ...

**4**

votes

**3**answers

987 views

### How to explain “Feller process” to an undergraduate student?

I had to explain in informal terms what a Feller process was, to undergraduate students who understand Markov property, Poisson processes and such. It was easy to define Levy process as generalisation ...

**4**

votes

**3**answers

365 views

### Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation
\begin{equation}
dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0,
\end{equation}
where ...

**4**

votes

**2**answers

250 views

### Anticoncentration of the convolution of two characteristic functions

Edit: This is a question related to my other post, stated in a much more concrete way I think.
I am interested in anything (ideas, references) related to the following problem:
Suppose that $A ...

**4**

votes

**2**answers

214 views

### Regarding Ricci curvature of Markov chains

In Ricci curvature of Markov chains on metric spaces Yann Ollivier, defines a coarse Ricci curvature for a Markov chain with transition kernels $\{m_x\}$ defined on a metric space $(X,d)$ as follows: ...