All Questions
16 questions
8
votes
1
answer
534
views
The cars problem, again
Consider the following simple problem: We are given $2n$ parking spots, labelled from 0 to $2n-1$. There are $n$ cars on the first $n$ spots, and the remaining $n$ spots are free. At every step, every ...
3
votes
0
answers
54
views
Multi-type Galton-Watson-like process where only majority-type is allowed to reproduce
Are you aware of any research papers that have explored a multi-type Galton-Watson process in which only particles of the majority type are permitted to reproduce in each generation?
I've been unable ...
- 56
6
votes
1
answer
361
views
Random walks on infinite directed regular graphs
Let us consider a directed graph $\Gamma=(V,E,s,t)$ ($V$ set of vertices, $E$ set of edges, $s,t: E \rightarrow V$ are the "source" and "target" maps).
Assume that $\Gamma$ is bi-regular, that is ...
- 26k
4
votes
2
answers
261
views
Probability question about random shuffling of piles of rocks
I have $k$ piles of rocks placed on a circle so that every pile has exactly two neighboring piles. We know that initially the piles have $x_1,\dots,x_k$ rocks in each respectively. A monkey plays the ...
- 41
3
votes
0
answers
115
views
Approximating the *conditional* probability of 1D discrete random walk not having revisited the origin given last position
I'm looking for a good closed form approximation to the following conditional probability, with provable approximation guarantees.
Consider a 1D random walk on the integers, starting at the origin, ...
- 101
12
votes
3
answers
1k
views
How to sample a uniform random polyomino?
A polyomino is formed by joining finitely many unit squares edge to edge. It may be regarded as a finite subset of the regular square tiling with a connected interior. In particular, for us, ...
- 7,857
0
votes
1
answer
414
views
Exact formula for computing n-step transition probability of random walks with self-transitions
Consider a semi-infinite random walks $X_n$, $n=0,1,2,\ldots$, whose state space is a set of consecutive integers and whose one-step transition probabilities are $P_{ij}=\mathrm{Pr}\{X_{n+1}=j|X_n=i\}$...
- 1
8
votes
1
answer
174
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 ...
- 183
1
vote
0
answers
46
views
Is there an effective algorithm for finding "minimal discovery times" for large graphs?
Consider a large, probably sparse graph with Markovian random walkers on it.
Define the discovery time as the expected time to first reach a vertex by
random walk from a uniform start. Are there ...
- 355
19
votes
5
answers
18k
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 ...
- 191
5
votes
2
answers
516
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 \...
- 696
14
votes
3
answers
9k
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 ...
- 173
4
votes
1
answer
587
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.
...
- 406
4
votes
5
answers
492
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 ...
- 5,191
18
votes
4
answers
3k
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. ...
- 85.6k
19
votes
9
answers
3k
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 ...
- 14k