2
votes
0answers
37 views

A random walk on the unit distance graph in $\mathbb{R}^n$

Define a graph $G_n$ whose vertices are the points in $\mathbb{R}^n$ with an edge connecting any two points that are one unit apart. Such a graph is called the unit distance graph in $\mathbb{R}^n$. ...
0
votes
2answers
56 views

How Do You Calculate Probabilities of Random Events Occuring in Sequence?

So I have a series: $f(x_{n+1})=x_n \pm t$ and $f(x_0)=W$ What I'd like to calculate is the probability in terms of $t$ and $W$ (assuming they're any constant $W>t$) that any $f(x_q)=0$ for all ...
5
votes
2answers
101 views

How can we directly see that the number of random walks starting and ending at the origin is ${n\choose n/2}^2$?

In an infinite two-dimensional square-shaped grid, we define four directions, north, south, east, west. We thus have $4^n$ random walks of length $n$. If we end where we started, for every north step ...
1
vote
2answers
189 views

Calculating expected value of random walk with one stop value.

I know that for a random walk with two stop values, the expected value of the number of steps needed is $ab$ where the stop values are $-a$ and $b$ and the initial position is at 0. What about for ...
2
votes
1answer
71 views

What is the probability that the robot steps on the bomb?

Suppose a robot is initially placed at $0$ on the number line, and is programmed to take steps of integer length in the positive direction between $1$ and $k$, inclusive, where $k$ is a positive ...
1
vote
1answer
65 views

Number non self avoiding closed walks surrounding some point

While studying some Peierls-like arguments in statistical physics I thought about the following problem: We have some 2d-integer lattice like this, for simplicity infinite in all directions. Now fix ...
1
vote
0answers
49 views

planar walks and catalan numbers

prove that following numbers are equal: (unordered) pairs of lattice paths with n+1 steps each, starting at (0,0), using steps (0,1) or (1,0), ending at the same point and only intersecting at the ...
12
votes
3answers
272 views

Select a new value from last $N$ values; how long until the last $N$ are all the same?

Say first we have N distinct numbers in a line, like 1,2,3,...,N, in each round, we choose a ...
6
votes
2answers
480 views

A question on calculating probabilities for the random walk

I am currently working on a high school project revolving around the 'Cliff Hanger Problem' taken from ”Fifty Challenging Problems in Probability with Solutions” by Frederick Mosteller. The problem ...
0
votes
2answers
165 views

Question from section 1.5 of Chung's Spectral Graph Theory

I'm (slowly) reading Fan Chung's Spectral Graph Theory. At the moment, I'm in section 1.5 which is about eigenvalues and random walks. There's a small technical point that puzzles me. The context ...
2
votes
1answer
95 views

$1$D bidirectional random walk question

In a $1$D random walk on x axis a particle can turn left with probability $\frac{3}{4}$ and right with probability $\frac{1}{4}$. What is the probability that $|x|\leq 1 $ for $1\leq ...