The random-walk tag has no usage guidance.

**9**

votes

**5**answers

8k views

### When do 3D random walks return to their origin?

The probability of a random walk returning to its origin is 1 in two dimensions (2D) but only 34% in three dimensions: This is Pólya's theorem. I have learned that in 2D the condition of returning to ...

**0**

votes

**1**answer

928 views

### Random walk question on 2D grid, probability of vertical line vs horizontal line hit

Hi,
I have a problem
(1) where I need to compute the ratio of probabilities of hitting and stopping at a positive vertical barrier x vs hitting and stopping at a negative horizontal barrier y ...

**6**

votes

**2**answers

870 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} ...

**8**

votes

**3**answers

1k views

### How to characterize a Self-avoiding path.

I cannot find any answer to that apparently simple problem :
On a square lattice, a path is given by a sequence of relative moves in {"move forward", "turn right" and "turn left"}.
Is there a rule ...

**24**

votes

**1**answer

1k views

### Random walk inside a random walk inside…

Let $G=(V,E)$ be a graph and consider a random walk on it. Let $G'=(V',E')$ be a subgraph consisting of the vertices and edges that are visited by the random walk.
Question 0: Is there a standard ...

**10**

votes

**3**answers

1k views

### A random walk on random lines

I am wondering if this random walk remains finite with positive probability.
Start with three lines $A,B,C$ that are extensions of an equilateral triangle.
Let $p_0$ be one corner. Generate a line ...

**14**

votes

**2**answers

758 views

### self-avoidance time of random walk

How many steps on average does a simple random walk in the plane take before it visits a vertex it's visited before?
If an exact formula does not exist (as seems likely), then I'm interested in good ...