The random-walk tag has no usage guidance.

**8**

votes

**1**answer

357 views

### Twisted random walks

Suppose the points of two random walks in $\mathbb{R}^2$ are given the
step number (or time) as a third coordinate, so that they become paths in $\mathbb{R}^3$.
Here are several pairs of walks of $n=...

**21**

votes

**5**answers

3k views

### A random walk with uniformly distributed steps

The following problem has bothered me for a long time.
Let us imagine a point on the real axis. At the beginning, it is located at point $O$. Then it will "walk" on the real axis randomly in the ...

**12**

votes

**3**answers

495 views

### An “inchworm-like” random walk on an integer interval

Imagine I place $k$ stones on an infinite one-dimensional integer interval $Z$ s.t. no stone is more than some distance $d$ from any other stone. For example, if $d=1$ and $k = 5$, we might place the ...

**20**

votes

**0**answers

603 views

### Time for Langton's ant to cover a “square” torus

Langton's ant is a cellular automaton running as follows:
Squares on a plane are colored variously either black or white. We
arbitrarily identify one square as the "ant". The ant can travel ...

**14**

votes

**1**answer

875 views

### In how many steps a random walk visits all the elements of a finite group, with a probability 1/2?

This question is a variation of the return to the origin problem.
Let $G$ be the finite group $\mathbb{Z}/n \times \mathbb{Z}/n$ and let the random transformation $T: G \to G$ such that $T(a,b) = (...

**17**

votes

**1**answer

512 views

### Longest of random worm-like paths in $\mathbb{Z}^2$

Imagine at each lattice point of $\mathbb{Z}^2$ within $[1,3n]^2$,
with coordinates
$\equiv 2 \bmod 3$,
we place, with equal probability, one of these six patterns:
The result ...

**7**

votes

**1**answer

235 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 $Z_{t} = (X_{t}[1], \ldots, X_{t}[k])$ ...

**5**

votes

**1**answer

504 views

### Stopping time of two dimensional random walk

Let $U_n=\sum_{i=1}^n X_i,V_n=\sum_{i=1}^n Y_i$, $n\geq 1$, be a two-dimensional random walk with i.i.d. increments $(X_n, Y_n)$, where $X_n, Y_n$ are discrete random variables with joint pmf $P_{X,Y}$...

**5**

votes

**1**answer

326 views

### Random walk with positive uniformly distributed steps

Let $U_1,U_2,\ldots$ be iid random variables distributed uniformly on $[0,1]$. I am interested in the random walk $X_i = \sum_{j \leq i} U_j$. In particular,
What is the expected number of points ...

**3**

votes

**1**answer

320 views

### Where does directed random walk hit the boundary of a region?

I have a problem that I more or less know the answer to (in an ad hoc way), but would really like to see it done in a systematic way. In spite of this, I will pose the question in quite a concrete way....

**2**

votes

**1**answer

422 views

### Probability that a “closable” self-avoiding random walk forms a polygon

Consider a self-avoiding random walk on an infinite graph (for concreteness, the grid of 2-dimensional lattice points $\mathbb{Z}^2$), in which on each step, the next position is chosen uniformly at ...

**1**

vote

**1**answer

135 views

### Scaling of First-passage times for Random Walk on integer lattices

Consider simple symmetric random walk $S_{n} = (S_{n}^{(1)},\dots,
S_{n}^{(d)})$ on the d-dimensional integer lattice with starting point the origin.
Let $\tau_{N}$ be the first time $S_{n}$ exits ...

**1**

vote

**1**answer

143 views

### Large deviations for maximizer of random walk with drift

Is it easy to write down the large deviations rate for the maximizer of a random walk with negative drift?
Let $X_i$ be the (iid, mean $-\mu$, variance $\sigma$, arbitrarily nice tails) jumps of a ...

**1**

vote

**1**answer

181 views

### Arc Sine law for Random Walk conditioned to non-absorption or not?

Let $S_n$ be simple symmetric Random walk on the integers in $[-N,N]$ with states $N$ and $-N$ absorbing. Let $\tau$ be the time to absorption when $S_0 = 0$.
Is the $E(S^{2}_{n}| \tau \geq n)$ known?...