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Questions tagged [random-walk]

For questions on random walks, a mathematical formalization of a path that consists of a succession of random steps.

A random walk is a type of stochastic process with random increments, and it is usually indexed by a continuous time variable or an equally spaced discrete time variable.

An elementary example of a random walk is the random walk on $\mathbb{N}_0$, which starts at $0$ and at each step moves $+1$ or $−1$ with equal probability. The path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the financial status of a gambler can all be approximated by random walk models, even though they may not be truly random in reality.

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Proving that $1$- and $2D$ simple symmetric random walks return to the origin with probability $1$

How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the origin with probability $1$? Edit: note that while…
Isaac
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Identity for simple 1D random walk

The question is to find a purely probabilistic proof of the following identity, valid for every integer $n\geqslant1$, where $(S_n)_{n\geqslant0}$ denotes a standard simple random walk: $$ E[(S_n)^2;S_{2n}=0]=\frac{n}2\,P[S_{2n-2}=0]. $$ Standard…
Did
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±1-random walk from 5 until 20 or broke

You play a game where a fair coin is flipped. You win 1 if it shows heads and lose 1 if it shows tails. You start with 5 and decide to play until you either have 20 or go broke. What is the probability that you will go broke?
koon93
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Random walk on $n$-cycle

For a graph $G$, let $W$ be the (random) vertex occupied at the first time the random walk has visited every vertex. That is, $W$ is the last new vertex to be visited by the random walk. Prove the following remarkable fact: For the random walk on…
benny
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Probability of completing a self-avoiding chessboard tour

Someone asked a question about self-avoiding random walks, and it made me think of the following: Consider a piece that starts at a corner of an ordinary $8 \times 8$ chessboard. At each turn, it moves one step, either up, down, left, or right,…
Brian Tung
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Expected Value of Random Walk

Can someone very simply explain to me how to compute the expected distance from the origin for a random walk in $1D, 2D$, and $3D$? I've seen several sources online stating that the expected distance is just $\sqrt{N}$ where $N$ is the number of…
Diego
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A random walk on a finite square with prime numbers

This question is following two similar questions that you can find here and here. The idea is to walk on a square of length $n\times n$, following some rules. We will identify the opposite sides. Formally, the square with the opposite sides…
E. Joseph
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Random Walk Without Repetitions

Suppose that we simulated a random walk on $\mathbb Z$ starting at $0$. At each step, we transition from position $x$ to position $x-3,\,x-2,\,x-1,\,x+1,\,x+2,$ or $x+3$ with equal probability. If we ever move to a position we have been at before,…
Milo Brandt
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What is the probability that a random walk on $\mathbb{Z}^2$ will hit $(1,0)$ before $(2,0)$?

Suppose we have a 2-dimensional simple random walk: we start at $(0,0)$, and at every step, we add a random unit vector in one of the four cardinal directions selected independently and uniformly. It is well-known that this procedure will with…
RavenclawPrefect
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Average swaps needed for a random bubble sort algorithm

Suppose we have $n$ elements in a random permutation (each permutation has equal probability initially). While the elements are not fully sorted, we swap two adjacent elements at random (e.g. the permutation $(1, 3, 2)$ can go to $(1, 2, 3)$ or $(3,…
Kyan Cheung
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A prime number random walk

This question came to my mind thanks to this question which I found really interesting (and beautiful! Like the mathematician Philippe Caldero said in his book Histoires Hédonistes de Groupes et de Géométries (roughly translated) "Let us stop for a…
E. Joseph
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Random walk: police catching the thief

This is a problem about the meeting time of several independent random walks on the lattice $\mathbb{Z}^1$: Suppose there is a thief at the origin 0 and $N$ policemen at the point 2. The thief and the policemen began their random walks independently…
zemora
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2D Random Walk Hitting Time

Suppose there is a grid $[1,N]^2$. A person standing at some initial point $(x_0,y_0)$ walk randomly within the grid. At each location, he/she walks to a neighboring location with equal probability (e.g., for an interior point, the probability is…
Hang Wu
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Take a random walk on Pascal's triangle, without revisits: Does the final number have infinite expectation?

Let's take a random walk on Pascal's triangle, starting at the top. Each number is in a regular hexagon. At each step, we can move to any adjacent hexagon with equal probability, but we cannot revisit a hexagon. The walk ends when we cannot…
Dan
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Probability on entering direction of a simple random walk

Let $X(n)$ be a simple random walk on $\Bbb{Z}^2$. Also we define $S_{R} = \inf\{n > 0 : X(n) \notin [-R, R]^2 \} $ : the exit time of the square $[-R, R]^2$, $T_{v} = \inf\{n > 0 : X(n) = v\}$ : the hitting time of the lattice point $v \in…
Sangchul Lee
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