Skip to main content
Mathematics

Return to Question

Rollback to Revision 1
Source Link
Steven Stadnicki
Steven Stadnicki
  • 52.4k
  • 9
  • 87
  • 151

Walking Random walks in a square$\mathbb{Z}^2$

Consider a random walk on the integer lattice in the plane. If a “particle” making a random walk arrives at a lattice point $p = (k_1,k_2)$ at the time $t$, then one of the four neighbors $(k_1±1, k_2 )$, $(k_1 , k_2 ± 1)$ of p is selected with equal probability $\frac{1}{4}$ . The particle moves to that neighbor at time $t + 1$. Let $D$ be a region in the plane (a square or a half plane for example), and let $B$ denote its boundary. Let $p$ be a point of $D$, and let $b$ be a boundary point. We’ll denote by $P_p(b)$ the probability that a random walk starting at $p$ exits at $b$, i.e., that $b$ is the first boundary point that is reached.

I was wondering if someone could help me answer some questions if the region in the plane that we are considering is a rectangle.

  1. What is the probability that a particle starting at $p$ never reaches the boundary?
  2. What is the “exit time”, the expected time for a particle starting at $p$ to reach the boundary?
  3. How does the exit time depend on $p$?

Walking in a square

I was wondering if someone could help me answer some questions if the region in the plane that we are considering is a rectangle.

Random walks in $\mathbb{Z}^2$

Consider a random walk on the integer lattice in the plane. If a “particle” making a random walk arrives at a lattice point $p = (k_1,k_2)$ at the time $t$, then one of the four neighbors $(k_1±1, k_2 )$, $(k_1 , k_2 ± 1)$ of p is selected with equal probability $\frac{1}{4}$ . The particle moves to that neighbor at time $t + 1$. Let $D$ be a region in the plane (a square or a half plane for example), and let $B$ denote its boundary. Let $p$ be a point of $D$, and let $b$ be a boundary point. We’ll denote by $P_p(b)$ the probability that a random walk starting at $p$ exits at $b$, i.e., that $b$ is the first boundary point that is reached.

I was wondering if someone could help me answer some questions if the region in the plane that we are considering is a rectangle.

  1. What is the probability that a particle starting at $p$ never reaches the boundary?
  2. What is the “exit time”, the expected time for a particle starting at $p$ to reach the boundary?
  3. How does the exit time depend on $p$?
deleted 2 characters in body; edited title
Source Link
amir
amir
  • 1.3k
  • 2
  • 15
  • 40

Random walks Walking in $\mathbb{Z}^2$a square

Consider a random walk on the integer lattice in the plane. If a “particle” making a random walk arrives at a lattice point $p = (k_1,k_2)$ at the time $t$, then one of the four neighbors $(k_1±1, k_2 )$, $(k_1 , k_2 ± 1)$ of p is selected with equal probability $\frac{1}{4}$ . The particle moves to that neighbor at time $t + 1$. Let $D$ be a region in the plane (a square or a half plane for example), and let $B$ denote its boundary. Let $p$ be a point of $D$, and let $b$ be a boundary point. We’ll denote by $P_p(b)$ the probability that a random walk starting at $p$ exits at $b$, i.e., that $b$ is the first boundary point that is reached.

I was wondering if someone could help me answer some questions if the region in the plane that we are considering is a rectangle.

  1. What is the probability that a particle starting at $p$ never reaches the boundary?
  2. What is the “exit time”, the expected time for a particle starting at $p$ to reach the boundary?
  3. How does the exit time depend on $p$?

Random walks in $\mathbb{Z}^2$

Consider a random walk on the integer lattice in the plane. If a “particle” making a random walk arrives at a lattice point $p = (k_1,k_2)$ at the time $t$, then one of the four neighbors $(k_1±1, k_2 )$, $(k_1 , k_2 ± 1)$ of p is selected with equal probability $\frac{1}{4}$ . The particle moves to that neighbor at time $t + 1$. Let $D$ be a region in the plane (a square or a half plane for example), and let $B$ denote its boundary. Let $p$ be a point of $D$, and let $b$ be a boundary point. We’ll denote by $P_p(b)$ the probability that a random walk starting at $p$ exits at $b$, i.e., that $b$ is the first boundary point that is reached.

I was wondering if someone could help me answer some questions if the region in the plane that we are considering is a rectangle.

  1. What is the probability that a particle starting at $p$ never reaches the boundary?
  2. What is the “exit time”, the expected time for a particle starting at $p$ to reach the boundary?
  3. How does the exit time depend on $p$?

Walking in a square

I was wondering if someone could help me answer some questions if the region in the plane that we are considering is a rectangle.

Notice removed Draw attention by amir
Bounty Ended with san's answer chosen by amir
Tweeted twitter.com/StackMath/status/701025264059420673
Notice added Draw attention by amir
Bounty Started worth 200 reputation by amir
Source Link
amir
amir
  • 1.3k
  • 2
  • 15
  • 40

Random walks in $\mathbb{Z}^2$

Consider a random walk on the integer lattice in the plane. If a “particle” making a random walk arrives at a lattice point $p = (k_1,k_2)$ at the time $t$, then one of the four neighbors $(k_1±1, k_2 )$, $(k_1 , k_2 ± 1)$ of p is selected with equal probability $\frac{1}{4}$ . The particle moves to that neighbor at time $t + 1$. Let $D$ be a region in the plane (a square or a half plane for example), and let $B$ denote its boundary. Let $p$ be a point of $D$, and let $b$ be a boundary point. We’ll denote by $P_p(b)$ the probability that a random walk starting at $p$ exits at $b$, i.e., that $b$ is the first boundary point that is reached.

I was wondering if someone could help me answer some questions if the region in the plane that we are considering is a rectangle.

  1. What is the probability that a particle starting at $p$ never reaches the boundary?
  2. What is the “exit time”, the expected time for a particle starting at $p$ to reach the boundary?
  3. How does the exit time depend on $p$?