The tag has no wiki summary.

learn more… | top users | synonyms

11
votes
1answer
551 views

Perimeters of random-walk polygons

I have a random walk on $\mathbb{Z}^2$ that takes a step with equal probability in the three directions that avoid retracing the previous step. The walk proceeds until it returns to a lattice point ...
6
votes
2answers
527 views

Optimally directing switches for a random walk

If you are sometimes called upon directing a random walk in a directed graph, how should you direct it so as to maximize the probability it goes where you want? Formal statement More specifically, ...
7
votes
2answers
504 views

Random walks on graphs: Cover time and blanket time

Winkler and Zuckerman conjectured that the blanket time is within a constant factor of the cover time. The conjecture was recently proved. The cover time $C$ is the expectation of the first time $t$ ...
0
votes
2answers
1k views

Two dimensional brownian motion first passage time

Hello, I am looking for information on how to solve/compute first passage time for two dimensional Brownian motion. any papers, references, books or web links for study will be helpful. thanks ...
4
votes
0answers
398 views

Monotonic properties of harmonic functions on graphs

I have a question concerning monotonic properties of "generalized harmonic functions" on graphs. I am a physicist and I didn't take any separate courses in neither graph theory nor discrete harmonic ...
15
votes
4answers
621 views

Sweep-segment bot: Will this random walk sweep the plane?

This model is inspired by the random behavior of the Roomba sweeping robot. Let a unit segment $ab$ in the plane be placed initially with $a=(0,0)$ and $b=(1,0)$. The segment is first rotated a ...
9
votes
5answers
6k 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
1answer
842 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
2answers
835 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} ...
7
votes
3answers
989 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 ...
22
votes
1answer
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
3answers
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
2answers
707 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 ...