The random-walk tag has no usage guidance.

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**1**answer

279 views

### Limit of a rescaled random sum of i.i.d. random variables

Consider a sequence of i.i.d. random variables $(X_i)_{i \in \mathbb N}$ and let $S_n=X_1+\dots+X_n$
For every $\alpha \in ]0,+\infty[$, let $N(\alpha)$ be a discrete random variable on $\mathbb N$, ...

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**1**answer

351 views

### Reference Request: Cover time for simple random walk on nxn torus

I'm putting the finishing touches on my masters thesis and need a reference for the following fact (which my advisor told me):
Let $G$ be the $n\times n$ grid and identify the sides to make it a ...

**21**

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**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 ...

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205 views

### $n$-th return of a random walk on $\Bbb Z^d$

Lets define $f_n = P(X_n =0 , X_k \ne 0, k< n)$ the first return distribution of the random walk $X_n$ on $\mathbb{Z}^d$, and lets go ahead and assume that $f_n \approx n^{-(1+\alpha)}$ for some ...

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**2**answers

428 views

### Gibbs sampling step size

I have some data generated using MCMC methods and in particular Gibbs sampling. I computed the autocorrelation but I'm unsure how to determine how many samples to skip.
I'd like to determine that ...

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**0**answers

253 views

### Random Walk in $\mathbb{R}^n$

Have there been papers dealing with random walks in $\mathbb{R}^n$ that are not on the lattice $\mathbb{Z}^n$? Instead of walking in one of the directions possible in $\mathbb{Z}^n$ with probability ...

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**1**answer

322 views

### Mixing time of unitary Brownian motion

Let $B_t$ be the unitary Brownian motion, i.e. Brownian motion on the unitary group $U(N)$. What is known about the mixing time of $B_t$, that is, how fast does the measure $B_t(\delta_{\{Id\}})$ ...

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**1**answer

1k views

### first passage time, brownian motion

Hi,
If X(t) is Brownian motion in 2D, where X(0) = 0, then we can ask what is the expected time required to first hit a circle of radius R, centered at the origin. This is a First Passage Time ...

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1k views

### Combinatorial proof for the number of lattice paths that return to the axis only at times that are a multiple of 4

Consider lattice paths consisting of $2n$ steps, each of which is either $(1,1)$ or $(1,-1)$. The number of such lattice paths that return to the horizontal axis only at times that are a multiple of ...

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**1**answer

311 views

### Maximum vertical distance for a lattice path when NSEW steps are allowed

Suppose we have a lattice path in 2D starting at the origin, in which north, south, east, and west steps are allowed. For a given path $L$, let $\max(L)$ be the maximum value of the $y$ coordinate ...

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1k views

### Simple Random Walk on a Locally Finite Graph - when is it recurrent?

I'm giving a talk tomorrow about a result in computer science which I recently proved. It's a recurrence-transience result on a random process which is related in spirit to a simple random walk. My ...

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**1**answer

1k views

### Pólya's Random Walk Constants at infinity

Let be the probability that a random walk on a d-D lattice returns to the origin. In 1921, Pólya proved that $p(1)=p(2)=1$
but $p(d)<1$ for $d>2$.
...

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**2**answers

440 views

### Limit shape for fixed-perimeter lattice polygons

Let $P$ be a simple polygon defined by $n$ unit-length segments
connecting lattice points of $\mathbb{Z}^2$.
I have two operations that preserve the perimeter of $P$.
The first is the "pop" of a ...

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**2**answers

882 views

### probability distribution of hitting nodes on a finite graph random walk

Consider a finite, undirected, scale-free graph $\{G}$, with uniform edge weights. We define a truncated random walk on $\{G}$ as a random walk that continues for exactly $\{k}$ steps. For an ...

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379 views

### Complexity of detecting a convex body in $\mathbb{R}^n$?

Let $K_0$ be a bounded convex set in $\mathbb{R}^n$ within which lie two sets $K_1$ and $K_2$. $K_0,K_1,K_2$ have nonempty interior. Assume that,
$K_1\cup K_2=K_0$ and $K_1\cap K_2=\emptyset$.
The ...

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**1**answer

407 views

### Decomposition of Haar measure other than Hurwitz's

Hurwitz defined a decomposition of the Haar measure on $SO(n)$ based on Given's rotation. So by left multiplication of Givens rotation one can always bring an orthogonal matrix into the identity. The ...

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186 views

### criterion for deciding whether the product of a sequence of Givens rotations can reach the full special orthogonal group

By Givens' rotation $R(1,2;\theta)$ I mean a matrix which has the
$$\begin{pmatrix} \hphantom{-}\cos \theta &\sin \theta \cr -\sin \theta & \cos \theta \end{pmatrix}$$
$2 \times 2$ block at ...

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**1**answer

246 views

### Walks that cannot hit the boundary

I've casually proved, as application of some ideas that I am developing, a result that might be of interest in itself. I am completely new in this field and then I would like to ask your help to ...

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879 views

### What is the maximum diameter of $N$ steps of a random walk?

Since probability is quite far away from my daily buisiness, please forgive me if my use of terminology is wrong or the question is too trivial. However, I was not able to find the right keyword to ...

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269 views

### Any approximation algorithms for self-avoiding walks?

I've a graph whose edges are weighted by probabilities, perhaps all equal. I would like to compute the overall probability of traveling between vertices x and y in the graph after I delete each edge ...

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944 views

### Probability of first return to starting vertex in Random walk on regular finite graph

Hi, this is related to this earlier question.
Given Random walk on a regular graph $G=(V,E)$. The Random walk is simple so that transition probabilities are $1/\text{deg}(v_i)$, and time is in ...

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**1**answer

426 views

### Comparing hitting probabilities for two different random walks

Let $p$ be a probability in $]0,1[$, and let $(X^p_i)_{i \geq 1}$ be a i.i.d. family
of variables with law $P(X=1)=p, P(X=-\frac{p}{1-p})=1-p$ (so that $E(X)=0$).
Set $S^p_n=\sum_{k=1}^{n} X^p_k$ for ...

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259 views

### How long does it take a Brownian particle to achieve a uniform probability distribution across a space?

Imagine I have a point-like Brownian particle, with diffusion constant $D$, and I place it at some initial coordinate in a cage of known geometry. Assuming the volume $V$ of the cage is "everywhere" ...

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428 views

### Averaging over random walk on binary lattice

I have a function $f$ defined over a bit vector of length $n$. Equivalently, this is a function defined on the set of integers $[0,\ldots,2^n-1]$. I would like to compute the mean or variance or some ...

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143 views

### Reference request for a result on subsets unlikely to be hit by random walks in a group

Suppose we are performing a random walk in a group. More precisely, we have a finite generating set $S$ of a group $G$ and the probability of walking along generator $s$ is given by $\mu(s)$ for some ...

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**1**answer

709 views

### How far can a particle travel from its origin if it exhibits self-avoiding Brownian motion in two-dimensions?

Let's say I have a point-like Brownian particle undergoing two-dimensional diffusion on an infinite plane with the caveat is that the particle can never return to a coordinate that it previously ...

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533 views

### Compute the expected value of the next step of a sorted random walk

Here's what I'm thinking about. If you have a random walk (move +1 or -1 at each step) of some fixed length, then if you're at the maximum of the walk, the next step you take is -1 with probability 1. ...

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286 views

### Cover time for a biased random walk on an 'N'-dimensional integer lattice

Imagine that I have a random walk on an $N$-dimensional integer lattice, $Z^N$, of finite dimensions, $(d_1, ..., d_N)$, where boundaries are fully reflecting and the walker is initialized at some ...

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591 views

### exactly simulating a random walk from infinity

In diffusion-limited aggregation on the square lattice, one lets a particle do "random walk from infinity" until it hits the current aggregate, at which point the site occupied by the particle is ...

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4k views

### Probability of a Random Walk crossing a straight line

Let $(S_n)_{n=1}^{\infty}$ be a standard random walk with $S_n = \sum_{i=1}^n X_i$ and $\mathbb{P}(X_i = \pm 1) = \frac{1}{2}$. Let $\alpha \in \mathbb{R}$ be some constant. I would like to know the ...

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3k views

### Rolling a random walk on a sphere

A ball rolls down an inclined plane, encountering horizontal obstacles, at which it
rolls left/right with equal probability. There are regularly spaced staggered gaps that let the ball
roll down to ...

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**4**answers

1k views

### Will a random walk on [0, inf) tend to infinity? [closed]

Consider a random walk on [0, inf) where you start at 0. With probability p = 0.5, you increase by 1. With probability (1-p) = 0.5, you decrease by 1, but not below 0.
As time goes to infinity, will ...

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1k views

### Probability of return at step $n$ of a Random walk to its starting vertex

Hi,
given a discrete simple Random walk on a symmetric graph, what is known about the probability of the random walker to return to a starting site at step $n$? Specifically, I am interested in the ...

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2k views

### What is the cover time of a random walk on a cube?

I can't quite figure this problem yet. There is an ant at one vertex of a cube. The ant goes from one vertex to another by choosing one of the neighboring vertices uniformly at random. What is the ...

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556 views

### Second stage of elliptic curve factorization via random walk/Pollard's rho in constant (or low) memory?

The second stage of elliptic curve factorization has the drawback of large memory usage.
Let $n=pq$, $E(\mathbb{Z}/n\mathbb{Z})$ is elliptic curve and $P$ point on $E(\mathbb{Z}/n\mathbb{Z})$.
On ...

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**2**answers

311 views

### Hausdorff dimension of non-recurrent walks

Preface: I am fairly new to the concept of Hausdorff dimension, so I don't know how interesting a question this is.
Identify walks on $\mathbb{Z}$ with infinite binary sequences (say $0$ means moving ...

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**1**answer

667 views

### Probability of return vs. probability of return in minimal number of steps

Consider a random walk on a connected graph $G=(V,E)$. That is, associate to each neighbouring nodes $a,b\in V\ $ transition probabilities $\mathbb{P}(a\rightarrow b), \mathbb{P}(b\rightarrow a) $ ...

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**1**answer

731 views

### Random walk on a simple finite network

Consider a graph $\Delta_N = \lgroup (x,y)\in\mathbb{Z}^2| x+y\leq N-1, x\geq 0,\ y\geq 0 \rgroup$ (set of edges is defined in a natural way): see here ).
Take a random walker that wonders around ...

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**1**answer

263 views

### Winding number bijection on graphs

Let $G=(V,E)$ be an isoradial graph. In other words the graph can be imbedded into the plane such that each face (plaquette) can be circumscribed a circle of radius 1 with the circle's center ...

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**1**answer

763 views

### A random walk on natural numbers

We are taking a random walk on the set of natural numbers. If we are at $M$, then with probability 1/4, we stay at $M$, with probability 5/12 we move to some random number less than or equal to $M/2$, ...

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**1**answer

935 views

### Intersection Probabilities for Random Walk in d>2

I'd like to get asymptotics on the probability that n independent random walks coalesce. Start with n independent walks. As soon as two walks intersect they become one walk and continue evolving as ...

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**1**answer

251 views

### Self Avoiding Walk Pair Correlation

Let $\gamma(i)$ be a self avoiding walk (SAW) on a 2D lattice $L$ (a square lattice for example) starting at a predefined origin ( $\gamma(0)=(0,0)$ ) and having length $n:=\ell(\gamma)$. Furthermore, ...

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681 views

### Random permutations of Z_n

In http://www.springerlink.com/content/y19u81675243r237/fulltext.pdf, the author states the following without proof (equation 3.1):
"Consider a random permutation $\pi$ of $\mathbb{Z}_n$. What is the ...

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**0**answers

287 views

### Self-avoiding Walk with next-nearest neighbors

Background
I study polymer physics and am doing experiments testing the model outlined in this paper. Basically, the polymers fall into an integer number of pits, and we create a partition function ...

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**1**answer

857 views

### Self Avoiding Walk Enumerations

Let $c(n)$ be the number of Self avoiding walks (SAW) of length $n$ on an infinite lattice $L$. Are there any known non-geometric interpretations of $c(n)$?. For example, is there a number theoretic ...

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585 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
...

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**2**answers

544 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, ...

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528 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 $t$ ...

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2k 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
...

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417 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 ...