The random-walk tag has no wiki summary.

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### Why is the density function of a sum of many iid random variables (i.e., the Gaussian) self-dual?

In the usual proof of the central limit theorem via characteristic functions, we note that if $X_1, \dots, X_n$ are independent and identically distributed, with probability density function $f(x)$, ...

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**1**answer

46 views

### N random walkers that hit node v in a graph

Consider a finite, undirected graph G, with uniform edge weights. Assume that there are n number of random walkers that will start at different nodes (lets say n=3, hence the random walkers will start ...

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194 views

### Mixing properties of random walks on graphs [migrated]

I have a question about this paper (not behind a pay wall) on the Cheeger inequality for graphs.
One of the main ideas of the paper is that random walks on graphs can be used to find sets with small ...

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**1**answer

304 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
$\equiv 2 \mod 3$,
we place, with equal probability, one of these six patterns:
The result ...

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**1**answer

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

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273 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 ...

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167 views

### Random walk in a convex body or convex polytope

Let $\Delta$ be a convex body (i.e. a compact convex subset) or a convex polytope in
$\mathbb{R}^n$. Let $x$ be a point inside $\Delta$ and consider a (uniform) random walk starting at $x$ inside ...

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**1**answer

149 views

### Diameter of $n$-unit-vector closed scribble

Suppose one creates a random, closed, likely self-crossing polygon
from $n$ unit-length vectors arranged head-to-tail,
randomly oriented except for the requirement
that their sum is zero (so the ...

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73 views

### 1d random walk probability of previous n positions

I have the following question.
(May be it is very simple, but I cannot find the answer).
Suppose I have a 1d random walk on integer numbers with equal pobabilities of unit step in either direction ...

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**1**answer

146 views

### Random walk on a Penrose tiling

Pólya proved that a random walk on $\mathbb{Z}^2$ almost surely returns to the
origin, or, equivalently, returns to the origin infinitely often.
It was subsequently established that in $\mathbb{Z}^3$, ...

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741 views

### Randomly walking a leashed dog

Let a human $h(t)$ random walk on $\mathbb{Z}^2$ by taking a unit-length step at every
time step $t$. A dog $d(t)$ on a leash of length $\lambda$ follows $h(t)$, also
taking a unit-length step at ...

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**1**answer

184 views

### Wander distance of self-avoiding walk that backs out of culs-de-sac

Suppose a self-avoiding walk on $\mathbb{Z}^2$, with random steps each one unit long,
backs out of culs-de-sac, but retaining the lattice points on which it stepped
marked as unavailable for future ...

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votes

**2**answers

608 views

### Do Random Walks on the Hexagonal Lattice have a limit?

For every positive integer $n$, consider a regular hexagon $\mathrm{H}_n$ such that
the distance of each vertex from the center is $\frac{1}{\sqrt{n}}$. That in turn
induces a tiling of ...

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141 views

### Does the concept of connective constant make sense for any tiling of the plane?

First let me define what is the "connective constant" of a two dimensional lattice.
Let $c_{n}$ denote the number of $n$ step self-avoiding walks starting from a fixed origin point in the lattice. ...

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421 views

### Reference requested: Random walk on groups

I am looking for a good reference to learn about random walks on groups (either finite groups or Lie groups). Ideally, I would like a reference for general theory of random walks on groups that is ...

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**1**answer

107 views

### Connection between degree of growth and return probabilities of random walks on Lie groups

Let $G$ be a finitely generated group of polynomial growth, let $\mu$ be a non-degenerate symmetric probability measure with finite support on $G$, and let $d$ be the degree of growth of $G$. ...

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108 views

### How much do the classes of geodesic metrics and Green metrics on a Gromov hyperbolic group differ?

Background
Let $G$ be a Gromov hyperbolic group. If $G$ acts properly discontinuously and cocompactly on a proper geodesic metric space $X$, then any bijective identification of $G$ with its orbit ...

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195 views

### Nonmonotonicity of expected distance of a random walk

What's the simplest example of a reversible random walk $X_n$ on an infinite vertex-transitive graph such that the expected distance from the origin is not increasing, i.e. there exists $n$ such that:
...

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**3**answers

364 views

### Hitting time for two out of three random walk particles

I'm imagining a simple random walk on $\mathbb{Z}$ with three independent particles (maybe add laziness so they don't jump over each other). Suppose the particles are initially placed at, say, $-10$, ...

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172 views

### Area covered by Brownian motion of 2D disc

I would like to know the expected value for the area covered by a disc of radius $R$ whose center undergoes Brownian motion (diffusion).
Specifically, let $\mathbf{X}_t$ represent a two-dimensional ...

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76 views

### Modification of a state in a random walk to be partially absorbing after a walker's position is well-approximated by a stationary distribution

Consider a random walk $(X_0, X_1, X_2, ...)$ on the interval $[0, N]$ starting from some position $k$, where $0$ and $N$ are reflecting barriers. The forward $+1$ transition probability is $p$, the ...

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544 views

### Random walk by simplex vertices

I apologize if this question is well-known, but I was unable to find it mentioned anywhere.
There exists a bug which moves around in $r$-space. The bug begins at the origin of this $r$-space. If the ...

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### Using Crump-Mode-Jagers processes to get logarithmic bound on a random tree height

I am currently pursuing my PhD degree and in my research I came across a family of random trees. I need to prove a logarithmic asymptotic bound for the heights of such trees as their size grows. I ...

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115 views

### Speed on recurrent graphs

Suppose that $G$ is a (locally finite) graph such that the simple random walk on $G$ is recurrent. Does this imply any upper bound on the speed $\mathbb{E}d(X_t,X_0)$ of such random walk?

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**1**answer

320 views

### Expectation of Maximum of Uniform Multinomial Distribution

Suppose we have a uniform multinomial distribution with $k$ buckets, i.e. we put $n$ items uniformly at random in $k$ buckets leading to $n_1, \dots, n_k$ items in each bucket respectively. Let $m = ...

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**1**answer

148 views

### Quadratic variation for discrete Martingale

Is there any analogue of continuous martingale quadratic variation for the discrete case? If so, are there any theorems which characterize simple random walk using quadratic variation - similar to ...

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votes

**1**answer

166 views

### Width of a random convex polygon

Consider a planar (2D) random walk comprised of N steps.
Consider the minimum convex polygon enclosing the N points visited by the random walker.
Assume the definition of the width of a convex ...

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votes

**1**answer

439 views

### Random walk in a circle

Suppose we have a circle of radius $R$ centered in the origin of a $x,y$ cartesian reference frame. A particle starting from the center of the circle is moving with a speed given by:
...

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**1**answer

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### Joint law for number of visits in transient simple random walk

Consider a simple $1$-dimensional random walk $X_n$.
Let $p$ and $q$ (with $p+q=1$) be the probability of transitions from $n$ to $n+1$ and from $n$ to $n-1$ respectively, and suppose that $p>q$ ...

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### Estimates for simple random walks in groups of intermediate growth

I'm looking for references for the rate of escape and return probability for a group of intermediate growth.
Let $0<\alpha < 1$. If the volume growth is $\succeq \mathrm{exp}(n^\alpha)$, then ...

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**2**answers

193 views

### Methods to approximate the betweenness centrality on large networks

To calculate the between centrality wiki def:
$g(v) = \sum_{s\neq v \neq t} \frac{\sigma_{st}(v)}{\sigma_{st}}$
of a node in a graph/network;$\sigma_{st}$ is the ...

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187 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 ...

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57 views

### Discretization model for Dirac equation in higher dimensions

I'm wondering if there are cases of discretization models (in lattices or graphs) that converge toward the non-linear Dirac equations? While there is a paper on the 2-d and 3-d cases ...

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### Estimate size of graph by taking random walks

Let $G$ be a connected, finite graph and let $v_0$ be a vertex of $G$. I'm interested in methods of estimating the number of vertices in $G$, based on local exploration only. What I have in mind is:
...

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350 views

### Random walk over a function

Let $\{X_n\}_{n\geq 0}$ be a random walk. Let us assume that $\mathbb{E}X_1 =0$ and $\mathbb{E}X_1^2=1$. Let also $\mathbb{E}\exp(c|X_1|)<+\infty$ for some $c>0$ and $X_1$ has a law with ...

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283 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 ...

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495 views

### random walk and Brownian motion on Riemannian manifold

As we know, the random walk on $\mathbb{Z}/n$ will converge(in some sense) to the Brownian motion on $\mathbb{R}$ when $n\to\infty$. I would like to know is there some higher dimensional analogy ...

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226 views

### Speed of random walks in groups

I've seen some estimates for the decay in $d$ of the probability a SRW makes a distance $d$ in time $n$, but is there any reference for the "speed" of a random walk in a group? I'm interested mostly ...

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271 views

### Random path in a graph

Consider a finite graph $G$. I would like to define a random path between two vertices $s$ and $t$ of the graph $G$ by looking at a measure $\mu$ on all spanning trees. Then the probability of a given ...

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498 views

### Random Walk on $\mathbb{R}$ with Uniformly Distributed Steps and “Reflective” Boundary at Origin

A particle lies on the real number line at the origin. For each step taken, the particle moves from its current position a distance (and direction) chosen equi-probably from range $[-1,r]$. However, ...

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286 views

### The number of lattice paths below y=n/m x for gcd(m,n) = 1

The motivation of my question is the recent preprent of Armstrong, Rhoades and Williams http://arxiv.org/abs/1305.7286 on rational Catalan combinatorics.
An important starting point of this paper is ...

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### 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 ...

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198 views

### Random walk on the hypercube

Consider the hypercube $Q_4$. I would like to know how to compute the number of steps of a random walk in this graph such that the probability to be at a vertex is a given number $x$. I think I just ...

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166 views

### Fundamental inequality of entropy in random walks

I'm looking for a reference for an inequality related to the "fundamental inequality" about entropy and rate of escape of random walks (on the Cayley graph of a group). Namely,
$\textbf{Question}$: ...

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**1**answer

273 views

### Probabilities of a random walk exiting a set

Let $F$ be a finite connected set in a graph (soon to be the Cayley graph of a group) and $\mathrm{Ex}_x^F$ be the function on the vertices in $F^c$ which are neighbour to vertices in $F$ defined as ...

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193 views

### Memory of Uniformly Random Dyck Paths

Let $D$ be the set of all Dyck paths on square grid of size $n\times n$. For any particular Dyck path, let $S(t)=X_1+X_2+\ldots +X_t$ store the path, where $X_i=\pm 1$. Being a Dyck path, we have ...

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### Hitting time probability in a Random Walk with possibility to die.

A Random Walker can move of one unit to the right with probability $p$, to the left with probability $q$ and it can jump again to the starting point with probability $r$ and die. Naturally $p+q+r=1$. ...

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241 views

### The minimal norm of a shifted stochastic matrix

Hello,
Given a row-stochastic matrix $M$ with singular values $\sigma_{1}\geq\ldots\geq\sigma_{n}$, I am looking for an upper bound on the expression: $\min_{\alpha}\parallel M- ...

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316 views

### Random walk on the hypercube

Let $H_N=\{0,1\}^N$ the N-dimensional hypercube. We define the following random walk $X_n$ on $H_N$:
start from a point $x \in H_N$
pick at random an integer $k$ in $[1,N-1]$ and exchange $x(k)$ and ...

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583 views

### asymmetric random walk, hitting time probability

Let's consider an asymmetric Random Walk on $Z$, with transition probabilities $p_{i, i+1}=p$, $~~p_{i, i+1}=q$, $\forall i \in \mathcal{Z}$, $p+q=1$ and $p>q$.
I am interested in the probability ...