Questions tagged [random-walks]
The random-walks tag has no usage guidance.
511
questions
17
votes
1
answer
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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
$\equiv 2 \bmod 3$,
we place, with equal probability, one of these six patterns:
The result ...
3
votes
1
answer
321
views
Maximizer of random walk with very small drift
This is an extended question based on
Large deviations for maximizer of random walk with drift.
Let $$S_k = X_1 + \ldots + X_k,$$ where $X_i$ are i.i.d. with mean $-\mu < 0$ and unit variance. ...
15
votes
2
answers
1k
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 ...
6
votes
3
answers
921
views
Probability that a self-avoiding walk on $\mathbb{Z}^3$ closes to a polygon
The probability that a random walk on $\mathbb{Z}^3$ returns to the origin is about 34%.
This is (part of)
Pólya's theorem.
I have been looking for an analogous (numerical) result for the probability
...
2
votes
2
answers
362
views
Speed and absence of non-constant bounded harmonic functions
For a (symmetric) random walks on countable groups generated by $\mu$, there is a "brute-force computation" argument of Avez (1974) that shows that if the entropy $h_\mu$ is trivial then there are no ...
2
votes
1
answer
359
views
Random walks with exponential decreasing steps
Let $g$ be the golden number (or another algebraic integer in $(0,1)$ that fullfills an equation with coefficients $\pm 1$). Consider the random walk on $\mathbb{R}$ starting with $0$ and walking $g^{...
5
votes
1
answer
681
views
Harmonic Crystal using Random Walk
Me and my advisor were looking for a specific proof of the disorder in $2d$ harmonic crystals. We could not find a paper or a textbook with it, so I thought trying my luck here.
Basically, it is a ...
4
votes
0
answers
298
views
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 $f(x)$, ...
2
votes
0
answers
79
views
Controlling fluctuations in a Markov chain
For $N>0$, consider the Markov chain $x_n$ on $\{0,1/N,...,1\}$ that moves up by $1/N$ at rate $(c-x_n/2)N$ and down by $1/N$ at rate $(c+x_n/2)N$. As $N\rightarrow\infty$ sample paths approach ...
1
vote
1
answer
126
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 ...
11
votes
3
answers
1k
views
Square filling self avoiding walk
I want to create an algorithm that fills a square grid with a random Hamiltonian path starting at a particular point. See this example.
One approach is to try a free direction as a next step, and ...
2
votes
2
answers
268
views
Distribution of a random walk on a directed line
Is there a closed formula for the distribution of $x_t$ in the following random process, describing a random walk on a directed line?
$x_0 = n$
$x_t$ is a uniformly random integer between 1 and $x_{...
12
votes
1
answer
730
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 ...
0
votes
1
answer
469
views
Brownian motion of every point in the plane
Suppose every point in the plane undergoes brownian motion for a time t. What is the probability n particles ended up at 0? For n finite, countable or uncountable?
What proportion of the plane does ...
7
votes
1
answer
326
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 $Z_{t} = (X_{t}[1], \ldots, X_{t}[k])$ ...
3
votes
1
answer
198
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 ...
6
votes
2
answers
408
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 $\...
4
votes
0
answers
156
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 (...
16
votes
2
answers
1k
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 ...
4
votes
2
answers
2k
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 $\mathbb{R}^...
11
votes
3
answers
840
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, $-10$, $...
4
votes
0
answers
177
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. ...
6
votes
2
answers
405
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:
...
4
votes
1
answer
199
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 $G$. ...
1
vote
1
answer
938
views
Stationary distribution for directed graph
I want to implement the algorithm of graph partitioning of sparse directed graph. In this algorithm after computing the transition matrix ,we should compute the stationary distribution of the random ...
8
votes
2
answers
865
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 ...
0
votes
0
answers
553
views
Probability density function of the node positions in a random walk after N time slots
Hello, my question basically is how do I find the probability density function of the position of the nodes in a given area after N discrete time slots when the nodes move following the 2D random walk ...
3
votes
0
answers
237
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 ...
4
votes
0
answers
156
views
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 ...
5
votes
2
answers
804
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 ...
7
votes
1
answer
2k
views
Generating function for Random Walk Hitting Time, taking the wrong root
In a calculation of the hitting time for a Bernoulli random walk we have to calculate the hitting time $\tau(1)=\inf\{n\ge 0:S_n=1\}$ to reach $+1$ and the generating function has the recursion ...
3
votes
1
answer
2k
views
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 $p+q+r=1$. ...
7
votes
0
answers
129
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?
0
votes
1
answer
139
views
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 $p>q$ ...
5
votes
2
answers
907
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 ...
6
votes
1
answer
822
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 ...
13
votes
1
answer
3k
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 ...
1
vote
0
answers
122
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 http://arxiv.org/...
4
votes
2
answers
710
views
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:
...
7
votes
1
answer
508
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 ...
9
votes
1
answer
618
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 $n=...
5
votes
1
answer
402
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 $S(0)...
3
votes
1
answer
384
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 ...
6
votes
1
answer
615
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 ...
2
votes
0
answers
240
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 $\...
7
votes
1
answer
853
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 ...
1
vote
1
answer
534
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,
$\textbf{Question}$: ...
5
votes
1
answer
737
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 ...
3
votes
1
answer
2k
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 ...
4
votes
0
answers
1k
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 $...