# Questions tagged [random-walks]

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### 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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### Prime number races in 2 dimensions

Is the mapping $$f: \ \mathbb{N} \rightarrow \mathbb{Z}[i], \ \ \ n \ \mapsto \sum_{2 < p \leq n \ {\rm prime}} e^{\frac{p-1}{4} \pi i}$$ surjective? In 1999, when I was an undergraduate student, ...
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### 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 ...
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### 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 ...
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### Annihilating random walkers

Suppose there are several walkers moving randomly on $\mathbb{Z}^2$, each taking a $(\pm 1,\pm 1)$ step at each time unit. Whenever two walkers move to the same point, they annihilate one another. ...
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### 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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### 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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### Random Walk on Pentagonal Tiling

I’ve recently been looking at closed walks on tilings of the plane in which the “player” can move from one tile to any of its edge-adjacent neighbors. In particular, I’m trying to find asymptotic ...
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### 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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### Estimate on currents in Cayley graphs

Take a Cayley graph $\Gamma$ (thought of as an electrical network with all edges having equal resistance) and break one edge $e$ and put a battery there. (Assume the graph has only one end* so that ...
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### 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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### 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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### The mean square distance of a random walk from the origin

I'm wondering whether the following type of problem is a standard one that has been studied by probabilists. The particular case needed (as a lemma that would help with a Polymath project) isn't quite ...
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### Is there a differentiable random walk?

Is there a random walk which is differentiable or smooth? Like brownian motion except smoothed out on small distances. I was wondering if there is a "natural" or "canonical" analogue of brownian ...
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### Growing a chain of unit-area triangles: Fills the plane?

Define a process to start with a unit-area equilateral triangle, and at each step glue on another unit-area triangle.                     $50$ ...
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### 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 ...
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### Biased random Fibonacci sequences

I have recently been toying (very superficially) with the random Fibonacci sequence, i.e., defined by $F_0=1=F_1=1$ and $$F_{n} = F_{n-1} + \varepsilon_n F_{n-2}$$ where $(\varepsilon_n)_{n\geq 2}$ ...
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### 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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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, $-10$, $... 3answers 1k views ### Random Walks on high dimensional spaces I've read on a paper that, in the two dimensional case, if you start from the origin and take steps of length one in arbitrary directions (uniformely on the unit sphere$S^1\$, not left-right-up-down), ...
Stirling's approximation is the following well-known asymptotic result: $$n! \approx \left(\frac{n}{e}\right)^n \sqrt{2 \pi n}$$ This result has several analytical proofs, for example via Laplace's ...