# Tagged Questions

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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?