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Problems 2 and 3 are equivalent by ignoring horizontal steps. Problems 1 and 2 are not equivalent. I believe the probabilities are known, but I don't know them. However, if the probability of hitting …

My interpretation of the problem is that you want a function $f(n)$ so that a walk of $n$ steps stays within $f(n)$ of the origin with probability $c+o(1)$ for some $0\lt c \lt 1$. If so, the right fo …

After rescaling (the variance is $1/3$ instead of $1$), the random walk approaches Brownian motion. The first hitting time of $c$ for Brownian motion follows a Lévy distribution ($\textrm{Levy}(0,c^2) …

These random walks are recurrent when $d\le 2$ and transient when $d \ge 3$. That behavior happens for a wide variety of random walks. The expected number of returns to the origin is $$\sum_{n=1}^\in …

The answer is the same for random walks and for Brownian motion. If you project a $3$-dimensional Brownian motion perpendicular to $x=y=z$, you get a $2$-dimensional Brownian motion. The projection o …

Here are two examples with bipartite graphs. Let the vertices be the integers. Take steps of $\pm(2n+1)$ with probability $2^{-n-2}$. This is bipartite and all vertices are connected to each vertex i …

For any vertex $v$ which is not in the all-left ray, there is some generation $n$ so that all descendants of $v$ in the $n$th generation are in the right half. (If the fraction of vertices to the left …

As $\delta \to 0$ you are approximating Brownian motion. The measure on the boundary of the first hitting location of a Brownian motion is called harmonic measure. If you fix a subset of the boundary …

I'll expand a bit on my comment. There are $n^2$ $3 \times 3$ tiles. From each, there are two directions you can follow the path. As you move along the path in one direction, you hit a new tile, a pre …

First, let me elaborate on my comment. If $X$ occurs with probability $\rho$, then the expected waiting time before you first see a streak of length $\ell$ $X$s in a row in independent trials is expon …

This is unfinished, but I'm not sure I will complete it and some of it may be useful. You can count the Dyck paths which pass through each point or set of points. The number of lattice paths from $(0 …

There is no such inequality even if we further restrict $c_k$ to be in $\{0,1\}$ and weaken the inequality to include a constant factor. (I think it is natural to add the condition that the $c_k$ valu …