The Fermi Paradox is the term used to describe the lack of evidence for extraterrestrial life in the face of a universe that should be, by the numbers, bursting with it. But we see no signs of alien technology, and our radio telescopes don’t pick up voices from other worlds.
For example, in our $3D$ world below, consider an alien starting at the origin $(0,0,0)$, and at each time step, moves to any of the 6 adjacent points (up, down, left, right, forward, backward) with equal probability. Suppose Earth is located at $(x,y,z)$ on this grid:
The total number of possible paths the alien could take after $n$ steps is $6^n$.
However, if we lived in a $2D$ world, the alien would eventually reach Earth since it has been proven that on a two-dimensional lattice, a random walk has unity probability of reaching any point (including the starting point) as the number of steps approaches infinity!
Questions
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For an unbiased $3D$ random walk, what's the probability of the alien reaching Earth, is it $\frac{1}{6^n}$? But, according to When do 3D random walks return to their origin?, visiting a specific point, such as Earth is $34\%$.
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For a biased $3D$ random walk, what's the probability of the alien reaching Earth? This is the most likely scenario since aliens would have a bias towards finding life.
For higher dimensions, such as $4D$ spacetime, the probability of reaching Earth decreases even further as the number of dimensions increases. If aliens do exist, seems like the chances of First Contact is much greater than 34%!
Best Answer
The probability of a particular path of length $n$ is indeed $6^{-n}$, however this is not the probability starting from $x$, you will be at point $y$ after $n$ steps; in fact, the latter probability is of order $n^{-3/2}$ (for fixed $x$ and $y$). As a result, by the Borel-Cantelli lemma the probability of hitting a particular site infinitely often is zero a.s. $0.34$ is the probability that a R.W. starting from $(0,0,0)$ returns (ever) to this point.