Random Walks in the High-Dimensional Limit I: The Wiener Spiral

We prove limit theorems for random walks with $n$ steps in the $d$-dimensional Euclidean space as both $n$ and $d$ tend to infinity. One of our results states that the path of such a random walk, viewed as a compact subset of the infinite-dimensional Hilbert space ${\mathrm{\ell }}^2$, converges in probability in the Hausdorff distance up to isometry and also in the Gromov-Hausdorff sense to the Wiener spiral, as $d,n\to \mathrm{\infty }$. Another group of results describes various possible limit distributions for the squared distance between the random walker at time $n$ and the origin.