Let $π$ be a permutation of $\{1,2,\ldots,n\}$. If we identify a permutation with its graph, namely the set of $n$ dots at positions $(i,π(i))$, it is natural to consider the minimum $L^1$ (Manhattan) distance, $d(π)$, between any pair of dots. The paper computes the expected value (and higher moments) of $d(π)$ when $n\rightarrow\infty$ and $π$ is chosen uniformly, and settles a conjecture of Bevan, Homberger and Tenner (motivated by permutation patterns), showing that when $d$ is fixed and $n\rightarrow\infty$, the probability that $d(π)\geq d+2$ tends to $e^{-d^2 - d}$. The minimum jump $mj(π)$ of $π$, defined by $mj(π)=\min_{1\leq i\leq n-1} |π(i+1)-π(i)|$, is another natural measure in this context. The paper computes the asymptotic moments of $mj(π)$, and the asymptotic probability that $mj(π)\geq d+1$ for any constant $d$.

20 pages, 3 figures. Minor changes throughout