  We show a remarkable similarity between strategies to realize a large
intersection or self-intersection local times in dimension five or more. This
leads to the same rate functional for large deviation principles for the two
objects obtained respectively by Chen and Morters, and by the present author.
We also present a new estimate for the distribution of high level sets for a
random walk, with application to the geometry of the intersection set of two
high level sets of the local times of two independent random walks.
