Lower complexity bounds without mutation Lower complexity bounds tend to be a very hard problem in general. Despite this, I was wondering if there are any theoretical results that relate lower complexity bounds for some class of problems in the case where mutation is allowed and in the case it's not allowed? In particular, what type of problems would be inherently harder to solve in a purely functional language on a RAM machine?

The following problem is why I started thinking about this: https://stackoverflow.com/questions/24105301/number-of-occurrences-in-list-of-each-element
See the section labelled "Negative result: you cannot avoid some slowdown, for some data structures" of https://cs.stackexchange.com/a/18266/755 for examples of problems whose solution in a functional language is inherently a bit slower than a solution in an imperative language with mutation.  One example is an array data structure; with mutation, you can achieve $O(1)$ worst-case time for both mutation and lookup, but with a functional language (with persistent data structures), you cannot.