Non-solutions to mixed equations in acylindrically hyperbolic groups coming from random walks

A mixed equation in a group G is given by a non-trivial element w(x) of the free product G∗Z, and a solution is some g∈G such that w(g) is the identity. For G acylindrically hyperbolic with trivial finite radical (e.g. torsion-free), we show that any mixed equation of length n has a non-solution of length comparable to log(n), which is the best possible bound. Similarly, we show that there is a common non-solution of length O(n) to all mixed equations of length n, again the best possible bound. In fact, in both cases, we show that a random walk of appropriate length yields a non-solution with positive probability.