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.