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For a free groupFrF_{r}of finite rankr≥2r\geq 2and a non-trivial elementw∈Frw\in F_{r}, the  primitivity rankπ⁢(w)\pi(w)is the smallest rank of a subgroupH≤FrH\leq F_{r}such thatw∈Hw\in Hand  is not primitive in  (if no such  exists, one putsπ⁢(w)=∞\pi(w)=\infty). The set of all subgroups ofFrF_{r}of rankπ⁢(w)\pi(w)containing  as a non-primitive element is denoted byCrit⁡(w)\operatorname{Crit}(w). These notions were introduced by Puder (2014). We prove that there exists an exponentially generic subsetV⊆FrV\subseteq F_{r}such that, for everyw∈Vw\in V, we haveπ⁢(w)=r\pi(w)=randCrit⁡(w)={Fr}\operatorname{Crit}(w)=\{F_{r}\}.

Primitivity rank for random elements in free groups