For this question I am taking a representation of a group $G$ to be a homomorphism from $G$ to $GL_n(K)$ for some field $K$. The degree of the representation is $n$.
I am trying to understand the minimal degree of a nontrivial representation for nonabelian finite simple groups. So suppose I define $f(n)$ to be the minimum of the degrees of nontrivial representations of all nonabelian finite simple groups of size at least $n$.
Is there some kind of asymptotic description of $f(n)$? Does any meaningful asymptotic information about $f(n)$ rely on the classification of finite simple groups?