Variational formulations in group theory?

Solution 1:

There are several strong theorems about random variables on groups, that might be something you were looking for:

Generation Formula:

Suppose $G$ is a finite group, $\{X_i\}_{i = 1}^{n}$ are i.i.d uniformly distributed random elements of $G$. Then $P(\langle \{X_i\}_{i = 1}^{n} \rangle = G) = \sum_{H \leq G} \mu(G, H) {\left(\frac{|H|}{|G|}\right)}^n$, where $\mu$ is the Moebius function for subgroup lattice of $G$.

Generalized Erdos-Turan theorem:

Suppose $G$ is a finite group. $\{X_i\}_{i = 1}^{n}$ are i.i.d uniformly distributed random elements of $G$. Then $G$ is nilpotent of class $n$ iff $P([ … [[a_0, a_1], a_2]… a_n] = e) > 1 - \frac{3}{2^{n + 2}}$

Law of large numbers for groups:

Suppose $G$ is a finitely generated group, $A$ is a finite set of generators of $G$. $d: G\times G \to \mathbb{N}$, is the metric on $G$ induced by the Cayley Graph $Cay(G, A)$. Suppose $\{X_i\}_{i = 1}^\infty$ is a sequence of i.i.d. random elements of $G$, such that $E(d(X_i, e)) < \infty$, and $Z_n = \Pi_{i=1}^n X_i$. Then $\exists \alpha \in \mathbb{R}, P(\lim_{n \to \infty} \frac{d(Z_n, e)}{n} = \alpha) = 1$.