  Driven periodic elastic systems such as charge-density waves (CDWs) pinned by
impurities show a non-trivial, glassy dynamical critical behavior. Their proper
theoretical description requires the functional renormalization group. We show
that their critical behavior close to the depinning transition is related to a
much simpler model, $O(n)$-symmetric $\phi^4$ theory in the unusual limit of
$n\to -2$. We demonstrate that both theories yield identical results to 4-loop
order and give both a perturbative and a non-perturbative proof of their
equivalence. As we show, both theories can be used to describe loop-erased
random walks (LERWs), the trace of a random walk where loops are erased as soon
as they are formed. Remarkably, two famous models of non-self-intersecting
random walks, self-avoiding walks (SAWs) and LERWs, can both be mapped onto
$\phi^4$ theory taken, with formally $n=0$ and $n\to -2$ components. This
mapping allows us to compute the dynamic critical exponent of CDWs at the
depinning transition and the fractal dimension of LERWs in $d=3$ with
unprecedented accuracy, $z(d=3)= 1.6243 \pm 0.001$, in excellent agreement with
the estimate $z = 1.624 00 \pm 0.00005$ of numerical simulations.
