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Driven periodic elastic systems such as charge-density waves (CDWs) pinned byimpurities show a non-trivial, glassy dynamical critical behavior. Their propertheoretical description requires the functional renormalization group. We showthat their critical behavior close to the depinning transition is related to amuch 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-looporder and give both a perturbative and a non-perturbative proof of theirequivalence. As we show, both theories can be used to describe loop-erasedrandom walks (LERWs), the trace of a random walk where loops are erased as soonas they are formed. Remarkably, two famous models of non-self-intersectingrandom 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. Thismapping allows us to compute the dynamic critical exponent of CDWs at thedepinning transition and the fractal dimension of LERWs in $d=3$ withunprecedented accuracy, $z(d=3)= 1.6243 \pm 0.001$, in excellent agreement withthe estimate $z = 1.624 00 \pm 0.00005$ of numerical simulations.

Depinning transition of charge-density waves: mapping onto $O(n)$ symmetric $\phi^4$ theory with $n\to -2$ and loop-erased random walks