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We map the problem of diffusion in the quenched trap model onto a newstochastic process: Brownian motion which is terminated at the coverage "time"${\cal S}_\alpha=\sum_{x=-\infty} ^\infty (n_x)^\alpha$ with $n_x$ being thenumber of visits to site $x$. Here $0<\alpha=T/T_g<1$ is a measure of thedisorder in the original model. This mapping allows us to treat the intricatecorrelations in the underlying random walk in the random environment. Theoperational "time" ${\cal S}_\alpha$ is changed to laboratory time $t$ with aL\'evy time transformation. Investigation of Brownian motion stopped at "time"${\cal S}_\alpha$ yields the diffusion front of the quenched trap model whichis favorably compared with numerical simulations. In the zero temperature limitof $\alpha\to 0$ we recover the renormalization group solution obtained by C.Monthus. Our theory surmounts critical slowing down which is found when $\alpha\to 1$. Above the critical dimension two mapping the problem to a continuoustime random walk becomes feasible though still not trivial.

Weak subordination breaking for the quenched trap model