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We investigate the biased quenched trap model on top of a two-dimensionallattice in the case of diverging expected dwell times. By utilizing thedouble-subordination approach and calculating the return probability in $2$d,we explicitly obtain the disorder averaged probability density function of theparticle's position as a function of time (for any given bias) in the limit oflarge times ($t \rightarrow \infty$). The first and second moments arecalculated, and a formula for a general $\mu$-th moment is found. The behaviorof the first moment, i.e. $\langle x(t)\rangle$, presents non-linear responseboth in time and in the applied external force $F_0$. While the non-linearityin time occurs for any measurement time $t$, the non-linearity in $F_0$ isexpected only when $t\gtrsim \big(F_0 \left| \ln (F_0)\right|\big)^{-2 /\alpha}$ where $\alpha=T/T_g$, for temperatures $T<T_g$. We support ouranalytic results by comparison to numerical simulations.

The case of the biased quenched trap model in two dimensions with diverging mean dwell times