therein). Thus the comparison question might seem a bit premature. Nevertheless, even when both speeds are well-defined, we do not expect to see anything like this in higher dimensions.*]{}\[rem\]

[*Our results can also be interpreted in terms of a closely related model of killed biased random walks conditioned to survive up to time $n$. The latter model exhibits a first order phase transition (in all dimensions) as the size of the bias increases (see, for example, [@MG02], [@Fl07], [@IV12b], and references therein). In dimension 1, there is a critical bias $b_{{\mathrm{cr}}}^{{\mathrm{ann}}}$ ($b_{{\mathrm{cr}}}^{{\mathrm{que}}}$) such that these random walks have the zero asymptotic speed when the bias is less than $b_{{\mathrm{cr}}}^{{\mathrm{ann}}}$ (resp., $b_{{\mathrm{cr}}}^{{\mathrm{que}}}$) and a strictly positive asymptotic speed when the bias is greater or equal to $b_{{\mathrm{cr}}}^{{\mathrm{ann}}}$ (resp., $b_{{\mathrm{cr}}}^{{\mathrm{que}}}$). The model of crossing random walks considered in the present article informally corresponds to the critical bias case. In particular, (\[aM\]) describes how the first order transition gap closes in on the second order transition in the pure trap model ($M=\infty$).*]{}

Discussion and an open problem.
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There are many papers concerning the relationship of the quenched and annealed Lyapunov exponents of a random walk in a random potential on ${\mathbb{Z}}^d$, $d\ge 