function as an initial condition.

-   The average $\langle \ln Y_t\rangle$ tends to a constant at large $t$. In addition, in any given event, the random variable $\ln Y_t$ itself tends to a constant, which has some distribution (which we do not know how to compute) which may be used to characterize the early-time fluctuations. Note that an appropriate generating function of the moments of Y\_t\_i e\^[x\_i(t)]{}=Y\_t \[eq:defYt\] also obeys the FKPP equation, but with the “critical” initial condition discussed in the Introduction. We will come back to the latter fact in Sec. \[sec:det\]. Also, in the context of directed polymers in random media, $\tilde Y_t$ is the partition function and $\langle\ln \tilde Y_t\rangle$ the free energy averaged over the disorder [@DS:1988].

-   $Z_t$ is the variable used by Lalley and Sellke in the theorem alluded to in the Introduction. However, we are not going to focus on this variable in the body of this paper, since we found it has many drawbacks for our purpose. First, a practical drawback: Although $Z_t$ tends almost surely to a positive constant when $t\rightarrow+\infty$ [@LS:1987], it takes negative values at finite times, with finite probability; $\ln Z_t$ is then undefined in these particular 