scale on the vertical axis). We see the bulk building up a smoother (more “deterministic”) distribution as time elapses, while the low-density tails remain noisy. Also, for this realization, the distribution is skewed towards negative values of $x$, due to an accidentally large drift in the initial stages, whose memory is kept throughout the evolution. ](plotdensity.eps "fig:"){width=".45\textwidth"}
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Several properties of BRW are known. In particular, in any given realization of the stochastic process, for large enough times, the forward part of the distribution of the particles looks like an exponential $e^{-x}$ (scaled by an appropriate time-dependent constant, also depending on the particular realization considered) up to fluctuations effectively concentrated at its low-density tip. We shall call this exponential part the “front.”

Then, one can also establish rigorously [@McK:1975; @B:1983] that the probability $Q(t,x)$ that all particles sit at a position smaller than $x$ obeys a nonlinear partial differential equation which reads \_t Q(t,x)=\_x\^2 Q(t,x)-Q(t,x)+Q\^2(t,x). \[eq:FKPP\] This is a version of the Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) equation [@F:1937; @KPP:1937]. (For an extensive review, see Ref. [@VS:2003], and for more applications of the FKPP equation, see e.g. Ref. [@Majumdar:2002]). If the BRW starts at time $t=0$ with a single particle 