of this paper. In Sections \[Sec:LBSB\] and \[Sec:UBSB\], we deal with the small bias case and prove respectively the lower and upper bounds of Theorem \[Th:smallbias\]. Here again, the lower bound is relatively short and follows from similar argument as those presented by Wilson [@Wil04] in the symmetric case, while the upper bound relies on a careful analysis of the area between the processes starting from the maximal and minimal configurations and under some grand coupling.

Preliminaries and technical estimates {#Sec:Prelim}
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Height function ordering and grand coupling
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To any configuration of particles $\xi\in\Omega_{N,k}^0$, we can associate a so-called height function $h=h(\xi)$ defined by $h(\xi)(0) = 0$ and $$h(\xi)(x) = \sum_{y=1}^x \big(2\xi(y) - 1\big)\;,\quad x\in {\llbracket}1,N{\rrbracket}\;.$$ For simplicity, we often abbreviate this in $h(x)$. The height function is a lattice path that increases by $1$ from $\ell-1$ to $\ell$ if there is a particle at site $\ell$, and decreases by $1$ otherwise. Its terminal value therefore only depends on $k$ and $N$. The set of height functions obtained from $\Omega_{N,k}^0$ through the above map is denoted $\Omega_{N,k}$.\
The particle dynamics can easily be rephrased in terms of height functions: every upward corner ($h(x)=h(x-1)+1=h(x+1)+1$) flips into a downward corner ($h(x)=h(x-1)-1=h(x+1)-1$) 