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I will explain joint work with Amol Aggarwal and Promit Ghosal inÌýwhich we construct the ASEP speed process that captures the joint behaviorÌýof perturbations of the asymmetric simple exclusion process under the basicÌýcoupling. In particular, consider an interacting particle system on $\mathbb{Z}$ inÌýwhich each site $i\in\mathbb{Z}$ initially starts with a class $i$ particle.ÌýIndependently and in continuous time, across each bond the neighboring sitesÌýswap their particles (and hence classes) with rate $p$ or $q$ (satisfying $p-q=1$)Ìýdepending on the ordering of the classes. Letting $X_i(t)$ denoteÌýthe locationÌýof the class $i$ particle at time $t$, we prove that $X_i(t)/t$ converges almostÌýsurely to a limit $U_i$ whose distribution is uniform on $[-1,1]$. The jointÌýprocess $\{U_i\}_i$ defines the ASEP speed process.