A note on the antisymmetry in the speed of a random walk in reversible dynamic random environment

Abstract

In this short note, we prove that v(−ε)=−v(ε). Here, v(ε) is the speed of a one-dimensional random walk in a dynamic reversible random environment, that jumps to the right (resp. to the left) with probability 1∕2+ε (resp. 1∕2−ε) if it stands on an occupied site, and vice-versa on an empty site. We work in any setting where v(ε),v(−ε) are well-defined, i.e. a weak LLN holds. The proof relies on a simple coupling argument that holds only in the discrete setting.