  We consider the two-opinion voter model on a regular random graph with n
vertices and degree $d \geq 3$. It is known that consensus is reached on time
scale n and that on this time scale the volume of the set of vertices with one
opinion evolves as a Fisher-Wright diffusion. We are interested in the
evolution of the number of discordant edges (i.e., edges linking vertices with
different opinions), which can be thought as the perimeter of the set of
vertices with one opinion, and is the key observable capturing how consensus is
reached. We show that if initially the two opinions are drawn independently
from a Bernoulli distribution with parameter $u \in (0, 1)$, then on time scale
1 the fraction of discordant edges decreases and stabilises to a value that
depends on d and u, and is related to the meeting time of two random walks on
an infinite tree of degree d starting from two neighbouring vertices. Moreover,
we show that on time scale n the fraction of discordant edges moves away from
the constant plateau and converges to zero in an exponential fashion. Our
proofs exploit the classical dual system of coalescing random walks and use
ideas from Cooper et al. (2010) built on the so-called First Visit Time Lemma.
We further introduce a novel technique to derive concentration properties from
weak-dependence of coalescing random walks on moderate time scales.
