  The Ronkin function was defined by Ronkin in the consideration of the zeros
of almost periodic function. Recently, this function has been used in various
research fields in mathematics, physics and so on. Especially in mathematics,
it has a closed connections with tropical geometry, amoebas, Newton polytopes
and dimer models.
  On the other hand, we have been investigated a new class of zeta functions
for various kinds of walks including quantum walks by a series of our previous
work on Zeta Correspondence. The quantum walk is a quantum counterpart of the
random walk. In this paper, we present a new relation between the Ronkin
function and our zeta function for random walks and quantum walks. Firstly we
consider this relation in the case of one-dimensional random walks. Afterwards
we deal with higher-dimensional random walks. For comparison with the case of
the quantum walk, we also treat the case of one-dimensional quantum walks. Our
results bridge between the Ronkin function and the zeta function via quantum
walks for the first time.
