  We are interested in the study of local and global minimizers for an energy
functional of the type $$ \frac{1}{4} \iint_{\mathbb{R}^{2 N} \setminus \left(
\mathbb{R}^N \setminus \Omega \right)^2} |u(x) - u(y)|^2 K(x - y) \, dx dy +
\int_{\Omega} W(u(x)) \, dx, $$ where $W$ is a smooth, even double-well
potential and $K$ is a non-negative symmetric kernel in a general class, which
contains as a particular case the choice $K(z) = |z|^{- N - 2 s}$, with $s \in
(0, 1)$, related to the fractional Laplacian. We show the existence and
uniqueness (up to translations) of one-dimensional minimizers in the full space
$\mathbb{R}^N$ and obtain sharp estimates for some quantities associated to it.
In particular, we deduce the existence of solutions of the non-local Allen-Cahn
equation $$ \mbox{p.v.} \int_{\mathbb{R}^N} \left( u(x) - u(y) \right) K(x - y)
\, dy + W'(u(x)) = 0 \quad \mbox{for any } x \in \mathbb{R}^N, $$ which possess
one-dimensional symmetry.
  The results presented here were proved in (Cabr\'e and Sol\`a-Morales, 2005),
(Palatucci, Savin and Valdinoci, 2013) and (Cabr\'e and Sire, 2015) for the
model case $K(z) = |z|^{- N - 2 s}$. In our work, we consider instead general
kernels which may be possibly non-homogeneous and truncated at infinity.
