  We study the occurrence of domain branching in a class of $(d+1)$-dimensional
sharp interface models featuring the competition between an interfacial energy
and a non-local field energy. Our motivation comes from branching in uniaxial
ferromagnets corresponding to $d=2$, but our result also covers twinning in
shape-memory alloys near an austenite-twinned-martensite interface
(corresponding to $d=1$, thereby recovering a result of Conti [Comm. Pure Appl.
Math. 53 (2000), 1448-1474. https://doi.org/cgkvm4 ]).
  We prove that the energy density of a minimising configuration in a large
cuboid domain $Q_{L,T}=[-L,L]^d\times [0,T]$ scales like $T^{-\frac{2}{3}}$
(irrespective of the dimension $d$) if $L\gg T^{\frac{2}{3}}$. While this
already provides a lot of insight into the nature of minimisers, it does not
characterise their behaviour close to the top and bottom boundaries of the
sample, i.e. in the region where the branching is concentrated. More
significantly, we show that minimisers exhibit a self-similar behaviour near
the top and bottom boundaries in a statistical sense through local energy
bounds: for any minimiser in $Q_{L,T}$, the energy density in a small cuboid
$Q_{\ell,t}$ centred at the top or bottom boundaries of the sample, with side
lengths $\ell \gg t^{\frac{2}{3}}$, satisfies the same scaling law, that is, it
is of order $t^{-\frac{2}{3}}$.
