  Reconfigurable intelligent surface (RIS) is a key technology to control the
communication environment in future wireless networks. Recently, beyond
diagonal RIS (BD-RIS) emerged as a generalization of RIS achieving larger
coverage through additional tunable impedance components interconnecting the
RIS elements. However, conventional RIS and BD-RIS can effectively serve only
users in their proximity, resulting in limited coverage. To overcome this
limitation, in this paper, we investigate distributed RIS, whose elements are
distributed over a wide region, in opposition to localized RIS commonly
considered in the literature. The scaling laws of distributed BD-RIS reveal
that it offers significant gains over distributed conventional RIS and
localized BD-RIS, enabled by its interconnections allowing signal propagation
within the BD-RIS. To assess the practical performance of distributed BD-RIS,
we model and optimize BD-RIS with lossy interconnections through transmission
line theory. Our model accounts for phase changes and losses over the BD-RIS
interconnections arising when the interconnection lengths are not much smaller
than the wavelength. Numerical results show that the performance of localized
BD-RIS is only slightly impacted by losses, given the short interconnection
lengths. Besides, distributed BD-RIS can achieve orders of magnitude of gains
over conventional RIS, even in the presence of low losses.
