  Unlike active array antennas, intelligent reflecting surfaces (IRSs) are
efficiently implemented at large dimensions. This allows for traceable
realizations of large-scale IRS-aided MIMO systems in which not necessarily the
array antennas, but the passive IRSs are large. It is widely believed that
large IRS-aided MIMO settings maintain the fundamental features of massive MIMO
systems, and hence they are the implementationally feasible technology for
establishing the performance of large-scale MIMO settings. This work gives a
rigorous proof to this belief. We show that using a large passive IRS, the
end-to-end MIMO channel between the transmitter and the receiver always
hardens, even if the IRS elements are strongly correlated.
  For the fading direct and reflection links between the transmitter and the
receiver, our derivations demonstrate that as the number of IRS elements grows
large, the capacity of end-to-end channel converges in distribution to a
real-valued Gaussian random variable whose variance goes to zero. The order of
this drop depends on how the physical dimensions of the IRS grow. We derive
this order explicitly. Numerical experiments depict that the analytical
asymptotic distribution almost perfectly matches the histogram of the capacity,
even in practical scenarios.
  As a sample application of the results, we use the asymptotic
characterization to study the dimensional trade-off between the transmitter and
the IRS. The result is intuitive: For a given target performance, the larger
the IRS is, the less transmit antennas are required to achieve the target. For
an arbitrary ergodic and outage performance, we characterize this trade-off
analytically. Our investigations demonstrate that using a practical IRS size,
the target performance can be achieved with significantly small end-to-end MIMO
dimensions.
