This post is courtesy of Johan and Nathanaël – a big thanks !

Today Olivier Levêque made a very nice presentation about results of
large random matrix theory applied to Multi Input Multi Output (MIMO)
antenna systems and large random networks.

He started with a short introduction to information theory, especially
to capacity of a communication channel, on the blackboard, with style.
Olivier then showed a very powerful result in the field of MIMO
communications, which states that when the number $\latex n$ of antennas
grows we get a capacity or order $\latex n$, or

C_n = n \int_0^4   \log ( 1 + Px) q(x) dx .

This result is powerful since it means that theoretically the capacity
grows linarly with the number of antennas (which is a significant
improvement of the previously believed limiting order

C = log(1 +   Pn^2)

for multi antennas systems).

Olivier also enphasized that we can achieve this capacity without
increasing the power P, but simply by distributing it in a clever way,
which can be done using water-filling techniques. He concluded the first
part of the lecture by reminding that, although this result is
demonstrated for n goes to infinity, it actually works already for small
values of n, which explains the joke common amongst larse random matrix
people which states : 8 = \infty .

The second main part of Oliver’s talk dealed with large self-organized
wireless communication networks. The main question proposed to answer
was: assuming that n users want to communicate in a given area with an
O(1) density, what is the rate R_n they can hope to achieve ?
And in relation to this statement, how does the rate scale with n ?

In a setup where we wish to establish n communication channels, we use a
model where the signals that are received depend on the gain between two
nodes divided by the distance, with an additive white gaussian noise.
Note that the gain is i.i.d. and follows a complex gaussian
distribution. Finally the power of emission per user is limited and has
a given value P. For this model we would like to achieve R_n = O(1)

Following this model, a very naive scheme is to dispatch the n different
pairwise communication instances into n different time slots (or
equivalently n different frequency bands). In this case we achieve
R_n = O(\frac{1}{n}) which is really bad for n large.

The second approach uses multi-hop communication. And in this case,
Olivier explained that due to an overload of the users in the center of
the network the limiting rate is R_n = O(\frac{1}{\sqrt{n}} ,
which is better but still not good.

Finally, Oliver presented a scheme coined Distributed MIMO
communications which proceeds in an iterative way. At each stage, a user
broadcasts its message to his neighborhood, which will then send the
message to the neighborhood of the destination user, which receives all
the messages from its own neighbors. We can then recursively use this
scheme for the communication within the neighborhood. In this case,
Oliver showed that the limiting rate for large n with k recursions is
R_n = O(n^{\frac{-1}{k+1}}) which tends to O(1) when
k goes to infinity.

Oliver concluded his lecture by explaining a controversy about this
model, which states that considering gains to be i.i.d. is in fact
unrealistic. Some authors indeed claims that for this setup we can only
hope to have O(2n) i.i.d. random variables (instead of the
O(n^2) that are needed to prove the limits presented above. And
this would bring down the rate back to O(\frac{1}{\sqrt{n}}).
The final word provided a way to avoid this problem simply by stating
that practical networks tend to have a lower density, and that we can
achieve interesting rates given that the area is large enough.