Applications of Random Matrix Theory
and Statistical Physics
in Communications and Networks
Date: November 17-18, 2014
Venue: Institut Henri Poincare,
11 Rue Pierre et Marie Curie, 75005 Paris, France.
Program with Abstracts
Monday, November 17, 2014
10:00 - 10:10 AM
A. Moustakas
Univ. Athens
Introduction – Welcome
10:10 - 10:50 AM
A. Tourin
ESPCI, Paris, France
Time Reversal for wireless communications
10:50 - 11:30 AM
R. Speicher
Saarland Univ., Saarbruecken, Germany
On deterministic equivalents from a free probabilistic perspective
I will explain how the notion of a deterministic equivalent can be made
precise in the setting of free probability theory and how operator-valued free
probability allows to deal effectively with such deterministic equivalents.
This is joint work with Carlos Vargas.
11:30 - 12:00 PM: Coffee Break
12:00 - 12:40 PM
X. Mestre
CTTC, Barcelona, Spain
Linear spectral statistics of large sample coherence matrices and their
application to the characterization of correlation tests for large
We consider linear spectral statistics (LSS) of sample coherence matrices
constructed from independent and identically distributed Gaussian
observations. Using random matrix theory tools, we derive the first and
second order behavior of these statistics assuming that both the observation
dimension and the number of samples tend to infinity at the same rate. These
results are then used to study the asymptotic performance of several binary
hypothesis correlation tests that are widely considered in the literature. In
particular, the Generalized Likelihoold Ratio Test (GLRT) and the
Frobenius Norm (FN) tests are characterized and compared in this
asymptotic framework.
12:40 - 1:20 PM
S. Majumdar
CNRS, Univ. Paris-Sud, France
Phase transitions and edge scaling of number variance in Gaussian
random matrices
We study the probability distribution of the number of eigenvalues of a
Gaussian random matrix that lie in a box [-L,L]. We show that the variance
of the number of eigenvalues of a large matrix, as a function of the box size
L, exhibits a dramatic non-monotonic behavior. It first increases with L
logarithmically, then falls sharply as L crosses a critical value. We derive
exact analytical results for large matrices using a Coulomb gas approach.
Incidentally, our exact results also solve an outstanding problem of the
counting statistics of free Fermions in a harmonic trap at zero temperature.
1:20 - 2:50 PM: Lunch Break
2:50 - 3:30 PM
M. Debbah
Huawei, France
Random Matrices for 5G Communications
3:30 - 4:10 PM
J. Najim
Univ. Marne La Vallée, France
Large Complex Correlated Wishart Matrices:
Fluctuations and Asymptotic Independence at the Edges
We study the asymptotic behavior of eigenvalues of large complex
correlated Wishart matrices at the edges of the limiting spectrum. For this
matrix model, the support of the limiting eigenvalue distribution may have
several connected components. Under mild conditions, we will show that the
extremal eigenvalue which converge almost surely towards the edges of the
support fluctuate according to the Tracy-Widom law at the scale N2/3.
Moreover, given several generic positive edges, we establish that the
associated extremal eigenvalue fluctuations are asymptotically independent.
Finally, when the leftmost edge is the origin, we prove that the smallest
eigenvalue fluctuates according to the hard-edge Tracy-Widom law at the
scale N2 (Bessel kernel). As an application, an asymptotic study of the
condition number of large correlated Wishart matrices is provided.
4:10 - 4:50 PM
G. Schehr
CNRS, Univ. Paris-Sud, France
Near extreme eigenvalues of large random Gaussian matrices
and applications
We study the phenomenon of ``crowding'' near the largest eigenvalue λmax of
random NxN matrices belonging to the standard Gaussian ensembles, with
Dyson index β. We focus on two distinct quantities: (i) the density of states
(DOS) near λmax, ρDOS(r,N), which is the average density of eigenvalues
located at a distance r from λmax and (ii) the probability density function
(PDF) of the gap between the first two largest eigenvalues, pGAP(r,N). For
GUE we compute these quantities exactly in terms of semiclassical orthogonal polynomials, which, as we show it, can be analyzed in
the large N limit in terms of a Lax pair associated to the Painleve XXXIV
equation. For generic index β, we provide a large N analysis of these
quantities, including their asymptotic behaviors, using scaling arguments. As
an interesting physical application, we show that, for GOE, these two
quantities ρDOS(r,N) and pGAP(r,N) control the late time dynamics of the
spherical Sherrington-Kirkpatrick spin-glass model.
4:50 - 5:20 PM: Coffee Break
5:20 - 6:00 PM
L. Zdeborova
CEA Saclay, France
Statistical physics insight on matrix factorization
6:00 - 6:40 PM
P. Vivo
King’s College, London
Universal Covariance Formula for Linear Statistics on
Random Matrices
8:00 PM
Tuesday, November 18, 2014
9:00 - 9:40 AM
R. Mueller
Univ. Erlangen, Germany
On the optimum asymptotic multiuser efficiency of randomly spread
9:40 - 10:20 AM
D. Challet
École Centrale Paris, France
Inference of implicit trader communication networks
Traders are idle most of the time. What triggers their activity is an important
open question, both theoretically and practically. We show here that their
global activity is predictable, hence that the activity of some them triggers
the activity of other traders. Our approach consists in clustering traders
according to the similarity of their past activity, which amounts to find the
implicit network that links their actions over time. We then use machine
learning methods to predict traders’ activity.
10:20 - 10:50 AM: Coffee Break
10:50 - 11:30 AM
R. Zecchina
Politecnico di Torino, Italy
Statistical physics of learning in large scale neural networks
11:30 - 12:10 PM
D. Saad
Aston Univ., UK
Belief propagation based utilisation in Cognitive Radio Networks
Cognitive Radio has been proposed as a key technology to significantly
improve spectrum usage in wireless networks by enabling unlicensed users
to access unused resource. We present new algorithms that are needed for
the implementation of opportunistic scheduling policies that maximize the
throughput utilisation of resources by secondary users, under maximum
interference constraints imposed by existing primary users. Our approach is
based on the belief propagation algorithm, which is advantageous due to its
simplicity and potential for distributed implementation. We examine
convergence properties and evaluate the performance of the proposed belief
propagation algorithms via simulations and demonstrate that the results
compare favourably with a benchmark greedy strategy.
12:10 - 12:50 PM
F. Krzakala
CNRS, ENS, Paris, France
Featuring the non-backtracking operator
12:50 - 2:30 PM: Lunch Break
2:30 - 3:10 PM
A. Tulino
Bell Labs, New Jersey, USA
3:10 - 3:50 PM
L. Pastur
Inst. for Low Temperatures, Kharkiv, Ukraine
On the Qubits Dynamics in Random Matrix Environment
We consider a quantum system consisting of two qubits and an environment,
whose Hamiltonian is modeled by a large random matrix. The arborenvironment interaction is described by another large random matrix. We
study the time evolution of the reduced density matrix of qubits, in particular
the corresponding negativity, concurrence and quantum discord. It is shown
that for various choices of the environment and initial entangled state of
qubits the negativity and concurrence become zero at a finite moment
(entangled sudden death phenomenon), while the discord vanishes only for t
= 1. The dependence of these entanglement characteristics on the parameters
of the model is also analyzed.
3:50 - 4:30 PM: Coffee Break
4:30 - 5:10 PM
M. Nowak
Univ. Krakow, Poland
Signal from noise retrieval from one and two-point Green’s functions:
A comparison
5:10 - 5:50 PM
R. Couillet
Supeléc, Gif-sur-Yvette, France
Random Matrix Theory and Robust Estimation of Scatter
In this talk, we will give an overview of recent findings in the large
dimensional statistics of robust estimators of scatter. We will show that these
non-conventional matrices behave asymptotically like standard random
matrix models, therefore more amenable to theoretical analysis. The role of
these estimators in outlier or impulsive noise rejection will be made clear
and will serve as a lever to introduce several novel detection and estimation
methods accounting for impulsive data.

Program with Abstracts - Large Networks and Systems Group