**YEP 2006 (Young European Probabilists)**

workshops

2004 --
2005 -- 2006

**Workshop on large deviations, random media,
and random matrices
**

EURANDOM, Eindhoven, The Netherlands

**ABSTRACTS**

**Disertori, Margherita**

**Rigorous Supersymmetric Approach to Random Matrix Problems. **

Supersymmetric approach has proved to be a powerful tool for the study of random systems in mesoscopic physics, where classical techniques do not apply. It seems also promising for a rigourous analysis. I'll give an introduction to the technique and its applications.

**Ferrari, Patrick**

**Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues
**

We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. Through the Robinson-Schensted-Knuth construction, one obtains the multilayer PNG model, which consists of a stack of non-intersecting lines, the top one being the PNG height. The statistics of the lines can be related to a particular class of Young tableaux and have the same structure as the eigenvalues of the GOE ensembles of random matrices. Finally, the edge scaling of the PNG and GOE converges to the same limit object, a Pfaffian point process.

**The expected area of the filled Brownian loop is Pi/5.**

Let B_t be a planar Brownian loop of time duration 1 (a Brownian motion
conditioned so that B_0 = B_1) and consider the compact hull obtained after
filling in all the holes, i.e., the complement of the unique unbounded component
of R^2\B[0,1]. The expected area of this hull is Pi/5. Even though the statement
of the problem is very classical, the computation of the expected area uses much
more recent techniques:

conformal restriction measures and SLE (Schramm Loewner Evolution). In this
talk, I will explain how one can use these new tools to get at the expected area
of the Brownian loop; I will give a brief overview of the necessary background.

If time allows I will also explain how one can use a result of Yor about the law
of the index of a Brownian loop to show that the expected areas of the regions
of non-zero index n equal 1/(2 Pi n^2). As a consequence, one finds that the
expected area of the region of index zero inside the loop is Pi/30; a quantity
that could not be computed directly using Yor's index description.

**Giardina, Christian**

**Factorization properties for the quenched measure of spin-glasses **

The expression for the free energy of the Sherrington-Kirkpatrick model, first obtained by G. Parisi, has been recently rigorously proven in the works of F. Guerra and M. Talagrand. Still many issues remains open to the mathematical analysis in the Parisi solution of the model. One of the main content of the theory is that the order paratemeter is a function on the unit interval, which is believed to describe completely the quenched equilibrium state. The overlap distribution satisfies a general set of identities (ultrametricity) which allow for the factorization of the equilibrium state. We will show the proof for a subclass of these identities (the so called Ghirlanda-Guerra identities) and discuss possible developments.

**Guionnet, Alice**

**Random matrices and combinatorics **

Large random matrices appear in different fields of physics and mathematics such as combinatorics, probability theory, statistics, number theory, operator algebra theory, quantum field theory etc In these lectures, we will focus on their relations with combinatorics within the scope to explore some of their uses in physics to model statistical models on random graphs. We will then discuss how such models can be studied thanks to their representations by matrix integrals.

**Khoruzhenko, Boris
**Joint work with Yan Fyodorov

**On absolute moments of characteristic polynomials of a
certain class of complex random matrices**

Let A be a fixed square matrix of size n and U be a unitary matrix picked up at random from the unitary group U(n). We express the integer moments of the spectral determinant |det(zI-AU)|^2 in terms of the characteristic polynomial of the matrix AA^*. Our work is motivated by studies of complex eigenvalues of random matrices and potential applications of the obtained results in this context will be discussed. We shall also discuss links between our work, Kaneko's generalization of the Selberg integral and Zirnbauer's color-flavor transformation.

**König, Wolfgang, Universität Leipzig**

**Probabilistic approaches to large Boson systems**

Consider a large system of $N$ Brownian motions with some non-degenerate initial measure on some fixed time interval $[0,\beta]$ with symmetrised initial-terminal condition. That is, for any $i$, the terminal location of the $i$-th motion is affixed to the initial point of the $\sigma(i)$-th motion, where $\sigma$ is a uniformly distributed random permutation of $1,\dots,N$. Such systems play an important role in quantum physics in the description of Boson systems at positive temperature $1/\beta$.

In work in progress (joint with Stefan Adams, Leipzig), we describe the large-$N$ behavior of the empirical path measure (the mean of the Dirac measures in the $N$ paths) and of the mean of the normalised occupation measures of the $N$ motions in terms of large deviations principles. The rate functions are given as variation formulas involving certain entropies and Fechel-Legendre transforms.

In a special case related to quantum physics, our rate function for the occupation measures turns out to be equal to the well-known Donsker-Varadhan rate function for the occupation measures of one motion in the limit of diverging time. This enables us to prove a simple formula for the large-$N$ asymptotics of the symmetrised trace of ${\rm e}^{-\beta \Hcal_N}$, where $\Hcal_N$ is an $N$-particle Hamilton operator in a trap. This is a preliminary step in the analysis of large Boson systems at positive temperature. Future goals concern the mutually repellent case, but this seems to lie far in the future.

As open problems that could be handled relatively soon, I propose to explore the cycle structure in the expectation over the permutations, to look a bit closer at the zero-temperature limit for the Boson system, and to attack the interacting case in the (hopefully, easier) case of random walks on a standard lattice.

**Kösters, Holger **

**Limit Theorems for the Diluted SK-Model **

In the last few years, various diluted versions of the Sherrington-Kirkpatrick spin glass model have been investigated by several authors. We consider a further version in this direction, namely the diluted SK-model on a Bernoulli random graph.

**Kuelske, Christof **

**On semigroups acting on rate-functions of mean-field systems **

We discuss semigroups acting on general mean-field spin systems. The latter are defined in terms of a Hamiltonian that depends on the empirical distribution. One example of such a multiplicative semigroup with parameter $r$ is obtained from the sampling of an extensive number $r N$ of spins from the original system. The action on the measure can be lifted to the level of rate-functions. While always well defined, it can lead to the loss of differentiability of the rate-function. We discuss connections between this phenomenon and the study of non-Gibbsian measures. Finally we describe new questions about Kac-limits coming up from this connection.

**Leveque, Olievier
**Joint work with Emmanuel Preissmann and Ayfer Ozgur.

**Determinants of random Cauchy matrices and capacity of wireless networks**

The development of wireless communications in the 90's has led to the
emergence of decentralized and self-organized wireless networks, that do not
require any fixed infrastructure in order to operate. In 2000, it has been shown
by P. Gupta and P. R. Kumar that under some realistic assumptions regarding
state of the art wireless communications, the capacity of such networks only
scales with the square root of the number of users, questioning therefore their
feasibilty on a large scale. With the evolution of technology, there is however
no reason to believe that the above mentioned realistic assumptions will still
be realistic in 10, 20 or 30 years. One should therefore try to obtain
information theoretic scaling laws, that do not rely on any particular
assumption on the way communication is established in the network.

In this talk, I will present an information theoretic approach to the problem,
that leads to the study of random Cauchy matrices. A precise estimate on the
determinant of such matrices leads to a new scaling law on the capacity of
one-dimensional networks (i.e., networks with users randomly distributed on a
straight line). For the more realistic two-dimensional case, one recovers a
scaling law close to Gupta-Kumar's square root law; this involves a reduction to
the one-dimensional problem, as well as the use of large deviations techniques.

**Maida, Mylene
**Joint work with J. Najim ans S.Pčchč.

**How to deal with outliers ?**

Our initial motivation for this work was to deal with problems similar to what Sandrine will explain in her talk on Wednesday, namely how adding finite rank perturbations can affect the largest eigenvalues of matrices from known ensembles, but in the real case.This leads us to large deviations questions for weighted empirical means with outliers and I will try to explain under which conditions we can understand the influence of these outliers on the rate function and also how it can help - or not -- answering the original question

**Merkl, Franz**

**Reinforced random walks: recent results and open problems**

Reinforced random walks are a class of non-Markovian random walks. Linearly edge-reinforced random walks show an intriguing structure, partly because they can be represented as random walks in a time-independent random environment, having a Gibbsian dependence structure.

In the talk, recent results on the theory of linearly edge-reinforced random walks will be reviewd, and the most important open questions in the field will be presented.

**Najim, Jamal**

**Large Deviations for the Trace of powers of a Wigner Matrix.
Exposition of the problem and a few suggestions to address special cases.**

In this open problem session, we will discuss the Large Deviations of $$
1/n^k Trace(X_n^k) $$

where k is fixed and X_n is a n*n matrix with Bernoulli i.i.d. entries (up to
the symmetry constraint). We will focus on the first non-trivial case (k=3). We
will try to raise some key issues as well as to compare this problem with other
well-known LDPs.

**Péché, Sandrine**

**The largest eigenvalue of small rank deformation of large Wigner matrices**

In this talk we will give some universality results for the fluctuations of
the largest eigenvalue of some non necessarily Gaussian complex Deformed Wigner
Ensembles. A $N\times N$ Deformed Wigner matrix can be written as a sum of a
standard Wigner matrix and a deterministic matrix, which we call the deformation.
The deformation is here of fixed rank, independent of $N$.

We first recall some preliminary results for the limiting behavior of the
largest eigenvalue of Gaussian Deformed ensembles, as $N$ goes to infinity. Then,
we extend these results to non Gaussian Deformed Ensembles, for some particular
type of deformation. This result is based on an approach due to A. Soshnikov. We
will finally explain that the type of the deformation might affect universality
results.

**Praehofer, Michael**

**Determinantal processes, random matrices, and nonequilibrium growth **

We give a basic introduction to determinantal point processes and explain the fermionic language often encountered in the more physical literature. The rest of the minicourse is devoted to applications in random matrix theory and in models for nonequilibrium crystal growth.

**Rolles, Silke
**Joint work with Franz Merkl

**Linearly edge-reinforced random walk**

Edge-reinforced random walk on any finite graph has the same distribution as a random walk in a random environment, where the environment is given by random, but time-independent weights on the edges. This result is well-known. In the talk I will present the result for arbitrary infinite graphs.The proof relies on a comparison with Polya urns. This comparison yields some bounds for the random environment.

**Stolz, Michael**

**Random matrices, symmetric spaces and mesoscopic physics**

In his seminal paper on the ``threefold way'', Dyson classified the spaces of
matrices that support the random matrix ensembles deemed relevant from the point
of view of classical quantum mechanics. Recently, Heinzner, Huckleberry, and
Zirnbauer have obtained a similar classification based on less restrictive
assumptions, thus taking care of the needs of modern mesoscopic physics.

Their list is in one-to-one correspondence with the infinite families of
Riemannian symmetric spaces as classified by Cartan.

The present talk, which is based on joint work with Peter Eichelsbacher
(University of Bochum), discusses the corresponding random matrix theories, thus
placing the Wigner-Dyson ensembles, the chiral ensembles, and the Bogoliubov-de
Gennes (superconductor) ensembles in a unified framework. Specifically, large
deviations of the empirical eigenvalue measures of these ensembles are studied
in the spirit of the well-known work of Ben Arous, Guionnet, Hiai, Petz, and
others.

**Vanlessen, Maarten**

**Universality for Laguerre type ensembles at the hard edge of the spectrum
**

Consider the Laguerre-type weights $|x|^\alpha e^{-Q(x)}$ on $[0,\infty)$ where $Q$ denotes a polynomial with positive leading coefficient. In this talk I will show that the local eigenvalue correlations of random matrices associated to these Laguerre-type weights have universal behavior (when the size of the matrices goes to infinity) at the hard of the spectrum. To get this result one needs the asymptotics of the corresponding orthogonal polynomials.

**Vermet, Franck**

**The capacity of q-state Potts neural networks with parallel retrieval
dynamics **

We define a Potts version of neural networks with q states. We give upper and lower bounds for the storage capacity of this model of associative memory in the sense of exact retrieval of the stored information. The critical capacity is of the order c N / log(N) where N is the number of neurons and the constant c increases quadratically with q.

*Last modified:
24-02-09
Maintained by Lucienne Coolen*