Schedule of the Workshop

Elie Aïdékon: Behaviour of the critical martingale in the branching random walk
We look at the critical martingale W_n in the branching random walk. It is known that W_n converges almost surely to zero. Using spine techniques, we prove that after a suitable renormalization, W_n converges in probability to a non-trivial limit. This is joint work with Zhan Shi.

Louis-Pierre Arguin: Some Properties of the Gibbs measure of Short-Range Spin Glasses
Physicists have conjectured that the Gibbs measure of spin glasses must exhibit a rich structure at low temperature (infinite number of states, ultrametric distance between states, ...). The purported structure for most Spin Glass Models is still elusive from a rigorous standpoint. Results of Panchenko, Talagrand and Arguin & Aizenman using general properties of the measure such as the stochastic stability and the Ghirlanda-Guerra identities has shed some light on the conjectures. The question of whether the sophisticate structure also holds for models with short-range interactions is wide open even from a heuristic point of view. In this talk, I will revisit the general properties of the Gibbs measure for the Edward-Anderson model, a short-range spin glass. We will see how the Gibbs measure is more restricted in this case compared to the Sherrington-Kirkpatrick model, its short-range counterpart. This is joint work with M. Damron. Slides

Alessandra Bianchi: Wavelet analysis of long-range dependent traffic: asymptotic normality of log-regression estimators of long memory
The network traffic is characterized by self-similarity and long-range correlations on various time-scales. The memory parameter of a related time series is thus a key quantity in order to predict and control the traffic flow.
In this talk we will present a wavelet-based analysis of the long memory behavior. After reviewing some of the main ideas and results concerning wavelet methods, we will analyze the efficiency of the log-regression wavelet estimator of the memory parameter.
Under some Gaussian assumptions on the discrete wavelet transform, we will show that this estimator is asymptotically normal in the size of the sample, and compute its asymptotic variance. Joint work with M. Campanino and I. Crimaldi.

Jirí Cerný: Complexity of spherical spin glasses and random matrices
The present rigorous understanding of the energy landscape of mean-field spin glasses is rather fragmentary. In order to consider physically relevant dynamics, one needs an information about the "structure" of its critical points, that is minima and saddle points of various indices.
In my talk I will present the first step in this direction [1], relating the mean number of critical points of the Hamiltonian of spherical spin glasses with the spectrum of GOE random matrices. This relation allows for rather precise asymptotic evaluation of the mean number of critical points. Slides
[1] Auffinger, Ben Arous, Cerný: Random matrices and complexity of spin glasses arXiv:1003.1129

Yan Fyodorov: Freezing transition in statistical mechanics with logarithmic correlations: 1/f noises, extreme values, and Burgers turbulence
The covariance of the Gaussian random free field in dimension D=2 depends logarithmically on the distance. Our goal is to to study the Boltzmann-Gibbs measures induced by the Gaussian fields with logarithmic correlations in the space of any dimension D. After shortly discussing the mean-field (D=infinity) case we will mainly concentrate on D=1 case where the corresponding random signals are the instances of the the so-called 1/f noise.
We will show how many explicit results can be obtained by heuristic methods of theoretical physics, including the (in)famous replica trick.
In particular, we will discuss multifractality of the Boltzmann-Gibbs measures and the phase transition ("freezing") which occurs in this type of models with decreasing temperature. We will conjecture what such a freezing should imply for the distribution of the free energies, and suggest a relation of freezing to the curious "duality relations" which such distributions show in the high-temperature phase. This will allow us to extract the explicit form of the distribution of the minimum of various regularized versions of 1/f noises. Such distributions are manifestly non-Gumbel and display a universal tail.
Finally, we will investigate a closely related problem of the so-called "Burgers turbulence", that is the solution of the Burgers equation with logarithmically-correlated random initial conditions. We will argue that the freezing transition should also happen in that model and will manifest itself via the formation of a non-Gaussian probability density of local velocities below some finite critical viscosity. The corresponding dualities are related to recently discovered relations in the context of theory of random matrices.
The lectures will be based on joint works with J. P. Bouchaud, P. Le Doussal and A. Rosso.
Slides-I, Slides-II, Slides-III, Slides-IV

Cristian Giardinà: Ferromagnetic models on random networks
We will present results for ferromagnetic spin system on random graphs. These disordered models are "simple" from the physical point of view, showing a second order ferromagnetic phase transition described by a replica-symmetric solution. Nevertheless, they pose interesting questions from the mathematical perspective.
In the first part of the talk (joint work with S. Dommers and R. van der Hosftad) we will review the formula for the free energy and show that the recent rigorous proof of Dembo and Montanari does apply also to the context of generic power law degree distribution random graph. In particular, the analysis includes graphs with finite mean degree and infinite variance.
In the second part (joint work with A. Bianchi, P.Contucci, S.Starr) we will discuss open problems.
In the context of the Erdos-Renyi random graph, we will discuss the property of monotonicity which is still unproved. Slides

Onur Gün: Universality of transient dynamics and ageing for spin glasses
We consider Random Hopping Time (RHT) dynamics of mean field spin glass models that can be seen as a (random) time change of the simple random walk on the state space. The simplest mean field spin glass is the Random Energy Model (REM) where energy landscape has no correlation structure. We study the dynamics of REM and prove that under a proper normalization the clock process (time change process) converges to an extremal process and the system exhibits ageing like behavior. Finally, we prove that the same is true for more complex models Sherrington - Kirkpatrick spin glass and p-spin models. Slides

Zakhar Kabluchko: Extremes of independent Gaussian processes
A stochastic process is called max-stable if the maximum of any finite number of independent copies of the process, taken pointwise, has the same distribution as the process itself up to an affine transformation. Max-stable processes appear as limits of maxima of n i.i.d. stochastic processes as n goes to infinity. In this talk we will be interested in a particular case of this setting: we will consider max-stable processes appearing as limits of maxima of independent Gaussian processes. The class of limiting processes will be completely described and its properties will be discussed. The limiting processes are related to a certain class of systems of "competing" particles. The starting positions of the particles are chosen according to a Poisson point process with intensity e^-x, and then the particles move independently according to the law of a fractional Brownian motion with some special negative drift. The position of the "leading" particle in such a system is a max-stable stochastic process. We will show that this process is stationary and appears as one observes a large number of stationary Gaussian processes near their extremes. We also discuss α-stable processes related to the particle systems mentioned above and show that they appear as limits of sums of independent geometric Brownian motions. This latter problem is motivated by the Random Energy Model.

Nicolà Kistler: Extreme Values, Branching Brownian Motion & Travelling Waves
Poisson Point Processes (and mixtures thereof) are ubiquitous in Extreme Value Theory. Over the last years, however, strong evidence has gathered suggesting that certain models "at criticality", such as the Directed Polymer on Cayley Trees, the 2-dimensional Gaussian Free Field, and certain Spin Glasses, may belong to a new universality class, that of Branching Brownian Motion. The latter is also intimately related to the issue of travelling waves in certain non-linear parabolic PDE’s. Despite enormous research over the last decades, the extremal process of Branching Brownian Motion remains however to these days unknown. I will give a brief account of the subject, and present some recent results obtained in collaboration with L.-P. Arguin and A. Bovier. Slides

Anton Klymovskiy: High-dimensional Gaussian fields with isotropic increments
We consider an arbitrary Gaussian field with isotropic increments indexed by a high-dimensional Euclidean space. How do the maxima of such field over the Euclidean balls of exponentially growing volume behave, as the dimension goes to infinity? We answer this question to the leading-order precision with respect to the dimension. It turns out that the extreme value behavior strongly depends on the range of correlations of the Gaussian field. In this context, three universality classes can naturally be singled out: short- (power or exponential decay), critical- (logarithmic growth) and long-range (power growth) correlations. Relation to Derrida s (generalized) random energy model and the so-called hierarchical replica symmetry breaking will be pointed out. Slides

Noemi Sibylle Kurt: Laplacian random interface models - maximum and entropic repulsion
We consider a model for a random interface, or semiflexible membrane, which is defined as a (Gaussian) random field field on the d-dimensional integer lattice, with a Hamiltonian depending on the discrete Laplacian. We will compare this model to the well-known gradient interface model, or discrete Gaussian free field, and explain the differences in the approach of the Laplacian model in the critical and supercritical dimensions. Our main result is to prove a repulsive effect of a "forbidden region" on the interface. We will show that the constraint for the field to take only positive values in a domain of side-length N results in an effect of "entropic repulsion". In the critical dimension, the height of the repelled field is determined by the maximum of the unconditioned field. Slides

Marcel Ortgiese: The critical window for polymers on disordered trees
We consider a model of a polymer interacting with a random environment. Instead of living on a lattice, here the polymer is a path in a regular tree. As the strength of interaction is increased, one can observe a phase transition. We ll concentrate first on the weak disorder regime and show which polymers are relevant for the free energy. Finally, we ll move on to the critical regime and in particular discuss what happens when we introduce a small perturbation of the critical temperature parameter.

Shannon Starr: Antiferromagnet on the Erdos-Renyi Random Graph
The Erdos-Renyi random graph is locally tree like, but with long loops. For the antiferromagnet, unlike for the ferromagnet, these loops may induce frustration and affect the phase structure at low temperatures. In particular, Krzakala and Zdeborova have argued that the antiferromagnet on a random graph behaves like a spin glass. We show how to use monotonicity to implement the interpolation scheme of Guerra and Toninelli for this model, and we state the extended variational principle. Then, using the Derrida-Ruelle random probability cascade, we discuss the replica symmetric solution and a replica symmetry breaking ansatz. This is joint work with Pierluigi Contucci, Sander Dommers and Cristian Giardina. Slides

Olivier Zindy: The Aldous conjecture on a killed branching random walk
We consider a branching random walk on R with an killing barrier at zero: starting from a nonnegative x, particles reproduce and move independently according to a certain point process, but are killed when they touch the negative half-line. In both critical and subcritical cases, the population dies out almost surely. We give the exact tail distribution of the total progeny of the killed branching random walk, which solves an open problem of D. Aldous. This is a joint work with Elie Aïdékon and Yueyun Hu.