Abstract: Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should in most circumstances be the same as in the random energy model. Here we give necessary conditions for this hypothesis to be true, which we show to hold in wide classes of examples: short range spin glasses and mean field spin glasses of the SK type. We also show that, under certain conditions, the conjecture holds even if energy levels that grow moderately with the volume of the system are considered. Finally, we show that the conjecture fails if levels near the extremes are considered, and show that a complex and rich structure emerges in this case in the example of Derrida's generalized random energy models.

Abstract: In the talk I formulate a set of conditons that lead to aging behaviour in so called Bouchaud's trap model on general graphs. If these conditions are satisfied one of the usual two-point functions considered for this model converges to the arc-sine law for Levy processes. I will then use these conditions to show the aging behaviour of the REM in some well chosen range of time-scales and temperatures.

Abstract: Finding the mean of the total number N_{tot} of critical points for N-dimensional random energy landscapes is reduced to averaging the absolute value of characteristic polynomial of the corresponding Hessian. For any finite N we provide the exact solution to the problem for a class of landscapes corresponding to the "toy model" of manifolds in random environment. For N >>1 our asymptotic analysis reveals a phase transition at some critical value \mu_c of a control parameter \mu from a phase with finite landscape complexity to the phase with vanishing complexity. The same value of the control parameter is known to correspond to an onset of glassy behaviour at zero temperature. Finally, we discuss a method of dealing with the modulus of the spectral determinant applicable to a broad class of problems. Reference: Physical Review Letters v. 92 (2004), 240601; Erratum: ibid. v.93 (2004),149901

Abstract: we present some recent advances on the localization/delocalization transition in a polymer model in the proximity of the interface between two solvents. The monomers carry a charge that determines their affinity for one or the other solvent. The charges are distributed along the chain in random (quenched) fashion. We will present some theoretical results on the phase diagram and on the scaling properties of the model. The region that for the moment is out of the reach of rigorous analysis is investigated in a numeric way. In this respect we will present in particular a statistical test for localization based on super--additivity and concentration bounds.

Abstract: We show how to build the probability cascades allowing to represent the free energy in Parisi spontaneous replica symmetry breaking, for mean field spin glasses. Then, we consider the problem of statistical distributions for the energy levels. Firstly, we take into account the random energy model, and its generalizations, where this problem can be explicitely solved. Then, we point out the difficulties for the usual Sherrington-Kirkpatrick model. The main result is that, contrary to the case of the random energy model, the probability cascades in the SK model are not directly connected with the statistical distribution of energy levels. Perspective of future developments are pointed out.

Abstract: We investigate the fluctuations of the overlpa parameter of the Hopfield model at criticality. We show that there is a non standard CLT with a random limit distribution provided the number of patterns is small enough. Moreover, we show a moderate deviations principle for a sufficiently small number of patterns "far away" from the CLT regime. Close to that regime the moderate deviations principle breaks down for any number of patterns larger than one.

Abstract: There are some models of mean fields glasses for which the basic theory is not complete. Our paper studies one of them. In this talk we consider a diluted version of the usual perceptron model.
- First we prove that the spin correlations vanish. In fact we obtain a more general property.
-Then we give the magnetization. Here the limiting law is a more complicated object than in the non-diluted perceptron model. One of the main tasks is to identify the order parameter of the model.
(This talk is based on a joint work with Carles Rovira from Universitat de Barcelona and Samy Tindel from Université de Nancy 1.)

Abstract: We present a theoretical framework for characterizing the geometrical structure of the space of solutions in constraints satisfaction problems, together with a practical algorithm for studying this structure on particular instances. We apply our method to the graph coloring problem, for which we obtain detailed informations such as the total number of solutions and the distribution of distances between solutions. (Work in collaboration with Marc Mezard and Olivier Rivoire, LPTMS Orsay).

Abstract: My aim in this talk is to discuss possible strategies for studying the occurrence of rare events. The background is the Ising model with Glauber dynamics. By looking at the system as a $d+1$ Gibbs state, with $d$ the space and time the extra dimension, after performing a suitable coarse graining transformation, the system is described by an effective hamiltonian at low temperatures, whose structure is determined by a large deviations rate function. The analysis of the minimizers of the effective hamiltonian under the appropriate boundary conditions and then a Pirogov-Sinai perturbative theory would then outline a possible strategy of proof. Supposing that the spin-spin interaction is given by a Kac potential, Comets has derived an explicit formula for the effective hamiltonian. In this talk I will discuss the minimizers of a non local functional which is a simplified version of Comet's large deviation rate function, thus accomplishing the second step of the above strategy. I will first report some recent results obtained by Bellettini, De Masi, Dirr and myself on tunnelling in a $d=2$ square of side $L$ for $L$ large enough. The ``penalty'' for tunnelling is explicitly found as the energy of a saddle point and optimizing sequences are characterized. It is found that the optimal tunnelling strategy is with functions having planar symmetry, so that for a while the orbit is macroscopically far from the Wulff shape. I will then discuss a result obtained recently in collaboration with De Masi and Dirr (it can be seen as a preprint of the Max Planck Institute in Leipzig, mis.mpg.de). In this case $d=1$ and we want to study the cost of moving by $r$ in a time $t$ an instanton (representing the stationary interface between the two equilibrium phases). In linear response theory the cost is given by $(V2/\mu) t$, $V=r/t$, $\mu$ a mobility coefficient. To investigate the question we notice that the cost is invariant under parabolic scaling and we study its limit value as $\eps\to 0$ after setting $r=\eps^{-1}R$ and $t=\eps^{-2}T$. We found that linear response theory is correct only if $V=R/T\le V_c$, $V_c$ a critical value. For larger values of $V$ it is more convenient to have nucleations ahead of the front, their number increasing with $V$.

Abstract: I will give an overview over recent work on linearly edge-reinforced random walks. On finite graphs, the process has the same distribution as a random walk in a random environment. This representation has been a powerful tool in analyzing the process on infinite graphs. Furthermore, it has applications in Bayesian statistics. The mentioned results were obtained in joined work with Persi Diaconis, Mike Keane, and Franz Merkl.

Abstract: The minority game is an idealisation of a stockmarket of speculators in which independent agents with no direct knowledge of one another's strategies make binary choices with the aim to be in the minority and learn from their success or failure. It has some analogies and some differences from infinite-ranged spin glasses or neural networks. It exhibits cooperative and scaling behaviour and an ergodic to non-ergodic phase transition. Using dynamical generating functional methods, and within the looser rigour of theoretical rather than mathematical physics, the Markovian microdynamics of the physical system of many independent agents interacting only through the common knowledge of real or artificial historical minority choices can be mapped into an ensemble of effective single agents subject to self-consistently determined coloured noise. Computer simulations of the original many-agent model, numerical iterations of the effective single agent system and ansatz-simplified analytic studies yield an ergodic to non-ergodic phase transition and varying degrees of comparability in appropriate macroscopic observables. In the ergodic region a mapping of the equilibrium behaviour to a spinglass-like combinatorial optimization problem has also been studied and shown be in good agreement with simulations in the ergodic region.

But several issues remain, certainly questioning the correctness of the extremal analysis employed and the need for the rigour of mathematical statistical physics such as has recently been applied successfully to the infinite-ranged spin glass, but also in the improvement of the analytic solution of the effective agent system, as well as several extensions of the simplest paradigm.

In this talk I shall review key aspects of the theoretical physics which exist to date and shall try to highlight some of the issues calling for the attention of mathematical rigour or even greater theoretical insight.

Abstract: I will consider a model of a random heterogeneous polymer in the proximity of an interface separating two selective solvents. This model exhibits a localization/delocalization transition. A positive value of the free energy corresponds to the localized regime and strong results on the polymer path behavior are known in this case. I will focus on the interior of the delocalized phase, which is characterized by the free energy equal to zero, and I will show in particular that in this regime there are O(log N) monomers in the unfavorable solvent (N is the length of the polymer). The previously known result was o(N). Our approach is based on concentration bounds on suitably restricted partition functions. The same idea allows also to interpolate between different types of disorder in the weak coupling limit. In this way we show the universal nature of this limit, previously considered only for binary disorder. (in collaboration with G. Giacomin)