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Workshop
Order, disorder, and double disorder Abstracts Louis-Pierre Arguin (Courant Institute New York University) Recent Progess on the Ultrametricity in Spin Glasses We will review recent results concerning the question of the ultrametricity of the Gibbs measure which is conjectured in the Parisi theory. We will first introduce the notion of Random Overlap Structures (ROSt) to study the Gibbs measure. We will then discuss the relation between the properties of the ROSt such as the Stochastic Stability and the Ghirlanda- Guerra identities and the ultrametricity of the measure. Noam Berger (Hebrew University of Jerusalem) Local limit theorems for random walk in random environment this talk I will present the result of my efforts for > proving a local CLT for ballistic RWRE, namely that under suitable ballisticity conditions, for almost every environment and every > $\alpha>0$, the probability of hitting a cube of size $k^\alpha$ which > is at distance $k$ from the origin is very close to the value given by > the Wiener distribution. > I will also present some challenges that are still open. Erwin Bolthausen (Zürich University) On the TAP equations, and a Morita type derivation of the replica symmetric solution in spin glasses The so-called replica symmetric solution for the free energy of the SK-model (high temperature, non-zero external field) is now mathematically well understood, and several different proofs exist, mainly by Talagrand, and Guerra-Toninelli. We still present another approach which is not simpler than the existing ones, but which hopefully sheds some new light on the formula. It is based on an analysis of the TAP equations, and a Morita type correction. The method avoids any interpolation technique, and also any induction on N. Anton Bovier (Institut für Angewandte
Mathematik Rheinische Friedrich-Wilhelms-Universität Bonn)
Metastability in the Random Field Curie-Weiss model A classical idea in the theory of metastability is the replacement of the long-term dynamics of a high-dimensional system by an effective diffusion process in a one-dimensional effective potential. In simple mean- field models this can be made rigorous by exploiting symmetries. The random field Curie-Weiss model is one of the simplest models where no such symmetries are present. We use a coarse graining proceedure and variational methods to compute precise asymptotics of metastable exit times which are seen to differ by a multiplicative constant from what if given by the naive one- dimensional approximation.
Nina
Gantert (University
of Münster) Biased random walks on percolation clusters of trees We consider a supercritical Galton-Watson tree, conditioned on survival (in particular, this includes the case of the infinite percolation cluster of a regular tree) and run a biased random walk on this tree. We investigate the distributions of the walker at time n and show that they are tight, but do in general not converge. Olle Häggstrom (Chalmers University Göteborg) Percolation beyond Z^d: the contribution of Oded Schramm Oded Schramm (1961-2008) had a crucial influence on the development of percolation beyond the usual Z^d setting, in particular the case of nonamenable lattices. In this talk I will review some of his work in this field. Remco van der Hofstad (Eindhoven University of
Technology) First passage percolation on random graphs We study the structure of minimal-weight paths in the configuration model with i.i.d. degrees with a fixed degree distribution. Here, each edge receives an independent exponential weight, leading to double randomness in the form of a stochastic process on a random graph. We consider the hopcount, which equals the number of edges of the minimal-weight path between two uniformly chosen vertices, and the weight of this minimal path. When the degrees obey a power law with degree power-law exponent \tau, then, whenever \tau>2, we see that the hopcount obeys a central limit theorem with asymptotically equal mean and variance proportional to \log{n}, where n is the size of the graph. A similar result was proved to hold on the complete graph, the difference being that the proportionality constant on the complete graph is equal to 1, whereas for the configuration model it is greater than 1 when \tau>3 and in (0,1) when \tau\in (2,3). This gives a remarkably universal picture for first passage percolation on random graphs. These results should be contrasted to the results when instead of i.i.d. exponential weights, each edge has a constant weight, so that the hopcount is equal to the graph distance on the graph. In the latter case, when \tau\in (2,3), the hopcount is asymptotically proportional to \log\log{n}, with uniformly bounded fluctuations. Thus, the addition of random edge weights has a marked effect on the structure of minimal-weight paths. We hope that these results shed light on the hopcount in Internet, which also obeys an asymptotic central limit theorem with roughly equal mean and variance. D. Ioffe (Technion - Israel
Institute of Technology) Finite connections for the super-critical Bernoulli bond percolation in 2D We derive sharp asymptotics of finite connections (connections via finite clusters) for the super-critical Bernoulli bond percolation in 2D. A well known analog of the resulting formula holds for the truncated two point function in the context of the exactly soluable 2D low temperature Ising model. Our approach relies on recent advances in the Ornstein Zernike theory which is built upon a stochastic geometric representation of long sub-critical clusters. The self-duality of the model plays, therefore, an essential role - the main contribution to the asymptotics comes from long open dual loop which encircle two distant directly connected points. Finn V. Jensen (Aalborg University) Probabilistic Graphical Models for Diagnosis and Decision Making Presentation 1 - 2 - 3 - 4 - 5 Nicola Kistler (Institut für Angewandte Mathematik Rheinische Friedrich-Wilhelms-Universität Bonn) A Perceptron version of the GREM, and some applications According to the variational principle of Aizenman, Sims and Starr, the Parisi Formula for the SK-model is obtained through a cavity field perturbation of the Derrida-Ruelle cascades. In a recent work with Erwin Bolthausen, we addressed the question of how these quantities arise in the thermodynamical limit. This naturally lead us to consider a Perceptron version of Derrida's GREM which shows a very intriguing behavior. In particular, for large but finite size, these systems organize in states that are all their own (state-dependent) random temperatures. In the thermodynamical limit, the random temperatures self-average around a deterministic value: the order parameter of the Parisi Theory. I will report on our findings. Anrea Montanari (Stanford University) Statistical mechanics on general graphs
Chuck Newman (New York
University) Metastates and Short-Range Spin Glasses (3 lectures) I. We introduce metastates as probability measures on the space of infinite-volume Gibbs states that may be used to study disordered systems with potentially many `competing' states. Extensions to metastates and excitation metastates for competing ground states may also be discussed. II. We discuss one or more partial results based on metastates, in the direction of proving uniqueness of infinite-volume ground state pairs for the two-dimensional Edwards-Anderson spin glass. III. We complete the discussion of lecture III (if necessary) and then present another partial result concerning the absence in any dimension of pure state decompositions into N states with N strictly between one and infinity. References for the overall series of lectures: 1. C.M. Newman, Topics in Disordered Systems, Birkhauser, 1997. 2. C.M. Newman and D.L. Stein, Are there incongruent ground states in 2D Edwards-Anderson spin glasses?, Commun. Math. Phys. 224 (2001), 205-208. 3. C.M. Newman and D.L. Stein, Metastates, translation ergodicity and simplicity of thermodynamic states in disordered systems: an illustration, submitted to Proceedings of the 2006 Inter. Congress of Mathematical Physics Vladas Sidoravicius (CWI Amsterdam and Rijksuniversiteit Leiden) Absorbing-State Phase Transition for Stochastic Sandpiles and Activated Random Walks We study long time behavior of one-dimensional conservative interacting particle systems on $\mathbb Z$. In particular, we consider the Activated Random Walk Model for reaction-diffusion systems, and Stochastic Sandpile Model. Our main result, based on Diaconis-Fulton representation of dynamics, is that the system locally fixates when the initial density of particles is small enough, establishing the existence of a non-trivial phase transition in the density parameter. This fact has been predicted by theoretical physics arguments and supported by numerical analysis. Floris Takens (Rijskuniversiteit Groningen) |
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