The Institute for Theoretical Physics offers the following seminars:
This talk describes recent joint work with Maurice Duits dealing with the fluctuations of the random empirical measure
for general orthogonal polynomial ensembles, on all scales, for both varying and fixed measures.
We obtain a general formula for the cumulants of linear statistics and use this to prove Laws of Large Numbers
(on both global and local scales) and Central Limit Theorems under fairly weak assumptions on the ensemble.
An important role in the analysis is played by notions arising in the spectral theory of orthogonal polynomials.
In the case of the Law of Large Numbers, this role is played by the classical Nevai condition.
In the case of the Central Limit Theorem, right limits take center stage.
ETH Science City HIT E 41.1 - Tue 11.02.2014 11:00
The notion of an orthogonal polynomial ensemble generalizes many important point processes arising in random matrix theory, probability and combinatorics. The most famous example perhaps is that of the eigenvalue distributions of unitary invariant ensembles (such as GUE) of random matrix theory. Remarkably, the study of fluctuations of these point processes is intimately connected to the study of one dimensional discrete Schroedinger operators. This talk will review recent work (partly joint with Maurice Duits) elucidating and exploiting this connection in the context of universality, laws of large numbers and central limit theorems.
ETH Science City HIT E 41.1 - Mon 10.02.2014 11:00
I will introduce the theory of quantum criticality in condensed matter and apply it to phase diagrams of high temperature superconductors. I will show how it leads to a description of the mysterious `pseudogap’ region in terms of the angular fluctuations of a multi-component order parameter, and give a comparison with numerous recent experiments.
ETH Science City HPV G 4 - Wed 18.12.2013 16:45
ETH Science City HIT F 41.1 - Tue 17.12.2013 11:00
The phenomenon of nucleation is associated with the non-equilibrium first-order phase transitions transforming a metastable parent phase to a thermodynamically stable daughter phase. The transformation proceeds through the creation of small clusters of molecules (nuclei) of the daughter phase out of the parent phase by thermal fluctuations [ref]. Recent advances in nucleation experiments made it possible to reach nucleation rates as high as 1016-1018 cm-3s-1. Such rates usually correspond to extremely small critical nuclei – containing only about 10 to 50 molecules. These nano-sized fractal-like objects can not be adequately treated within the purely phenomenological models (Classical Nucleation Theory and its modifications) based on the so called capillarity approximation. The latter assumes that a cluster is a sufficiently big spherical object with a homogeneous density and a rigid boundary; it also assumes that the cluster surface energy can be described in terms of the plain layer (macroscopic) interfacial tension. Obviously, for small clusters the concept of macroscopic interfacial tension loses its meaning and this assumption fails. A challenging task for a theoretician is to propose a nucleation model which treats clusters of all sizes on the same footing. In my lecture I will describe a realization of this program – the Mean-field Kinetic Nucleation Theory (MKNT). This model treats small clusters using statistical mechanical considerations and provides a smooth interpolation to the limit of big clusters obeying the capillarity approximation. Comparison of MKNT with experiment and computer simulations (molecular dynamics) for various microscopically diverse substances will be presented. Among various important features, the model leads to the Generalized Kelvin Equation signaling the pseudo-spinodal (as opposed to its classical analogue). I will briefly discuss the extension of these ideas to binary nucleation where the treatment of small clusters has to be complemented with the treatment of adsorption effects.
UZH Irchel Y 16 G 05 - Mon 16.12.2013 16:45-18:00