The minimal energy of the nonrelativistic one-electron Pauli-Fierz model within the class of quasifree states is studied. It is shown that this minimum is unchanged if one restricts the variation to pure quasifree states, which simplifies the variational problem considerably. Existence and uniqueness of minimizers are established under the…

I will discuss a general strategy to derive Hartree's theory for the ground state of a generic interacting many-bosons system with mean-field scaling. The validity of the mean-field approximation is interpreted as a consequence of the structure of the set of bosonic density matrices with large number of particles, in particular of the so-called…

The classical Shannon-McMillan theorem states that an ergodic system has typical sets satisfying the asymptotic equipartition property. This theorem demonstrates the significance of entropy which gives the size of the typical sets. There has recently been great progress in the quantum version of the Shannon-McMillan theorem .In particular,…

l also show how methods from statistical mechanics can be used to prove the full scale invariance of this QFT. (Joint work with Abdelmalek Abdesselam and Gianluca Guadagni.)

We consider a two-dimensional Ising model with nearest neighbor ferromagnetic and long-range, power-law decaying, antiferromagnetic interactions. We assume that the decay exponent of the antiferromagnetic part is larger than 4. The ground state displays a transition as the strength of the ferromagnetic interaction J is increased: if J is smaller…

In this talk, we will discuss the role of parafermionic observables in the study of several planar statistical physics models. These objects have been introduced recently by Smirnov and Cardy and have been instrumental in Smirnov's proof of conformal invariance of the Ising model. We will explain how they can be combined with combinatorial and…

We investigate a non solvable two-dimensional ferromagnetic Ising model with nearest neighbor plus weak finite range interactions. We rigorously establish one of the predictions of Conformal Field Theory (CFT), namely the fact that at the critical temperature the finite size corrections to the free energy are universal. The corresponding…

We consider a random two-dimensional surface satisfying a Lipschitz constraint. The surface is uniformly chosen from the set of all real-valued Lipschitz functions on a two-dimensional discrete torus. Our main result is that the surface delocalizes, having fluctuations whose variance is at least logarithmic in the size of the torus. Our technique…

I give a summary of recent progress in establishing the diffusion approximation for random band matrices. We obtain a rigorous derivation of the diffusion profile in the regime W > N^{4/5}, where W is the band width and N the dimension of the matrix. As a corollary, we prove complete delocalization of the eigenvectors. Our proof is based on a new…

We study the effects of random scatterers on the ground state of the one-dimensional Lieb-Liniger model of interacting bosons in the Gross-Pitaevskii regime. We prove that Bose Einstein condensation survives even a strong random potential, but the character of the wave function of the condensate depends in an essential way on the interplay…

Quantum information functionals and their algebraic properties play an important role in quantum information theory. Recent developments in the theory of operator monotone and operator convex functions and related topics have simplified earlier results and also led to new insights. One example is the convexity of chi-square divergences. We…

Since the days of Chernoff, the hypothesis testing has played an important role in theoretical and applied statistics. In the last decade, the mathematical structure and basic results of classical hypothesis testing have been extended to the non-commutative setting and are an important branch of quantum information theory. It was recently…

I'll describe a new link between supersymmetric gauge theories, with gauge group G, and the Yangian for the Lie algebra of G. The main result is that a certain twisted, deformed N=1 supersymmetric gauge theory, with gauge group G, is "controlled" by the Yangian built from G, in the same way that Chern-Simons theory is controlled by the quantum…