Extended States in a Lifshitz Tail Regime for Random Operators on Trees
We will discuss the spectral properties of random operators on regular tree graphs. The models have have been among the earliest studied for Anderson localization, and they continue to attract attention because of analogies with localization issues for many particles. The talk will focus on the location of the mobility edge. Somewhat surprisingly, a resonance mechanism will be proven to cause the appearance of absolutely continuous spectrum in a regime extending well beyond the energy band of the operator's non-random hopping term. For weak disorder, this includes a Lifshitz tail regime of very low density of states.

Oct. 26

The energy density field of the Ising model
We consider the planar Ising model on bounded domains from a conformal invariance point of view. We are interested in the scaling limit of the model at critical temperature. Physics theories, notably Conformal Field Theory, predict the existence of two conformal local fields describing the model in the continuum: the spin and the energy density. We have recently proved the conjectured formulae for the energy field on bounded domains, with an improved precision, using discrete complex analysis techniques, thanks to the introduction of holomorphic spinors, that exhibit a fermionic structure.

We relate the correlation functions of the energy to special values of the spinors, and prove convergence of the latter to continuous holomorphic spinors, giving scaling formulae for the correlation functions. Partly based on joint work with Stas Smirnov.