MATHEMATICAL PHYSICS SEMINAR

Fall 2007 Lectures

Regular meeting time: Tuesdays 4:30--5:30
Place: Jadwin 343

Date Speaker Title
Sept. 4 Raffaele Esposito,
University of Rome		 Uniqueness and stability of the front for a binary fluid
We consider a kinetic system of two species of particles interacting via a long range (Vlasov) repulsive force between particles of different species, and in contact with a thermal reservoir modeled by a Fokker-Plank operator. The system undergoes a phase transition at low temperature consisting in the separation of the two species. The study of the one dimensional, infinite volume front, interpolating between the equilibrium densities, is an important tool for the construction of multidimensional interfaces separating regions indifferent phases. We prove the uniqueness of the front solution via displacement convexity techniques and its stability with respect to a suitable class of initial perturbations by means on an energy method based on the linearization of the free energy.
Nov. 13 Ilya Gruzberg,
University of Chicago		 Critical curves in conformally invariant statistical systems
We consider conformally invariant curves that appear at critical points of two-dimensional statistical mechanical systems. We show how to describe these curves in terms of the Coulomb gas formalism of conformal field theory (CFT). We also provide links between this description and the Schramm-Loewner evolution (SLE). The connection appears in the long-time limit of stochastic evolution of various SLE observables related to CFT primary fields. We show how the multifractal spectrum of harmonic measure and other fractal characteristics of critical curves can be obtained

Note special date:
Thurs., Nov.
29		 Bertrand Duplantier,
Theoretical Physics , SACLAY		 Large Deviations and KPZ Relation in Quantum Gravity
KPZ (Knizhnik, Polyakov, Zamolodchikov) formula (1988) from conformal field theory is a well-known relation between critical exponents of statistical systems in the plane and the corresponding ones in quantum gravity, i.e., on a fluctuating lattice or random metric. After an introduction from the physical perspective, a probabilistic approach will be proposed. One considers critical Liouville quantum gravity measures of the form $\mu = e^h \lambda_D$, where $\lambda_D$ is Lebesgue measure on a bounded planar domain $D$ and $h$ is a multiple of the Gaussian free field. Given a random subset $X$ of $D$ one can define the fractal scaling dimension of $X$ using either the Euclidean measure or the quantum metric derived from $\mu$. A large deviations principle allows to derive a general quadratic relation between these two numbers, which can be viewed as a probabilistic formulation of the KPZ relation. (Work in progress with Scott Sheffield.)
Dec. 4 Alessandro Pizzo,
ETH, Switzerland		 Solution of the infrared catastrophe problem in non-relativistic QED
Within the framework of non-relativistic QED, we construct the scattering states of an electron interacting with the quantized electromagnetic field. The generic scattering state \psi_{h,\kappa}^{out/in} represents an electron with a wave function h in the momentum variable, with support in a region corresponding to small (asymptotic) velocities, accompanied by a cloud of real photons described by a Bloch-Nordsieck factor, and with an upper photon frequency cutoff \kappa. This is a joint work with T. Chen and J. Froehlich.