Nathan Hayford: The Ising model on a random planar lattice: exact genus zero free energy

Speaker
Nathan Hayford, KTH Royal Institut of Technology

Abstract
The 2D Ising model is one of the most celebrated examples of an exactly solvable lattice model. Motivated by problems in statistical mechanics and 2D quantum gravity, in 1986 Vladimir Kazakov considered the Ising model on a random planar lattice using techniques from random matrix theory. He was able to derive a formula for the free energy of this model, and made the first prediction of the Kniznik-Polyakov-Zamolodchikov (KPZ) formula for the shift of the critical exponents of a conformal field theory when coupled to quantum gravity. Unfortunately, his derivation was not mathematically rigorous, and the formula he obtained for the free energy was somewhat unwieldy. In this talk, I will review some of the details regarding both the Ising model and random matrices, and sketch a rigorous proof of Kazakov’s formula for the free energy. If time permits, we will also discuss the multicritical behavior of this model. This is joint work with Maurice Duits and Seung-Yeop Lee.