Conformal Geometry Program Lecture Series: Dmitry Chelkak, Steklov Institute

Dmitry Chelkak (Steklov Institute (PDMI RAS) & Chebyshev Lab, St.Petersburg)

Title: Spin Correlations in the Planar Ising Model via Fermionic Observables. Part I.
Speaker: Dmitry Chelkak (Steklov Institute (PDMI RAS) & Chebyshev Lab, St.Petersburg)
Date: Wednesday, February 27, 2013
Time: 4:00pm – 5:30pm
Place: Seminar Room 313, Simons Center




Abstract: Based on recent joint projects with Clement Hongler and Konstantin Izyurov. The main goal of the minicourse is to discuss the rigorous proof of existence and conformal covariance of scaling limits of spin correlations in the critical Ising model in bounded planardomains.

We introduce the main tool – discrete holomorphic spinor (aka fermionic) observables, and show their relation to spatial logarithmic derivatives of spin-spin expectations in the Ising model. Essentially, this construction allows us to reformulate the analysis of spin-spin expectations on the language of discrete complex analysis. We then illustrate our approach considering the single spin expectations (aka magnetization) in the half-plane with `+’ boundary conditions: in this case, one can further reformulate the question on the language of orthogonal polynomials which leads to explicit formulae for those expectations in terms of some Toeplitz+Hankel determinants.