Title: Magnetization in the layered Ising model.

Abstract:
In recent years, a number of rigorous convergence results for the critical 
2D Ising model correlation functions has been established via a careful 
analysis of various boundary value problems for fermionic observables, which 
satisfy a version of the discrete Cauchy-Riemann relations at the critical 
temperature. Since similar linear relations hold true for arbitrary 
interaction constants, one can also use them to derive some information on 
the 2D Ising model in more general setups. Following this route, we present 
a new formula for the magnetization (average value of a particular spin) in 
the `layered' Ising model considered in the discrete half-plane (above, 
'layered' means that interaction constants depend on the distance to the 
boundary only). The answer is given in terms of truncated determinants of 
the square root of a simple Jacobi matrix constructed from a sequence of 
interaction constants, and leads to some natural conjectures on the decay 
of the magnetization at infinity depending on the choice of a sequence of 
coupling constants. Interestingly, this formula also gives an explicit answer 
at the criticality, which seems to have been unknown before. This is a joint 
work with Clement Hongler (EPFL, Lausanne).