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).