Abstract: The Ising Model is a classical model in statistical physics describing the behavior of ferromagnetic moments on a lattice interacting via a local interaction. When the lattice is one-dimensional and in the case of homogeneous nearest-neighbor interaction, the model is known to be exactly solvable (and simple).
However, the disordered version of the one-dimensional Ising Model (called the Random Field Ising Chain), where the chain interacts with an i.i.d environment, is a much more challenging model. In particular, it exhibits a pseudo-phase transition as the strength Gamma of the inner-interaction goes to infinity. A description of the typical configurations when Gamma is large has been given in the physical literature in terms of a renormalisation group fixed point.
In this talk, we will present and discuss the RFIC model, on the level of the free energy and on the level of configurations. We will consider the cases of both centered and uncentered external fields. The notion of Gamma-extrema of the Brownian motion, introduced by Neveu and Pitman, will play a crucial role in our analysis.