The Random Field Ising Chain in the case of centered disorder

Orphée Collin

Our understanding of the behaviors appearing in statistical physics has been built firstly on the well-known Ising Model. Here, we will consider a disordered version of the one-dimensional Ising model: we will present and study the ferromagnetic Ising model on a line graph interacting with an external magnetic field, sampled from an i.i.d. distribution. We will be interested in the regime where the intensity of the disorder is fixed and the spin-spin interaction goes to infinity.

We will also introduce a continuous version of the model, which naturally arises from a weak disorder limit. For this continuous model, various quantities (such as the free energy) can be computed explicitely, thus yielding precise information on the typical configurations of the system.

The free energy of the discrete model can easily be expressed as the Lyapunov exponent of a random product of 2x2 matrices, which we estimate using Furstenberg’s theory: we will present recent results on the asymptotics of the free energy, in the regime we consider.

Furthermore, in both the discrete and the continuous models, we will caracterise the behaviour of the system at the level of configurations. In agreement with predictions in the physics literature, we will show that the configurations are typically close to one given configuration, determined by the environment (the external field), thus showing that the disorder is strongly relevant.

Our discussion will concern mainly the critical case, i.e., the case where the disorder is centered, but we may also address the non-critical case.

 All talks  Summer '25: Spin systems and phases of matter