Asymptotic Properties Of Disordered Systems

This thesis considers asymptotic behaviors of high-dimensional disordered systems, including Ising model and mean-field spin glass models. We first study the decay rate of correlations in the two-dimensional random field Ising model (RFIM). Second, we study the limit free energy of disordered systems. For RFIM, we are interested in the two-dimensional case where the external field is of i.i.d centered Gaussian variables. We show that under nonnegative temperature, the effect of boundary conditions on the magnetization in a finite box decays exponentially in the side length of the box. On the side of mean-field models, we use the Hamilton-Jacobi equation (HJE) approach, initiated by Jean-Christophe Mourrat, to characterize limiting free energy in many models from statistical inference problems and mean-field spin glass models. We now investigate infinite-dimensional models including many spin glass models and inference problems where the rank of the signal matrix increases as $n$ is sent to infinity. We give an intrinsic meaning to the Hamilton--Jacobi equation arising from mean-field spin glass models in the viscosity sense, and establish the corresponding well-posedness.This will shed more light on the mysterious Parisi formula as the limit of free energy in the Sherrington--Kirkpatrick model.