Abstract

This is survey on some probabilistic problems motivated by the strong coupling limit of the Fr\”{o}hlich polaron (after H. Fr\”{o}hlich 1952). This model describes the movement of a charged particle in a lattice with polarized nodes. In physics, the description is usually given in terms of quantum mechanics and field theory. Feynman in his lectures on Statistical Mechanics 1972 gave in a purely probabilistic description of some of the key quantities. In particular he predicted that the effective mass is proportional to $\alpha^{4$ where $\alpha$ is the coupling parameter. The effective mass then equals the inverse of the variance parameter of a 3-dimensional Brownian motion with an interaction which is long range in time, and given by an attractive Coulomb force in space. The model is in essence a delicate continuous time statistical model with a singular Kac type interaction. The latter means that there is a small Kac parameter $\lambda$, and for $\lambda \rightarrow 0$, the interaction becomes weak but long range in time, so that it should become in the limit a mean-field type model. A key feature is that there is a continuous symmetry in space. In the polaron, $\lambda $ equals $\alpha} {-2}.$ Based on this translation, there is a short and very appealing, but far from rigorous, old argument by Spohn 1987 that predicts the correct asymptotic behavior. The result itself has, after many efforts, recently been proved by two groups: Bazaes, Mukherjee, Varadhan, Sellke, and Brooks, Seiringer, the latter in the quantum mechanical setup. The first group uses a delicate expansion of the Brownian interaction, which bypasses the Kac-picture advocated by Spohn. The latter addresses a situation that is much more general than the one appearing in the polaron, and there still remain interesting open problems on which there is some recent progress together with Amir Dembo.