TBA

In this talk, we derive effective dynamics governed by PDEs (nonlinear Schrodinger equation, Vlasov-Poisson equation) from the N-body Schrödinger equation with interactions in the large N limit. For the derivation, we first visit quantum mechanics. Then, using the knowledge of many-body quantum mechanics given in the talk and with the suitable conditions for each situation, we derive PDEs describing the effective motion of the system.

The Discrete Gaussian model is a type of integer-valued random height function. In the 2D setting, it exhibits a phase transition between a localised phase and a delocalised phase. This phenomenon is also called the Kosterlitz-Thouless phase transition, whose terminology originates from its dual counterpart, the planar XY model. Motivation for studying the Discrete Gaussian model is multifold. Due to its duality relations with a number of 2D mathematical physics models, such as the XY model or the Coulomb gas, studies on integer-value height functions are capable of proving a number of conjectures usually not accessible using classical methods. Other discrete height functions also have dualities with a number of different interesting models, so it will be of vast interest to develop a general framework that deals with discrete height functions. Also, discrete height functions are considered to be appropriate test cases for recently developed techniques from probability theory. In this talk, we discuss a particular method called the renormalisation group method, which is believed to serve as a general framework for studying random fields. We also discuss briefly how the renormalisation group method can be used to prove that the scaling limit of the 2D Discrete Gaussian model is a 2D Gaussian free field.