Multi-scale analysis of mass-less random fields with non-convex interaction energies - a Renormalisation group approach

Outline and aims: The overall theme is interacting particle systems, their critical phenomena and scaling limits. The novelty is to develop new multi-scale methods to tackle mass-less random fields with non-convex energies. The primary focus is on mass-less bosonic random fields. The aim is to better understand critical phenomena for non-convex energies, and then to analyse applications to material science with non-convex free energies, random surfaces (interfaces), and scaling limits to novel continuum objects as well as space-time functional integral representations of bosonic systems. The idea is to go beyond the continuum Gaussian Free Field which has seen results in the last decade. The objectives are to analyse the (a) infinite volume Gibbs measures, (b) the scaling limits (continuum), (c) thermodynamic limits of the free energies for various settings (boundary conditions and interaction energies) and the corresponding large deviation principles, (d) concentration of measure phenomena and phase transitions.
Risk: The develop novel multi-scale methods is very challenging but may lead to fundamental improvements in existing theories. The project allows a way that the objectives can be tackle with recent multi-scale methods to mitigate the risk. Background. Main techniques to be used are variants of renormalisation group techniques, infinite-dimensional Laplace integral techniques, large deviation analysis, stochastic analysis and concentration inequalities.
(1) As a warm-up into the area of research, the project derives a characterisation of the infinite volume Gibbs measures in spatial dimension one for non-convex energies of gradient and Laplacian type.
(2) Investigate the conditional large deviation techniques in (1) for their applicability for higher dimensions. What types of large deviation principles
exist? What can they say about the thermodynamic limit of free energy?
(3) The second step is to study the scaling limit of the random field with nonconvex energies to prove the conjecture that the diffusion matrix of the continuum Gaussian Free Field is the Hessian of the thermodynamic limit of the free energy. Here, the main novelty will be non-constant boundary conditions
(tilts).
(4) Steps (1) and (2) are preparations to achieve novel versions of multi-scale methods - here the idea is to either develop further the infinite-dimensional
Laplace integral method or, to analyse the so-called hyper-contractivity of the random field with entropy methods. The two approaches may replace the finite
range decomposition used recently.
(5) Once step 4 proves to be successful, critical phenomena of mass-less models can be thoroughly analysed (scaling limits, phase transition (non-uniqueness of Gibbs measures), correlation decay).
(6) Can one use the strict convexity of the thermodynamic limit of the free energy to prove the concentration of measure? This question applies to (a) finite-dimensional Gibbs distributions, and (b) infinite volume Gibbs measures. A vital conjecture is that in case the free energy is strictly convex, all configurations are concentrated in a neighbourhood of the tilted plane. Open questions concern non-constant tilts as in (3).