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

大纲和目标：总体主题是相互作用的粒子系统，它们的临界现象和尺度限制。新颖之处在于发展新的多尺度方法来处理具有非凸能量的无质量随机场。主要的焦点是无质量玻色子随机场。目的是更好地理解非凸能量的临界现象，然后分析非凸自由能在材料科学中的应用，随机表面（界面），新型连续体物体的缩放限制以及玻色子系统的时空泛函积分表示。这个想法是超越连续的高斯自由场，这在过去的十年里已经取得了成果。目的是分析(a)无限体积吉布斯测度，(b)尺度极限（连续体），(c)各种设置（边界条件和相互作用能）下自由能的热力学极限和相应的大偏差原理，(d)测量现象和相变的浓度。风险：开发新的多尺度方法是非常具有挑战性的，但可能会导致现有理论的根本性改进。该项目提供了一种方法，可以用最近的多尺度方法来解决目标，以降低风险。背景。主要使用的技术有重整化群技术的变体、无限维拉普拉斯积分技术、大偏差分析、随机分析和浓度不等式。(1)作为进入研究领域的预热，该项目导出了梯度和拉普拉斯型非凸能量的空间维1无限体积Gibbs测度的特征。(2)研究(1)中的条件大偏差技术对高维的适用性。存在哪些类型的大偏差原理？关于自由能的热力学极限他们能说些什么呢？(3)第二步研究非凸能量随机场的标度极限，证明连续高斯自由场的扩散矩阵是自由能热力学极限的Hessian的猜想。这里，主要的新奇之处将是非恒定边界条件（倾斜）。步骤(1)和(2)是为实现新版本的多尺度方法做准备——这里的想法是进一步发展无限维allaplace积分方法，或者用熵方法分析所谓的随机场的超收缩性。这两种方法可以代替最近使用的有限范围分解。(5)一旦步骤4被证明是成功的，就可以彻底分析无质量模型的关键现象（缩放极限、相变（吉布斯测度的非唯一性）、相关衰减）。(6)能否用自由能热力学极限的严格凸性来证明测度的集中？这个问题适用于(a)有限维吉布斯分布和(b)无限体积吉布斯测度。一个重要的猜想是，在自由能是严格凸的情况下，所有构型都集中在倾斜平面的邻近区域。开放式问题涉及(3)中的非恒定倾斜。

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 principlesexist? 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-dimensionalLaplace integral method or, to analyse the so-called hyper-contractivity of the random field with entropy methods. The two approaches may replace the finiterange 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).