Nam-Gyu Kang, Nikolai Makarov
In these mostly expository lectures, we give an elementary introduction to conformal field theory in the context of probability theory and complex analysis. We consider statistical fields, and define Ward functionals in terms of their Lie derivatives. Based on this approach, we explain some equations of conformal field theory and outline their relation to SLE theory.
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Conformal welding of independent Gaussian multiplicative chaos measures
本文研究了由独立高斯乘法混沌测度生成的两个随机保形映射的共形焊接问题，证明了存在共形映射使得归一化测度在公共边界上一致。
Conformal field theory of Gaussian free fields in a multiply connected domain
该研究基于高斯自由场构建多连通域下中心荷$c≤1$的共形场论框架，引入广义Eguchi-Ooguri方程推导Ward方程显式形式，借助BPZ方程实现对应SLE的共形场论表述并构造其鞅可观测量。
We study the extremal process associated with the Discrete Gaussian Free Field on the square lattice and elucidate how the conformal symmetries manifest themselves in the scaling limit. Specifically, we prove that the joint process of spatial positions ($x$) and centered values ($h$) of the extreme local maxima in lattice versions of a bounded domain $D\subset\mathbb C$ converges, as the lattice spacing tends to zero, to a Poisson point process with intensity measure $Z^D(dx)\otimes e^{-αh}d h$, where $α$ is a constant and $Z^D$ is a random a.s.-finite measure on $D$. The random measures $\{Z^…
We develop a version of dipolar conformal field theory in a simply connected domain with the Dirichlet-Neumann boundary condition and central charge one. We prove that all correlation functions of the fields in the OPE family of Gaussian free field with a certain boundary value are martingale-observables for dipolar SLE(4).
Level lines of the Gaussian free field and $c=1$ degenerate conformal blocks
研究具有分段常数狄利克雷边界数据的单连通域上的高斯自由场，证明其水平线的穿越概率由中心荷 $c=1$ 的共形场论中的退化初级场共形块决定，并给出度量图高斯自由场在缩放极限下的收敛性。
We consider dimer models on graphs which are bipartite, periodic and satisfy a geometric condition called {\em isoradiality}, defined in \cite{Kenyon3}. We show that the scaling limit of the height function of any such dimer model is $1/\sqrtπ$ times a Gaussian free field. Triangular quadri-tilings were introduced in \cite{Bea}; they are dimer models on a family of isoradial graphs arising form rhombus tilings. By means of two height functions, they can be interpreted as random interfaces in dimension 2+2. We show that the scaling limit of each of the two height functions is $1/\sqrtπ$ times a…
We develop a version of dipolar conformal field theory based on the central charge modification of the Gaussian free field with the Dirichlet boundary condition and prove that correlators of certain family of fields in this theory are martingale-observables for dipolar SLE. We prove the restriction property of dipolar SLE(8/3) and Friedrich-Werner's formula in the dipolar case.
Two Decades of Probabilistic Approach to Liouville Conformal Field Theory
本文综述了刘维尔共形场论概率方法二十年来的发展，涵盖基于高斯自由场和高斯乘性混沌的严格定义、DOZZ公式推导及共形自举的完整表述，并指出了开放问题。
This is the second of two papers devoted to the proof of conformal invariance of the critical double random current on the square lattice. More precisely, we show convergence of loop ensembles obtained by taking the cluster boundaries in the sum of two independent critical currents (both for free and wired boundary conditions). The strategy is first to prove convergence of the associated height function to the continuum Gaussian free field, and then to characterize the scaling limit of the loop ensembles as certain local sets of this Gaussian Free Field. In this paper, we derive crossing prope…
We present analytical implementation of conformal field theory on a compact Riemann surface. We consider statistical fields constructed from background charge modifications of the Gaussian free field and derive Ward identities which represent the Lie derivative operators in terms of the Virasoro fields and the puncture operators associated with the background charges. As applications, we derive Eguchi-Ooguri's version of Ward's equations and certain types of BPZ equations on a torus.
We construct a family of random Jordan curves in the plane by welding together two disks on their boundaries using a random homeomorphism. This homeomorphism arises from a random measure whose density, in a generalized sense, is the exponentiated Gaussian Free Field at criticality. We also introduce a representation of the Gaussian Free Field in terms of vaguelets, which may be of separate interest. The result extends a theorem of Astala, Jones, Kupiainen and Saksman (\cite{AJKS09}) to criticality.
Conformal invariance of double random currents I: identification of the limit
本文作为两篇系列论文的第一篇，通过证明高度函数收敛于高斯自由场并唯一识别环系综的子序列极限，为正方格子上临界双随机电流模型的共形不变性证明奠定基础。
We prove that crossing probabilities for the critical planar Ising model with free boundary conditions are conformally invariant in the scaling limit, a phenomenon first investigated numerically by Langlands, Lewis and Saint-Aubin. We do so by establishing the convergence of certain exploration processes towards SLE$(3,\frac{-3}2,\frac{-3}2)$. We also construct an exploration tree for free boundary conditions, analogous to the one introduced by Sheffield.
We determine the exact threshold of the extended Seiberg bound for the existence of correlation functions in the boundary Liouville conformal field theory in the unit disk. In probabilistic terms, our result is a toolbox yielding the threshold for the existence of positive moments of Gaussian multiplicative chaos measure, appliable to the case where singularities of arbitrary (co-)dimension in the background metric are present. We improve previous results of this type for 0-dimensional singularities in [DKRV16] and a sufficient condition for the 1-dimensional singularity in an unpublished appe…
In a remarkable paper in 2008, Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of (sub-critical) Gaussian multiplicative chaos (GMC) associated to the Gaussian free field (GFF) on the unit circle. In this paper we will give a proof of this formula. In the mathematical literature this is the first occurrence of an explicit probability density for the total mass of a GMC measure. The key observation of our proof is that the negative moments of the total mass of GMC determine its law and are equal to one-point correlation functions of Liouville conformal field…
There are two natural notions of Lévy processes in free probability: the first one has free increments with homogeneous distributions and the other has homogeneous transition probabilities. In the two cases one can associate a Nevanlinna function to a free Lévy process. The Nevanlinna functions appearing in the first notion were characterised by Bercovici and Voiculescu. I give an explicit parametrisation for the Nevanlinna functions associated with the second kind of free Lévy processes. This gives a nonlinear free Lévy-Khinchine formula.
We prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain Markov property is a multiple of the Gaussian free field. This result holds subject only to a fourth moment assumption.
Inverse of the Gaussian multiplicative chaos: Lehto welding of Independent Quantum disks
本文利用随机共形焊接框架，证明了在γ<0.1818条件下高斯乘性混沌的逆及其独立副本的Lehto焊接存在性，并得到粘合两个GMC边界圆盘的共形不变环Γ，但尚未证明其为SLE环。
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