arXiv 1206.02122012-06-01math.PRmath-phmath.COmath.MP

量子引力与KPZ公式

Quantum gravity and the KPZ formula

Christophe Garban

备注：: Bourbaki seminar (after Duplantier-Sheffield), 38 pages

摘要：

本文是对B.Duplantier和S. Sheffield的“Liouville量子引力和KPZ”一文的综述（Bourbaki研讨会）。二维（d=2）统计物理模型在其临界点的研究通常是一个相当困难的问题（更不用说d=3的情况了）。八十年代，三位物理学家Knizhnik，Polyakov和Zamolodchikov（KPZ）在\cite{\KPZ}中提出了一种新颖而深远的方法，以理解这些模型的临界行为。其中，人们发现例如随机游动，渗流以及伊辛模型。他们的方法的主要基本思想是沿着如下的两步过程来研究这些模型：a/首先，不是考虑平面的一些规则晶格上的模型（例如$\Z^2$），而是将其定义在精心选择的“随机平面晶格”上。这样做相当于研究{ \it量子引力}形式的模型。在渗流的情况下，随机格的适当选择与所谓的平面映射相匹配。b/然后返回到实际的{ \it Euclidean}设置。这要归功于著名的{ \bf KPZ公式}，它给出了量子引力公式中模型的几何性质与欧几里得情况下模型的类似物之间非常精确的对应关系。这种强大的对应关系的性质和起源在很长一段时间内都是相当神秘的。事实上，KPZ公式仍然没有严格建立，仍然是一个几何对应。这项调查的目的是解释如何最近的工作Duplantier和谢菲尔德能够解释一些神秘隐藏在这个KPZ公式。用一句话来总结他们的贡献，他们的工作意味着通过随机晶格的均匀化对KPZ对应进行了美丽的解释，被视为黎曼曲面。

英文摘要：

This text is a survey (Bourbaki seminar) on the paper "Liouville quantum gravity and KPZ" By B.Duplantier and S.Sheffield. The study of statistical physics models in two dimensions (d=2) at their critical point is in general a significantly hard problem (not to mention the d=3 case). In the eighties, three physicists, Knizhnik, Polyakov et Zamolodchikov (KPZ) came up in \cite{\KPZ} with a novel and far-reaching approach in order to understand the critical behavior of these models. Among these, one finds for example random walks, percolation as well as the Ising model. The main underlying idea of their approach is to study these models along a two-step procedure as follows: a/ First of all, instead of considering the model on some regular lattice of the plane (such as $\Z^2$ for example), one defines it instead on a well-chosen "random planar lattice". Doing so corresponds to studying the model in its {\it quantum gravity} form. In the case of percolation, the appropriate choice of random lattice matches with the so-called planar maps. b/ Then it remains to get back to the actual {\it Euclidean} setup. This is done thanks to the celebrated {\bf KPZ formula} which gives a very precise correspondence between the geometric properties of models in their quantum gravity formulation and their analogs in the Euclidean case. The nature and the origin of such a powerful correspondence remained rather mysterious for a long time. In fact, the KPZ formula is still not rigorously established and remains a conjectural correspondence. The purpose of this survey is to explain how the recent work of Duplantier and Sheffield enables to explain some of the mystery hidden behind this KPZ formula. To summarize their contribution in one sentence, their work implies a beautiful interpretation of the KPZ correpondence through a uniformization of the random lattice, seen as a Riemann surface.

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摘要：

英文摘要：

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