Probability and Mean Field Models for Spin Glasses

七十年代，G.Parisi发现了自旋玻璃平均场模型的意想不到的行为。这些模型是简单的数学对象。值得注意的是，他们表现出如此微妙的行为，他们的严谨研究是如此具有挑战性。PI相信，阐明这些模型将给概率论带来一个全新的方向的发现。10年来，他一直在努力建立研究它们的新方法。其中一个基本目标(证明Parisi公式给出了Sherrington-Kirkpatrick模型的自由能)已经实现，但这一成功不能掩盖这样一个事实，即总体上我们的理解仍然相当有限。特别是，两个核心问题(称为超测距猜想和混沌问题)似乎一如既往地困难。它们可能与分析问题有关。PI计划解决这些非常困难的问题，但也将努力在不那么令人望而生畏的问题上取得渐进进展，特别是对展示新型复制对称方程的新的数学正则平均场哈密顿的研究。至少从伽利略开始，数学和物理学就在根本上相互影响，这种相互影响比以往任何时候都更强。在七十年代，物理学家发现某些合金(称为自旋玻璃)以一种非传统的方式对磁模拟做出反应。他们引入了新的模型来解释这些行为。这些模型是简单而自然的数学对象。然而，在很长一段时间里，没有数学方法来研究它们。物理学家的早期工作使用的启发式方法不一定是完全可靠的。通过从数学中开发新的工具，特别是概率论，以完全理解这些基本模型，本项目是消除这种差异的长期计划中的一步。这个项目的长期成果可能是概率论的新推动力，概率论是数学中最重要的应用分支之一。

The unexpected behavior of mean-field models for spin glasses was discovered in the seventies by G. Parisi. These models are simple mathematical objects. It is remarkable that they exhibit such a subtle behavior, and that their rigorous study is so challenging. The PI believes that elucidating these models will bring the discovery of an entire new direction in probability theory. He has been trying for 10 years to build new methods to study them. One of the fundamental objectives (the proof of the Parisi formula giving the free energy of the Sherrington-Kirkpatrick model) has been attained, but this success cannot hide the fact that overall our understanding remains rather limited. In particular, two central questions, (known as the Ultrametricity conjecture and the Chaos problem) seem as hard as ever. They are possibly related to questions of analysis. The PI plans to work on these very hard questions, but also will strive to make incremental progress on less daunting issues, in particular the study of new mathematically canonical mean field Hamiltonians that exhibit replica symmetric equations of a new type. Mathematics and Physics have influenced each other in fundamental ways since at least Galileo, and this mutual influence is stronger than ever. In the seventies, Physicists discovered that certain alloys (called spin glasses) responded in an unconventional way to magnetic simulation. They introduced new models to explain these behaviors. These models are simple and natural mathematical objects. Yet for a long time there existed no mathematical method to study them. The early work of the physicists uses heuristic methods that need not be completely reliable. The present project it a step in the long range program of eliminating this discrepancy, by developing new tools from mathematics, and in particular probability theory, to completely understand these fundamental models. The long rang outcome of this program could be a new impetus on probability theory, one of the most important branch of mathematics for applications.