Two problems in Probability

在“自旋玻璃模型”的名义下，物理学家用非严格的方法研究了许多基本的数学对象。这些是相当典型的具有光滑的“高维”相关结构的非常大的相关随机变量族，与经典概率论研究的结构非常不同。物理学家预测在非常普遍的条件下会出现一种被称为超对称性的自组织。我们怀疑这种现象的根源在于几年前由Ghirlanda和Guerra发现的一系列概率恒等式，我们计划调查这是否真的是这样。该提案的一个稍微不同的方向计划集中在随机过程的一般理论上。有迹象表明，一些新的结构定理是可能的，这将在某种精确意义上肯定，如果某些类型的随机过程根据它们的样本有界性质表现良好，那么它们的良好行为就有简单的“见证”。这意味着，当试图确定一个特定的随机过程是否具有良好的行为时，这只能通过找到适当的证人而不是其他方法来完成。提案的两个部分都代表了概率论的一个真正的新方向。越来越多的大规模操作（电网安全、核电站安全、金融）本质上依赖于概率论的工具。本文研究了这一理论的两个新方向。第一个问题涉及理论物理学家发现的在某些大型随机结构中出现的一种显著的自组织。第二种是试图证明存在一种通用方法来控制某些随机数量的“最坏情况”的大小，因此，除了通用方法规定的方法之外，任何其他方法都是无用的。

Under the name of ``spin glass models" physicists have studied by non-rigorous methods a number of fundamental mathematical objects. These are rather canonical very large families of correlated random variables with a smooth ``high-dimensional" correlation structure, which are very different from the structures investigated by classical probability theory. The physicists predict the emergence under very general conditions of a kind of self-organization called ultrametricity. We suspect that the root of this phenomenon lies in a family of probabilistic identities discovered a few years ago by Ghirlanda and Guerra and we plan to investigate whether this is really the case. A somewhat different direction of the proposal plans to focus on the general theory of stochastic processes. There are indications that some new structure theorems are possible, that would in some precise sense affirm that if certain types of stochastic processes are well-behaved according to their sample boundedness properties, then there is simple ``witness'' of their good behavior. This would imply that when trying to decide whether a specific stochastic process has a good behavior, this can be done only by finding the appropriate witness and by no other means. Both part of the proposal represent a genuinely new direction of Probability theory. A growing number of large scale operations (safety of electrical networks, safety of nuclear plants, finance) essentially depend on tools from Probability theory. This proposal investigates two new directions in this theory. The first concerns the emergence in certain large random structures of remarkable kind of self-organization discovered by theoretical physicists. The second attempts to prove that there is a kind of universal method to control how large can be the ``worst occurrence'' of certain random quantities, and that therefore it is useless to attempt to do this in any other way than as prescribed by the universal method.