Two problems in Probability

在“自旋玻璃模型”的名义下，物理学家已经用非严格的方法研究了一些基本的数学对象。这些是相当典型的非常大的相关随机变量家族，具有平滑的"高维"相关结构，这与经典概率论研究的结构非常不同。 物理学家预测 在一种叫做超对称性的自组织的一般条件下。我们怀疑这种现象的根源在于几年前Ghirlanda和Guerra发现的一个概率身份家族，我们计划调查是否真的是这样。 一个有点不同的方向的建议计划集中在随机过程的一般理论。有迹象表明，一些新的结构定理是可能的，这将在某种精确的意义上肯定，如果某些类型的随机过程是良好的行为，根据其样本有界性的性质，那么有简单的“成功”，他们的良好行为。这意味着，当试图决定一个特定的随机过程是否具有良好的行为时，只能通过找到适当的证据来完成，而没有其他方法。这两个部分的建议代表了一个真正的新方向的概率论。越来越多的大规模操作（电网安全，核电站安全，金融）基本上依赖于概率论的工具。这一建议探讨了这一理论的两个新方向。第一个问题是理论物理学家发现的某些大型随机结构中出现了一种引人注目的自组织。第二部分试图证明，存在一种普适的方法来控制某些随机量的“最坏情况”的大小，因此，除了普适的方法所规定的方法之外，试图用任何其他方法来做这件事都是无用的。

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.