Probabilistic methods in quantum spin systems and spin glasses

PI将研究两个问题，这两个问题都位于概率论和数学物理的交叉点。第一个是尝试分类“准平稳随机重叠结构”。随机重叠结构出现在平均场自旋玻璃的研究中，如谢灵顿-柯克帕特里克模型。它们类似于随机分区结构，但更一般。准平稳性是随机重叠结构的不变性，从物理学家的“腔台阶”中抽象出来。它类似于确定性平均场自旋系统的交换自旋。因此，我们的建议是找到并证明一个自旋玻璃版本的de Finetti定理。与此无关，PI将研究铁磁海森堡量子自旋系统。这个量子模型的哈密顿量等于（一减）对称排斥过程（SEP）的马尔可夫生成元。物理学家的自旋波论证为这些算符的谱提供了重要的理论依据。例如，一个这样的猜想（事后）被证明是等价于奥尔德斯的猜想，即SEP的谱间隙等于随机游走的谱间隙。自旋玻璃是一种特殊的磁性材料，在短时间尺度上可以形成稳定的构型，但在长时间尺度上可以从一种构型跳到另一种构型。因此，根据观察时间的长短，它们看起来像固体或液体。玻璃做同样的事情：例如，非常古老的教堂的窗户底部比顶部厚，据说是因为玻璃随着时间的推移而流下。20多年前，物理学家乔治·帕里西（Giorgio Parisi）对最简单的自旋玻璃模型提出了一个独特的解决方案。直到最近，他的直觉猜测才被数学家们真正证明。这导致了有趣的新数学和新物理。但仍有许多工作要做。Parisi还对计算机科学中的问题做出了其他猜测。对于许多涉及寻找最优解的计算问题，众所周知，任何算法都是“NP完全的”。这意味着（人们强烈认为）一个解决这些问题的程序将需要非常长的时间来运行。另一方面，Parisi和其他人提出了许多这些问题的随机版本的解析解。据推测，仅次于万无一失的解决方案的最佳方案是一种高概率的解决方案。准平稳性的属性似乎是所有这些启发式解决方案所固有的。因此，理解它对于全面了解这个主题是很重要的。

The PI will investigate two problems, both of which lie at the intersection of probability theory and mathematical physics. The first is to try to classify ``quasi-stationary random overlap structures''. Random overlap structures arise in the study of mean-field spin glasses such as the Sherrington-Kirkpatrick model. They are analogous to random partition structures, but more general. Quasi-stationarity is an invariance property for random overlap structures, abstracted from the physicists' ``cavity step''. It is akin to exchangeability for deterministic mean-field spin systems. So, the proposal is to find and prove a spin-glass version of de Finetti's theorem. Unrelated to this, the PI will study the ferromagnetic Heisenberg quantum spin system. The Hamiltonian of this quantum model equals (one minus) the Markov generator of the symmetric exclusion process (SEP). Physicists' spin wave arguments suggest important conjectures for the spectrum of these operators. For example, one such conjecture turned out (after the fact) to be equivalent to Aldous's conjecture, that the spectral gap for the SEP equals the spectral gap for the random walk. The proposal is to try to prove these conjectures.Spin glasses are unusual magnetic materials, which settle into stable configurations on short time scales, but hop from configuration to configuration on long times scales. So, depending on the length of time observed, they look either like solids or liquids. Glass does the same thing: for example, windows in very old churches are thicker at the bottom than the top, supposedly because the glass flowed down over time. Over 20 years ago, Giorgio Parisi, a physicist, proposed a unique solution to the simplest spin glass model. Only recently was his intuitive guess really proved by mathematicians. This led to interesting new math as well as new physics. But much remains to be done. Parisi made other guesses for problems in computer science. For many computational problems which involve finding an optimal solution, it is known that any algorithm is ``NP complete''. This means (it is strongly believed) that a program solving such problems would take inordinately long to run. On the other hand, Parisi and others proposed analytical solutions to random versions of many of these problems. Presumably, the next best thing to a foolproof solution is one that works with high probability. The property of quasi-stationarity seems to be inherent to all these heuristic solutions. So understanding it is important for an overall picture of the subject.