Probabilistic methods in quantum spin systems and spin glasses

PI将调查两个问题，这两个问题都位于概率论和数学物理的交叉点上。第一种是尝试对“准平稳随机重叠结构”进行分类。在研究像Sherrington-Kirkpatrick模型这样的平均场自旋玻璃时，会出现随机重叠结构。它们类似于随机分区结构，但更具一般性。准平稳性是随机重叠结构的一种不变性，从物理学家的“空穴台阶”中抽象出来。它类似于确定性平均场自旋系统的互换性。因此，我们的建议是找到并证明德·菲内蒂定理的一个自旋玻璃版本。与此无关，PI将研究铁磁海森堡量子自旋系统。该量子模型的哈密顿量等于(1-)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对这些算符的光谱提出了重要的猜想。例如，一个这样的猜想(事后)被证明等同于奥尔多斯的猜想，即SEP的光谱间隙等于随机游动的光谱间隙。这个提议是为了试图证明这些猜想。自旋玻璃是一种不寻常的磁性材料，在短时间尺度上稳定地形成构型，但在长时间尺度上从一个构型跳跃到另一个构型。因此，根据观察到的时间长短，它们要么看起来像固体，要么看起来像液体。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，据推测是因为随着时间的推移，玻璃会往下流。20多年前，物理学家乔治·帕里西为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，他的直觉猜测才真正得到数学家的证实。这导致了有趣的新数学和新物理。但仍有许多工作要做。帕里西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)一个解决这类问题的程序将需要非常长的时间才能运行。另一方面，Parisi和其他人提出了许多此类问题的随机版本的解析解决方案。大概，仅次于万无一失的解决方案的是一种概率很高的解决方案。准平稳的性质似乎是所有这些启发式解所固有的。因此，理解这一点对于全面了解这一主题是很重要的。

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.