Probabilistic methods in quantum spin systems and spin glasses

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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.