8 Chaos , Concentration , and Multiple Valleys

Disordered systems are an important class of models in statistical mechanics, having the defining characteristic that the energy landscape is a fixed realization of a random field. Examples include various models of glasses and polymers. They also arise in other areas, like fitness models in evolutionary biology. The ground state of a disordered system is the state with minimum energy. The system is said to be chaotic if a small perturbation of the energy landscape causes a drastic shift of the ground state. We present a rigorous theory of chaos in disordered systems that confirms long-standing physics intuition about connections between chaos, anomalous fluctuations of the ground state energy, and the existence of multiple valleys in the energy landscape. Combining these results with mathematical tools like hypercontractiv-ity, we establish the existence of the above phenomena in eigenvectors of GUE matrices, the Kauffman-Levin model of evolutionary biology, directed polymers in random environment, a subclass of the generalized Sherrington-Kirkpatrick model of spin glasses, the discrete Gaussian free field, and continuous Gaussian fields on Euclidean spaces. We also list several open questions.