High-Dimensional Random Systems: Geometry of Landscapes, and Life Without One

High-dimensional random landscapes (random fields defined on high-dimensional manifolds) have been widely studied as models of the potential energy landscapes of complex systems. Elizabeth Gardner played a key role here, starting from her 1985 study of a prototypical landscape model, the Ising p-spin glass. In recent years these problems have come back into focus, as landscapes appear as cost functions in many high-dimensional problems in optimization, inference, machine learning and control. Understanding the structure of these landscapes has been, and still is, important for understanding the dynamics of associated systems, especially out of equilibrium. In this talk, I will summarize how to obtain results on the "geometry" of prototypical random landscapes: how their stationary points (local attractors of the dynamics) are distributed in configuration space, how connected they are, what the barriers and bottlenecks look like, and how this information can help us understand how local dynamics explores the landscape. In the second part of the talk, I will ask similar questions for dynamics that no longer come from an energy function: systems with random, non-reciprocal interactions, where the landscape picture breaks down. This is motivated by applications in biology, for example biological neural networks: a direction Elizabeth Gardner herself, once again ahead of her time, had begun to explore shortly before her death in 1988.