A central goal in the study of quantum chaos is being able to make universal statements about the dynamics of generic Hamiltonian systems. Under time evolution, an initially local operator progressively explores the Hilbert space of a system becoming increasingly non-local in the process. We will see that this idea lends itself to a natural notion of operator complexity measured (in the Hilbert space of operators) by the overlap of a time-evolving operator with a basis naturally adapted to time evolution and stratified by the growth in the operator's support. The information contained in this so-called Krylov basis is encoded in a sequence called the Lanczos coefficients which quantify the rate at which an operator is "pushed" along the Krylov basis to successively more complex elements. The universal operator growth hypothesis is then the conjecture that the Lanczos coefficients grow asymptotically linearly in any quantum chaotic system. In this talk, I will present an overview of these ideas and see how they manifest in the example of the well-studied SYK model. This talk is primarily based on 1812.08657.

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TBA

I will review how the large N expansion can be used in the context of the renormalisation group to probe some strongly coupled regimes. In particular, I will discuss a work by Gawedzki and Kupiainen where the authors study the three-dimensional non-Gaussian infrared fixed point of Phi^4 in the case of a hierarchical model of rank-one covariance, and explain how their approach could generalise to more realistic models. 

This is a joint work with Ajay Chandra.