Bootstrapping universal operator growth from locality, unitarity, and hydrodynamics

The eigenstate thermalization hypothesis (ETH) posits a universal form for the matrix elements of local operators in the energy eigenbasis of a chaotic many-body Hamiltonian. However, it is an outstanding question to determine which microscopic properties we expect to lead to the validity of the ETH. In this talk, I will outline a way to `bootstrap` universal behavior of such matrix elements from the fundamental principles of unitarity and locality. The nature of the universality is akin to that found in random matrix theory, with notions of a 'bulk' and 'hard' or 'soft' edges, and there are extra modifications if the local operator overlaps with a conserved quantity. I will also show how the recently conjectured 'Operator Growth Hypothesis' implies that correlation functions of local operators in chaotic systems can be generically identified with the critical point of a confinement transition of a classical Coulomb gas, and explore the consequences of this criticality. To prove our results we employ powerful Riemann-Hilbert techniques, originally developed to study orthogonal polynomials, which we adapt to the problem of operator growth in Krylov space, allowing for a controlled expansion in the dimension of the Krylov space. Finally, we elucidate how hydrodynamics affects the growth of 'Krylov complexity', and develop a numerical algorithm which exploits this universality to compute hydrodynamic transport coefficients.