Explore the advanced mathematical concept of Loop Vertex Expansion (LVE) in this 39-minute conference talk by Vasily Sazonov from Paris-Saclay University and CEA List. Delve into the development of LVE for constructing Quantum Field Theory (QFT) models with both local and non-local interactions. Learn how LVE can be used to prove the analyticity of free energies and cumulants in various vector, matrix, and tensor-type models within a finite cardioid-like domain in the complex plane of the coupling constant. Discover Sazonov's innovative approach of selecting an initial approximation based on the coupling constant, which enables the analytic continuation of the free energy of the quartic matrix model beyond the standard LVE cardioid. Gain insights into cutting-edge research in theoretical physics and advanced mathematical techniques applied to quantum field theory.