The M-Layer construction: a diagrammatic framework for critical behavior in lattice models

This thesis presents and applies the M-layer construction, a recent diagrammatic method for studying the critical behavior of lattice models in finite dimensions. The construction provides a unified tool to compute correlation functions and critical exponents through a perturbative expansion in topological loops, whose leading order corresponds to the behavior on a Bethe lattice. The first application of the method is on well-known problems, such as the Ising model and the percolation problem, where it successfully reproduces standard results, thus validating its efficiency. It is then applied to the Edwards–Anderson spin glass model in a magnetic field at zero temperature, a problem whose finite-dimensional physics remains widely debated. The remarkable new result of the M-layer construction is the identification, for the first time through analytical computations, of a phase transition in a field below the upper critical dimension. The results, obtained through analytical computations and partially checked numerically, show that this approach provides a promising alternative to standard Renormalization Group techniques for investigating phase transitions in disordered and frustrated systems.