Supermathematics and its applications in statistical physics

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This text introduces Grassmann variables and supersymmetry to physicists interested in disordered and critical systems, as well as statistical physics. Drawing from the author's extensive teaching experience, it provides a systematic tutorial on the subject. Part I covers the algebra and analysis of Grassmann variables, applying these concepts to random matrix models, fermionic path integrals, dimer models, and the two-dimensional Ising model. Part II delves into supermathematics, exploring the properties of supervectors and supermatrices that include both commuting and Grassmann components, along with integral theorems. In Part III, the focus shifts to supersymmetric physical models, originally introduced in particle physics to illustrate symmetry between bosons and fermions. The text extends this concept to statistical physics, linking states with equal energies, and touches upon applications in quantum mechanics. Various models are examined, leading to the representation of the random matrix model through the nonlinear sigma-model, as well as the determination of the density of states and level correlations. Finally, the discussion includes mobility edge behavior and a brief overview of the ten symmetry classes of disorder, two-dimensional disordered models, and superbosonization.