Phase transitions in two-dimensional classical lattice systems and random matrix theory

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Our understanding of the thermodynamics of phase transitions in the matter comes mainly from the study of simple idealized models that are mathematically tractable but still retain the main physical features that characterize real systems. These systems are the two-dimensional classical spin models, of which the best known example is the Ising model. This project proposes to investigate two of the most interesting properties that have attracted the interest of physicists and mathematicians over the past forty years: universality and the scaling hypothesis. Thermodynamical variables in proximity of critical temperatures obey power laws whose exponents seem to depend only on the dimensionality and symmetries of the system; this phenomenon is known as universality. The scaling hypothesis, instead, asserts that the thermodynamical properties of a macroscopic system depend only on few relevant variables that characterize its behaviour on a particular time or length scale.The main part of this project will focus on a new approach that will use mathematical techniques, which go under the name of random matrix theory, to prove universality and the scaling hypothesis for a class of classical spin models whose symmetries can be put in one-to-one correspondence with ensembles of random matrices. Few rigorous results are available in this area. The second part of the project is devoted to using random matrix techniques to compute entanglement of the ground state of families of one-dimensional quantum spin chains associated to appropriate matrix ensembles. Random matrix theory allows a rigorous approach to these important problems that works for families of systems and symmetry classes to which previous methods, involving the renormalization group approach and conformal field theory, do not apply.