Lighthouse in Ischia Porto. Photo by Miriam Fiore.
Universality is a central concept in several branches of mathematics, mathematical physics and theoretical physics. In the broad context of statistical mechanics, universality usually refers to the exact independence of certain observable quantities from the microscopic details appearing in the definition of the model.
Some remarkable examples of universality phenomena are:
the universality of the scaling theory (critical exponents, universal amplitudes, and even finite-size correction coefficients) at a second order phase transition point;
the universality of the eigenvalue statistics of random matrices;
the ubiquitous emergence of massless Gaussian free field fluctuations in the theory of dimers, random surfaces, etc.
Many of these phenomena are predicted on the basis of a treatment of the model in terms of low-energy effective theories which exhibit universality properties, for example on the basis of approximate renormalization group arguments. The mathematical understanding of these phenomena is largely open. However, in recent years, different and complementary methods have been introduced to attack these problems in a mathematically rigorous framework, and in certain cases have made it possible to prove the expected universality relations, and even to predict new ones. Some examples are:
the probabilistic and functional analytic methods used to prove the universal behavior of the eigenvalue distribution of Wigner random matrices;
the constructive renormalization group methods used to compute exact critical exponents in scalar field theories, dimer and spin models;
the functional analytic methods used for deriving the effective dynamics of stochastic growth interface models;
the probabilistic and large deviation methods used to control the localization/delocalization transition in pinned polymer models;
the probabilistic and combinatorial techniques used to control the logarithmic fluctuations of non-harmonic discrete random surfaces.
This school is intended to introduce students and young postdocs to this these sophisticated methods, and to strengthen the connections among the sub-communities that have developed and used them in recent years.