Seminars

Abstract  In this talk, I will show results on the spectral properties of an infinitely renormalizable random network model. This model is an 'anti-benchmark’ for Random Matrix Theory techniques because many working assumptions fail, such as moment matching methods, the low rank hypothesis for the signal matrix, and the non-self-averaging nature of the ESD. Despite these challenges, I will show an ansatz for the eigenfunctions based on the Laplace transform of the weights distribution. This allows to solve the eigenvalue equation for the signal matrix, which is a good proxy for the largest eigenmodes of the network realisations. The leading eigenvalues are all of order √n, alternate in sign, and lie at the intersection between the real axis and a logarithmic spiral in the complex plane. I will also show haow to calculate the associated eigenvectors, which display complex-valued scaling exponents and log-periodicity, indicating discrete scale invariance.

The renormalization group is one of the most important tools for the description of critical points in statistical mechanics and other areas of theoretical physics.  Most mathematically rigorous treatments are based on implementing the renormalization group as a discrete dynamical system, which introduces a number of theoretical complications, and in particular obscures the emergence of full scale invariance, which is one of the most important features of a critical point.  In the last few years, however, there have been several new results about solutions of the Polchinski equation (a nonlinear differential equation which implements the renormalization group as a continuous dynamical system) for Fermionic systems.  I will present one in particular, the construction of a nontrivial fixed point of a family of continuous renormalization group flows corresponding to certain weakly interacting Fermionic quantum field theories with a parameter in the propagator allowing the scaling dimension to be tuned in a manner analogous to dimensional regularization.