GGI Post-Doc Day

This talk explores the enumeration of classical closed lattice random walks by their algebraic area (or signed area), with connections to the Hofstadter model and exclusion statistics. We begin with a review of the algebraic area of closed random walks on a square lattice and its relation to the Hofstadter model. We then focus on the coefficients of the secular determinant of the Hofstadter Hamiltonian, which can be interpreted in terms of partition functions with the exclusion parameter g=2. The algebraic area enumeration is obtained in terms of the associated cluster coefficients. Extending this analysis to triangular lattice random walks, we establish a correspondence to a system of particles obeying a mixture of g=1 (fermions) and g=2 exclusion statistics. Finally, we briefly discuss applications of exclusion statistics and random walks in exactly solvable models and topological string theory.

In this talk, we examine quantum corrections to the spectrum and background of U(1) relativistic superfluids, where the finite charge density stored in these configurations spontaneously breaks the internal and Lorentz symmetry. By addressing inconsistencies in the literature, we explicitly show that the two-loop corrected superfluid background is stationary, and the one-loop spectrum of perturbation has a Goldstone mode, in agreement with non perturbative results. Also, we explicitly build the superfluid state in terms of the fundamental degrees of freedom of the theory and show how the aforementioned results depend on the non-gaussian nature of this state. Lastly, when examining the non-relativistic theory of the Goldstone, we find that this mode develops a cutoff-sensitive gap at one-loop, suggesting shortcomings in the low-energy effective theory when quantum corrections are considered.

Cosmological perturbation theory models the Large-Scale Structure (LSS) of the Universe by treating deviations from homogeneity as small perturbations. However, nonlinear corrections and higher-order perturbation theory are crucial to describe matter clustering in recent cosmic times. The first project presents an iterative method to construct nonlinear relations between cosmological observables, with applications to time lapse, radial distortions, and deflection angles, relevant for upcoming LSS surveys. The second project, inspired by DESI observations, uses mock data from the Boltzmann solver CLASS-PT to study the full-shape power spectrum and constrain dynamical dark energy models, including one-loop corrections with the Effective Field Theory of LSS. Finally, we explore f(R) and f(R,T) modified gravity theories, incorporating scalar fields into effective redshift-dependent cosmological parameters to address the Hubble and matter clustering tensions. These projects collectively offer novel insights into perturbative methods, dark energy, and modified gravity.