David Pérez-García - Universidad Complutense de Madrid Projected Entangled Pair States (PEPS) constitute an important family of Tensor Networks, since they approximate well ground states and thermal states of quantum many body systems. PEPS come together with a parent Hamiltonian. In this talk I will focus on the problem of estimating its spectral gap, i.e. the difference between the two smallest eigenvalues. I will introduce the problem and motivate it from several perspectives. Then I will show some new results, and an application to bound the convergence time of quantum Gibbs-sampling algorithms motivated by the study of self-correcting quantum memories.