Mathematical Physics 2023 - projects Project 1 Consider a classical lattice-gas model with translation-invariant finite-range non-frustrated interactions, without periodic ground-state configurations and with a unique translation-invariant ground-state measure. We would like to construct a non-perodic Gibbs measure - a small perturbation of a non-periodic ground-state configuration (recall the Peierls contour method used for the Ising model). Hypothesis: Such a non-periodic Gibbs state does not exist if the ground-state measure does not satisfy the Strict Boundary Condition. pdf Try to prove the hypothesis using the entropy-energy argument presented in the paper pdf As an example consider The Robinson's tiling pdf Zero-temperature non-stability of The Robinson's ground state is shown in pdf Project 2 Try to prove that the Tsirelson's construction provides an example of a lattice-gas model which satisfies the Strict Boundary Condition. pdf pdf Project 3 Try to prove that the Fibonacci ground state is stable against a small chemical potential (one-site interaction) favoring the presence of particles. pdf Project 4 Prove the existence of the Devil's staircase - Theorem 0 on page 596 of JSP paper below. pdf pdf Project 5 Enlarge the family of lattice-gas models for which you can prove the existence of periodic ground-state configurations.