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. 

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Try to prove the hypothesis using the entropy-energy argument presented in the paper   

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As an example consider The Robinson's tiling

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Zero-temperature non-stability of The Robinson's ground state is shown in

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  Project 2  

Try to prove that the Tsirelson's construction provides an example of a lattice-gas model which satisfies 
the Strict Boundary Condition.

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  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.

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  Project 4  

Prove the existence of the Devil's staircase - Theorem 0 on page 596 of JSP paper below.

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  Project 5  

Enlarge the family of lattice-gas models for which you can prove the existence of periodic ground-state configurations.