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The mini workshop is focusing on ordered structures of complex systems at low temperatures. It intends to gather many fields from dynamical systems, computability theory, Gibbs theory, tilings and quasicrystals.

• Schedule

  	Tuesday	Wednesday	Thursday	Friday
  9:00 - 10:00	Computability (lecture 1/3) Daniel Graça	Gibbs measure (lecture 1/3) Jacek Miekisz	Tilings (lecture 1/3) Yaar Salomon	William da Silva
  10:00 - 10:30	break	break	break	break
  10:30 - 11:30	Computability (lecture 2/3) Daniel Graça	Gibbs measure (lecture 2/3) Jacek Miekisz	Tilings (lecture 2/3) Yaar Salomon
  11:30 - 12:00	break	break	break	break
  12:00 - 13:00	Computability (lecture 3/3) Daniel Graça	Gibbs measure (lecture 3/3) Jacek Miekisz	Leo Paviet Salomon	??
  13:00 - 14:30	lunch	lunch	lunch	lunch
  14:30 - 15:30	Manon Blanc	??	Solène Esnay
  15:30 - 16:00	mini break	mini break	mini break
  16:00 - 17:00	Matteo d'Achille	??	Sébastien Ferenczi
  17:00 - 17:15	break	break	break
  17:15 - 18:15	??	??

• Speakers:
  • Speaker: Matteo d'Achille (Université de Lorraine)
    Title: Uncountably many extremal Ising Gibbs states on Lobachevsky lattices
    Abstract: Aizenman and Higuchi famously proved that, at low temperatures, any Gibbs state of the Ising model on $\mathbb{Z}^2$ is a convex combination of two extremal states.
    
    In this talk I will exhibit uncountably many extremal low-temperature Gibbs states for the Ising model on the graph given by a regular tessellation of the hyperbolic plane (a.k.a. Lobachevsky lattice).
    
    The proof combines two main ingredients:
    
    - An excess energy lemma (which holds for the Ising model defined on any non-amenable graph) providing a uniform lower bound to the cost of a spin flip via a linear combination of the number of frustrated bonds and of the Cheeger constant of the graph;
    
    - A layer decomposition specific to these lattices, due to Rietman–Nienhuis–Oitmaa and Moran, which allows us to build Dobrushin-like interfaces by a suitable gluing of two infinite trees in the dual graph. En passant, I will also prove that certain "regular balls" built via this layer decomposition solve the isoperimetric problem at fixed volume.
    
    This partially solves a conjecture of Series-Sinai.
    
    Talk mostly based on 10.1214/25-ECP724 in collaboration with Loren Coquille (Grenoble) and Arnaud Le Ny (Paris-Est Créteil); and on 2504.14080 (to appear) with Vanessa Jacquier (Padova) and Wioletta M. Ruszel (Utrecht).
  
  
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