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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 (University of Algarve)	Quasicrystals (lecture 1/3) Jacek Miȩkisz (University of Warsaw)	Tilings (lecture 1/2) Yaar Solomon (Ben-Gurion University )	William da Silva (University of Vienna)
  10:00 - 10:30	break	break	break	break
  10:30 - 11:30	Computability (lecture 2/3) Daniel Graça (University of Algarve)	Quasicrystals (lecture 2/3) Jacek Miȩkisz (University of Warsaw )	Tilings (lecture 2/2) Yaar Solomon (Ben-Gurion University )	Jean Vereecke (University Grenoble Alpes)
  11:30 - 12:00	break	break	break	break
  12:00 - 13:00	Computability (lecture 3/3) Daniel Graça (University of Algarve)	Quasicrystals (lecture 3/3) Jacek Miȩkisz (University of Warsaw )	Leo Paviet Salomon (ENS de Lyon)	Benjamin Hellouin (University Paris-Saclay)
  13:00 - 14:30	lunch	lunch	lunch	lunch
  14:30 - 15:30	Manon Blanc ( ITU i Kobenhavn)	Gaël Meignan (Universty of Bordeaux)	Solène Esnay
  15:30 - 16:00	break	break	break
  16:00 - 17:00	Matteo d'Achille (University of Lorraine)	Philipp Göhlke (University of Vienna)	Sébastien Ferenczi (I2M Marseille)
  17:00 - 17:15	mini break	mini break	mini 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).
  
  
  • Speaker: Manon Blanc
    Title: Robustness in Dynamical Systems: when allowing noise makes the undecidable decidable
    Abstract: Reachability for dynamical systems evolving over the reals is undecidable. However, various results in the literature have shown that decision procedures exist when restricting to robust systems, provided a suitably chosen notion of robustness is employed. When infinitesimal perturbations are allowed, decision procedures for state reachability exist. More fundamentally, while all these statements are only about computability issues, we also consider complexity theory aspects. We prove that robustness to some precision is inherently related to the complexity of the decision procedure. We also give and prove two notions of robustness for tilesets in order to have a decidable Domino problem.
  
  
  • Speaker: Solène Esnay
    Title: Bootstrap Percolation on Rhombus Tilings
    Abstract: In this talk, we introduce dynamical (bootstrap) percolation, where a polygonal tiling of the plane has each tile independently colored 0 or 1 following a measure mu; then the 1 state propagates step by step to tiles adjacent to at least two 1-colored tiles. We are interested in the asymptotic behavior of this system, which is a cellular automaton with a spreading 1 state: is the entire tiling asymptotically invaded by 1s from mu-almost every initial configuration?
    
    After summarizing how this is known on $\mathbb{Z}^2$, we answer this question positively for a wide variety of tilings (all rhombus tilings, with a focus during the presentation on Penrose tilings) and measures (a large class that includes nontrivial Bernoulli measures). We do so with a careful study of what stable non-invading patches of 1s would look like, then use probabilistic arguments to explain why this situation mu-almost never happens.
    
    We end this presentation with possible future extensions of this work (directed propagation, other adjacency graph structures…).
    
    This talk is based on a joint work with Victor Lutfalla and Guillaume Theyssier.
  
  
  • Speaker: Sébastien Ferenczi
    Title: Languages of interval exchanges and the clustering property
    Abstract: We investigate various interactions between the codings of interval exchange transformations and the property of clustering for the Burrows-Wheeler transform, a lossless algorithm used in data compression. We state two main results linking these objects through a combinatorial property of languages we call the order condition; then we use this theory to answer some questions on the clustering of return words of interval exchanges, and on language related features distinguishing between standard and generalised interval exchange transformations.
  
  
  • Speaker: Philipp Göhlke
    Title: Multifractal analysis of generalized Thue-Morse measures
    Abstract: The Thue-Morse measure is a spectral (diffraction) measure of the classical Thue-Morse sequence, and a prototypical example of a singular continuous measure. It can also be described as the equilibrium measure of a singular potential over the doubling map. This allows us to capture the local dimensions of the measure via tools from thermodynamic formalism. Some anomalies occur due to the singularity in the potential, including a super-polynomial decay around dyadic points.
    
    These observations can be extended to a family of generalized Thue-Morse measures, each of them related to a 2-multiplicative sequence that is the fixed point of a substitution on a compact alphabet. We show that several thermodynamic quantities depend sensitively on the parameter and discuss the implications on the multifractal analysis of these measures. (based on joint work with M.Baake, M.Kesseböhmer, G.Lamprinakis, T.Schindler and J.Schmeling)
  
  
  • Speaker: Daniel Graça
    Title: Theory of computation and dynamical systems
    Abstract: In this talk we will explore interconnections between the theory of computation and dynamical systems. We will begin by providing simple examples and their underlying motivation to demonstrate the interest of this approach, and we will then proceed to review concepts and notions from the theory of computation and from dynamical systems theory.
    
    We continue by taking two complementary perspectives to connect the theory of computation with dynamical systems theory, with an emphasis on continuous-time dynamical systems. We start by analyzing several results from dynamical systems theory from a computational perspective, where the objective is to understand how complex dynamical phenomena can be from a computational perspective. We will see that algorithmic hardness is an independent source of complex behavior for dynamical systems. In the second perspective, we consider classes of dynamical systems that are able by themselves to perform computation, and we relate their computational power to classical (discrete) models of computation, to gain further insights between the connections of the theory of computation and dynamical systems theory.
  
  
  • Speaker: Benjamin Hellouin de Menibus
    Title: Surjective cellular automata and probability measures
    Abstract: I will be presenting a variety of results about the typical behaviour of surjective cellular automata when the initial configuration is drawn at random (in particular their limit probability measures or invariant probability measures).
    
    For general cellular automata, uncomputable behaviour is possible and any nontrivial question is undecidable. By contrast, the hypothesis of surjectivity seem to enforce a lot of combinatorial and thermodynamical structure, with the uniform measure playing a special role. It is largely open what behaviours exactly can happen.
    
    I will cover a few examples of well-understood behaviours (conservation laws and invariant measures, randomisation), as well as some more artificial counterexamples and open questions.
  
  
  • Speaker: Gaël Meignan
    Title: Dichotomy theorem for minimizing configurations and minimizing measures in the discrete Aubry-Mather model
    Abstract: We present results over existence and characterization of minimizing configurations and minimizing measures on the one dimensional lattice $\mathbb{Z}$. In order to generalize what is known for nearest neighbor interactions to finite range interactions, we first provide dichotomy results over the set of minimizing configurations. We prove for $\phi \in \mathbb{R}^\mathbb{Z}$ minimizing configuration the equivalence between
    • $\liminf_n \frac{1}{(2n+1)} \sum_{B \cap [ -n,n ]\neq \emptyset} H_B(\phi) < + \infty$
    • $|\phi_i - \phi_j| \leq K|i-j|$ for some $K > 0$
    • $\sup_{j \in \mathbb{Z}} \sup_{i \in \mathbb{Z}} |\phi_j - \phi_i - \omega (j-i)| \leq L$ for some $\omega \in \mathbb{R}$ and $L > 0$
    Then, we discuss the existence and characterization of invariant minimizing measures defined on the space of configurations. In this direction, we give a characterization of the ground state energy of an interaction. This will allow us to assert that the support of such minimizing measures is included in the minimizing configurations of our model. Finally, thanks to our first dichotomy result, these supports are included in configurations with strong rotation number or even satisfying the Birkhoff property depending on the interaction.
  
  
  • Speaker: Jacek Miȩkisz (ENS de Lyon)
    Title: Quasicrystals, a meeting place for mathematical physics, ergodic theory, and topology
    Abstract: We will address a fundamental question how global order arises from local rules. This is a story of quasicrystals which are non-periodic structures but nevertheless are uniform in certain senses. We present a brief history of quasicrystals and a short introduction to classical lattice-gas models of interacting particles and show how ergodic theory and topology enter mathematical physics.
    
    We will discuss uniquely ergodic measures of non-periodic tilings and one-dimensional non-periodic sequences which form symbolic dynamical systems. Symbols can be interpreted as particles and we are concerned with configurations which minimize interaction energy of particles, the so-called ground-state configurations. One of the main questions is whether ground-state configurations are stable with respect to small perturbations of interactions. Such stability may be interpreted as a sort of rigidity of non-periodic structures such as tilings, substitution systems, and other uniquely ergodic non-periodic systems. We introduce a property of bounded fluctuations of finite patterns of non-periodic configurations, the so-called Strict Boundary Property. We discuss how relevant this property is for stability of non-periodic structures. Examples of Robinson's tilings, Thue-Morse, Fibonacci, and general Sturmian systems are presented. Some open problems are formulated.
    
    Bibliography
    
    J. Miekisz, Quasicrystal problem – on rigidity of nonperiodic structures from statistical mechanics point of view, in Proceedings of Geometric Methods in Physics XLI Workshop, Bialystok, Poland, 2024
    editors: Piotr Kielanowski, Alina Dobrogowska, David Fernandez, Tomasz Golinski, Trends in Mathematics, Birkhäuser (2025) https://arxiv.org/pdf/2412.19594
  
  
  • Speaker: Léo Paviet Salomon (ENS de Lyon)
    Title: Variants of the Mozes theorem and graph subshifts
    Abstract: This talk presents some variations on the "Mozes Theorem" [Mozes, 1989], which can be summarized as "Under mild conditions, substitutive subshifts are sofic". We present the different contexts in which similar statements hold: $\mathbb{Z}^d$ subshifts, $\mathbb{R}^d$ subshifts, some Cayley graphs, Schreier graphs of automata groups ... The two main goals of the talk are the following:
    
    First, try to derive a meta-theorem, precising the "mild conditions" under which the theorem holds in all those contexts. We claim that they are mostly of combinatorial nature, and that the different variants of the theorem can then be proven in a very similar way.
    
    Then, try to give another proof of this result, in a new setting: graph subshifts. We continue the work of [Arrighi, Durbec, Guillon, 2023] and try to define a model of substitutions on the relevant class of graphs; we finally obtain a (here stated informally) theorem which generalizes some of the previously known results:
    
    If $s$ is a sufficiently-connected graph-substitution, and $X_s$ is the graph-subshift it generates, then the set of graph-coverings of $X_s$ is sofic.
  
  
  • Speaker: William da Silva (University of Vienna)
    Title: The self-dual point of Fortuin--Kasteleyn planar maps is critical
    Abstract: Fortuin-Kasteleyn (FK) maps form a classical model of planar maps decorated with a percolation-like configuration, depending on a weight $q > 0$. This model is also bijectively equivalent to the fully packed (bicoloured) loop-O(n) model on planar triangulations. These have been traditionally studied using either techniques from analytic combinatorics (based in particular on the gasket decomposition of Borot, Bouttier and Guitter) or probabilistic arguments (based on Sheffield's hamburger-cheeseburger bijection). In this work, we establish a dictionary relating quantities of interest in both approaches, which allows us to derive precise asymptotics for geometric features of the self-dual Fortuin--Kasteleyn planar map model when $ 0 < q < 4$, such as the exact polynomial tail behaviour of cluster and loop sizes. I will explain how this can be used to establish that Fortuin--Kasteleyn maps undergo a sharp phase transition at the self-dual point, therefore proving that the self-dual point of Fortuin--Kasteleyn maps is the critical point. This talk is based on joint work with Nathanaël Berestycki (University of Vienna).
  
  
  • Speaker: Jean Vereecke (University Grenoble Alpes)
    Title: Uncountably many Series--Sinai states are extremal on Lobachevsky lattices
    Abstract: Starting from the family of extremal states constructed by D'Achille--Coquille--Le~Ny for the Ising model on Lobachevsky lattices (see the talk by Matteo D'Achille), I will present a way to extract from this family uncountably many ''interface states'' when the tessellation is of high degree. The latter interface states are analogues of Dobrushin states in the hyperbolic setting, i.e. weak limits of finite-volume Ising measures with boundary condition $+1$ on one side of a geodesics of the Poincaré disk $\mathbb{H}^2$, and $-1$ on the other side. This partially solves a conjecture by Series and Sinai who proved in 1990 that any two such states are mutually singular.
    
    The main tool we use is the Morse--Mostow lemma of hyperbolic geometry. We think there should be a (potentially stronger) alternative proof using Bowen-Series codings.
  
  
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