Since the discovery of quasicrystals in 1984, one of the fundamental problems in statistical physics is to prove stability of nonperiodic ground-state measures (measures which minimize energy of hamiltonians of systems of interacting particles). This concerns the stability with respect to small perturbations of energy functionals and low-temperature stability against thermal fluctuations. The main open problem is to construct a classical lattice-gas model on Z 2 with translation-invariant interactions and with the unique translation-invariant ground-state measure supported by nonperiodic ground-state configurations. Two-dimensional (2d) models with finite-range interactions are based on non-periodic tilings of the plane (for example Robinson’s tilings). One-dimensional (1d) models with infinite-range interactions are based on substitution dynamical systems (for example Thue-Morse sequences) or Sturmian systems defined by rotations on the circle (for example Fibonacci sequences which are also produced by substitutions). In the case of Sturmian systems obtained by rotations by badly approximable irrationals some results were recently obtained in the framework of NCN grant, the current applicant was its principal investigator. We proved fast convergence in ergodic theorem and use it to show non-stability of non-periodic ground states for fast decaying interactions. We would like to pursue this approach to prove stability for slower decaying interactions. In both cases unique ground-state measures correspond to uniquely ergodic measures of corresponding symbolic dynamical systems. 

Non-periodic tilings enforced by matching rules give rise to interactions between particles, where particles correspond to tiles (or in general symbols in nonperiodic sequences) and interactions to matching rules which are violated. These are so-called systems of finite-type. In one-dimensional models we try to construct systems of minimal infinite type (it is known that one cannot force non-periodicity in 1d by finite rules/finite-range interactions). 

It seems that one of the properties of non-periodic tilings (or non-periodic configurations in general) is the rapid convergence to equilibrium of frequencies of local patterns, the so called Strict Boundary Property (fluctuations are bounded by the perimeter of the considered finite subset of the lattice). We will analyze the relation of this property to topological and ergodic properties of nonperiodic configurations or sequences. 

The non-periodic long-range order is an effect of finite-range interactions. This means that relevant information should flow without disruption over arbitrarily long distances. This somehow corresponds to actions at a distance in quantum systems, That is one of the reason to explore connections between classical and quantum models of interacting particles. 

In all our examples we have an uncountably many ground-state configurations. This is reminiscent of particle systems with continuous symmetry. The famous Mermin-Wagner theorem states that two-dimensional systems with continuous symmetries do not exhibit phase transitions. We will study if such a theorem is applicable in our systems. 

It is known that d-dimensional probabilistic cellular automata models give rise to d+1 equilibrium lattice-gas models. Non-periodic tilings can be incorporated in PCA and might give rise to nonperiodic Gibbs states. Some results were obtained recently within the framework of NCN grant. We will explore further this connection. 

Our primary efforts will be to analyze infinite system on nonperiodic structure by analytical means. However, we are aware that such problems are very hard. We will plan to supplement our analysis by numerical and stochastic simulations and elements of machine learning (recently used to locate phase transitions in systems of interacting particles).

 Our project involves systems of many (infinitely many) interacting particles. Sets of configurations of our models are uncountable, therefore the data generated in computer simulations (in finite systems) is enormous. Hence these might appropriate systems to be treated by methods of machine learning.

Our aim is to develop connections between various approaches to mathematical analysis of nonperiodic structures. This is an active research area present in many scientific meeting in various branches of mathematics, see for example recent workshop https://conferences.cirm-math.fr/3002.html

 We hope that our project will facilitate cooperation between mathematicians and physicists of the University of Warsaw and also between mathematicians in our mathematics faculty, and it will attract students to this research. We plan to organize at MIMUW a school for master and PhD students of both mathematics and physics and a weekly working seminar. We also plan to organize a workshop outside Warsaw. We expect that around 30 students, PhD students, and University of Warsaw scientists will participate in our programme.