Selected Publications

Papers are bundled according to subject area with short intros. The groups are in roughly inverse chronological order i.e. latest first.

Long range exclusions

In Symbolic Dynamics resolving the size of a subshift is not trivial even in 1-d. For finite subshifts one can utilize the matrix formulation, but for infinite ones no such tool exists. Here is an illustration what can happen, elaboration on a problem originally proposed in two different contexts by P. Erdös and M. Keane.

Dense and entropic packings on graphs

Motivated by the spectacular progress by Hales in solving the Kepler conjecture as well as other results on packing especially in large spaces (higher dimensional Euclidean spaces, the hyperbolic space etc.) we have investigated the geometry of densest packings and their construction with a probabilistic cellular automaton on Archimedean and certain other geometric graphs. There is an interesting trichotomy in the packing type that seems correspond with the type of local/non-local action allowed and the criticality/non-criticality in the generating PCA. In the positive temperature regime the geometric structure at large vanishes and other aspects of this model come to the forefront. In another paper we establishing entropy lower bounds (which are tight i.e. converging unbiassed estimates) without the usual matrix numerics. This method also gives some insight into the structure of the measure of maximal entropy.

Vertex models on planar lattices

The project on bounded lattice systems should be viewed as a part of the program initiated by Conway, Lagarias and Thurston on the tileability of finite planar domains with given primitives (dominoes, polyominoes, nearest neighbor arrows, Wang tiles etc.). In some models the shape of the domain imposes subtle conditions on the fill-in and in some others this is further complicated by the choice of the configuration on the boundary. Overall the problem is more combinatorial and algebraic in character than the infinite problem that has been worked out for many models in the equilibrium Statistical Mechanics. The aim of the project is to extend the earlier findings in the infinite/unbounded case and to formulate general principles that unify aspects of multidimensional symbolic dynamics, tilings and Statistical Mechanics.

The project also has a definite computational aspect: an underlying connectivity principle allows efficient computation of all configurations compatible with the given boundary using PCAs.

Criticality in deterministic cellular automata

The notion of partial permutivity (see below) works in any dimension. In these papers we explore this extension by studying the two dimensional case since already there critical phenomena appear. We compare deterministic and probabilistic c.a. and find that in a strong statistical sense the partially permutive c.a. are extremely close to corresponding PCAs.

Permutive cellular automata

Every one-dimensional cellular automaton rule can been rewritten as a binary operation. If invariant subalphabets are found this leads to an useful tiling interpretation of the bi-infinite evolution. If the subalphabet is non-trivial the cellular automaton has partial permutivity properties. The property of partial permutivity is far more prevalent among c.a. than the (full) permutivity of Hedlund (even just one-sided). The existence of non-trivial permutive subalphabets typically manifests in the evolutions as defects (i.e. Bloch walls or kinks). Their motion can be exactly characterised.

Statistical stability of billiards and Markov processes

Structural stability was extended to the useful notion of statistical stability by D. Ornstein and B. Weiss. It incorporates a natural measure of the system, invariant measure, into the picture and thereby enables generic stability results. Moreover the notion is applicable far beyond the set-up of smooth dynamics (like the hyperbolic geodesic flow) Two applications:

Other dynamics works

Miscellaneous research

Lecture notes

Popular exposition