Tiling Problems on Baumslag-Solitar groups

We exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove that the domino problem is undecidable on these groups. A consequence of our construction is the existence of an arecursive tile set on Baumslag-Solitar groups.Comment: In Proceedings MCU 2013, arXiv:1309.104

Six vertex model with domain-wall boundary conditions in the Bethe-Peierls approximation

We use the Bethe-Peierls method combined with the belief propagation algorithm to study the arctic curves in the six vertex model on a square lattice with domain-wall boundary conditions, and the six vertex model on a rectangular lattice with partial domain-wall boundary conditions. We show that this rather simple approximation yields results that are remarkably close to the exact ones when these are known, and allows one to estimate the location of the phase boundaries with relative little effort in cases in which exact results are not available.Comment: 19 pages, 14 figure

Two-by-two Substitution Systems and the Undecidability of the Domino Problem

10+6 pagesThanks to a careful study of elementary properties of two-by-two substitution systems, we give a complete self-contained elementary construction of an aperiodic tile set and sketch how to use this tile set to elementary prove the undecidability of the classical Domino Problem

Exact solution of the six-vertex model with domain wall boundary condition. Critical line between ferroelectric and disordered phases

This is a continuation of the papers [4] of Bleher and Fokin and [5] of Bleher and Liechty, in which the large $n$ asymptotics is obtained for the partition function $Z_n$ of the six-vertex model with domain wall boundary conditions in the disordered and ferroelectric phases, respectively. In the present paper we obtain the large $n$ asymptotics of $Z_n$ on the critical line between these two phases.Comment: 22 pages, 6 figures, to appear in the Journal of Statistical Physic

The arctic curve of the domain-wall six-vertex model

The problem of the form of the `arctic' curve of the six-vertex model with domain wall boundary conditions in its disordered regime is addressed. It is well-known that in the scaling limit the model exhibits phase-separation, with regions of order and disorder sharply separated by a smooth curve, called the arctic curve. To find this curve, we study a multiple integral representation for the emptiness formation probability, a correlation function devised to detect spatial transition from order to disorder. We conjecture that the arctic curve, for arbitrary choice of the vertex weights, can be characterized by the condition of condensation of almost all roots of the corresponding saddle-point equations at the same, known, value. In explicit calculations we restrict to the disordered regime for which we have been able to compute the scaling limit of certain generating function entering the saddle-point equations. The arctic curve is obtained in parametric form and appears to be a non-algebraic curve in general; it turns into an algebraic one in the so-called root-of-unity cases. The arctic curve is also discussed in application to the limit shape of $q$-enumerated (with $0<q\leq 4$) large alternating sign matrices. In particular, as $q\to 0$ the limit shape tends to a nontrivial limiting curve, given by a relatively simple equation.Comment: 39 pages, 2 figures; minor correction