Partition Function Zeros of a Restricted Potts Model on Lattice Strips and Effects of Boundary Conditions

We calculate the partition function $Z(G,Q,v)$ of the $Q$-state Potts model exactly for strips of the square and triangular lattices of various widths $L_y$ and arbitrarily great lengths $L_x$, with a variety of boundary conditions, and with $Q$ and $v$ restricted to satisfy conditions corresponding to the ferromagnetic phase transition on the associated two-dimensional lattices. From these calculations, in the limit $L_x \to \infty$, we determine the continuous accumulation loci ${\cal B}$ of the partition function zeros in the $v$ and $Q$ planes. Strips of the honeycomb lattice are also considered. We discuss some general features of these loci.Comment: 12 pages, 12 figure

Shortcut sets for the locus of plane Euclidean networks

We study the problem of augmenting the locus N of a plane Euclidean network N by in- serting iteratively a finite set of segments, called shortcut set , while reducing the diameterof the locus of the resulting network. There are two main differences with the classicalaugmentation problems: the endpoints of the segments are allowed to be points of N as well as points of the previously inserted segments (instead of only vertices of N ), and the notion of diameter is adapted to the fact that we deal with N instead of N . This increases enormously the hardness of the problem but also its possible practical applications to net- work design. Among other results, we characterize the existence of shortcut sets, computethem in polynomial time, and analyze the role of the convex hull of N when inserting a shortcut set. Our main results prove that, while the problem of minimizing the size of ashortcut set is NP-hard, one can always determine in polynomial time whether insertingonly one segment suffices to reduce the diameter.Ministerio de Economía y Competitividad MTM2015-63791-

Unbounded sl3\mathfrak{sl}_3-laminations and their shear coordinates

We study the space $\mathcal{L}_{\mathfrak{sl}_3}^x(\Sigma,\mathbb{Q})$ of rational unbounded $\mathfrak{sl}_3$-laminations on a marked surface $\Sigma$. We introduce an $\mathfrak{sl}_3$-analogue of the Thurston's shear coordinates associated with any decorated triangulation, which gives rise to a natural identification $\mathcal{L}_{\mathfrak{sl}_3}^x(\Sigma,\mathbb{Q}) \cong \mathcal{X}_{\mathfrak{sl}_3,\Sigma}^\mathrm{uf}(\mathbb{Q}^{\mathrm{trop}})$. We also introduce the space $\mathcal{L}^p(\Sigma,\mathbb{Q})$ of rational unbounded $\mathfrak{sl}_3$-laminations with pinnings, which possesses the frozen coordinates as well. Then we give a tropical anologue of the amalgamation maps [FG06b] between them, which is indeed a procedure of gluing $\mathfrak{sl}_3$-laminations with "shearings". We also investigate a relation to the graphical basis of the $\mathfrak{sl}_3$-skein algebra [IY21], which conjecturally leads to a quantum duality map.Comment: 67 pages, 39 figure