Related papers: Delocalization of uniform graph homomorphisms from…
We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical Riemann-Hurwitz formula, study the functorial maps on Jacobians and…
A {\em $(d,h)$-decomposition} of a graph $G$ is an order pair $(D,H)$ such that $H$ is a subgraph of $G$ where $H$ has the maximum degree at most $h$ and $D$ is an acyclic orientation of $G-E(H)$ of maximum out-degree at most $d$. A graph…
We study gradient models for spins taking values in the integers (or an integer lattice), which interact via a general potential depending only on the differences of the spin values at neighboring sites, located on a regular tree with d + 1…
In this paper we determine all singular endomorphisms of the Hamming graph and other related graphs. The Hamming graph has vertices $\mathbb{Z}^{m}_n$ where two vertices are adjacent, if their Hamming distance is $1$. We show that its…
$2$-form abelian and non-abelian gauge fields on $d$-dimensional hypercubic lattices have been discussed in the past by various authors and most recently by Lipstein and Reid-Edwards. In this note we recall that the Hamiltonian of a…
Girth-regular graphs with equal girth, regular degree and chromatic index are studied for the determination of 1-factorizations with each 1-factor intersecting every girth cycle. Applications to hamiltonian decomposability and to…
We investigate the existence of Boolean degree $d$ functions on the Grassmann graph of $k$-spaces in the vector space $\mathbb{F}_q^n$. For $d=1$ several non-existence and classification results are known, and no non-trivial examples are…
We propose an exact Hamiltonian lattice theory for (2+1)-dimensional spacetimes with homogeneous curvature. By gauging away the lattice we find a generalization of the ``polygon representation'' of (2+1)-dimensional gravity. We compute the…
We study a class of $\Z^{d}$-substitutive subshifts, including a large family of constant-length substitutions, and homomorphisms between them, i.e., factors modulo isomorphisms of $\Z^{d}$. We prove that any measurable factor map and even…
In this paper we derive a Large Deviation Principle (LDP) for inhomogeneous U/V-statistics of a general order. Using this, we derive a LDP for two types of statistics: random multilinear forms, and number of monochromatic copies of a…
We prove the existence of a finitely dependent proper colouring of the integer lattice Z^d that is fully isometry-invariant in law, for all dimensions d. Previously this was known only for d=1, while only translation-invariant examples were…
Suppose that $f$ is a homomorphism from the mapping class group $\mathcal{M}(N_{g,n})$ of a nonorientable surface of genus $g$ with $n$ boundary components, to $\mathrm{GL}(m,\mathbb{C})$. We prove that if $g\ge 5$, $n\le 1$ and $m\le g-2$,…
We study the first uniformly finite homology group of Block and Weinberger for uniformly locally finite graphs, with coefficients in $\mathbb{Z}$ and $\mathbb{Z}_2$. When the graph is a tree, or coefficients are in $\mathbb{Z}_2$, a…
In this paper, we study the graph classification problem from the graph homomorphism perspective. We consider the homomorphisms from $F$ to $G$, where $G$ is a graph of interest (e.g. molecules or social networks) and $F$ belongs to some…
We show that the six-vertex model with parameter $c\in[\sqrt 3, 2]$ on a square lattice torus has an ergodic infinite-volume limit as the size of the torus grows to infinity. Moreover we prove that for $c\in[\sqrt{2+\sqrt 2}, 2]$, the…
We study a class of duality transformations in generalised Z(2) gauge theories and Ising models on two- and three-dimensional compact lattices. The theories are interpreted algebraically in terms of the structure constants of a…
We study images of equilibrium (Gibbs) states for a class of non-invertible transformations associated to conformal iterated function systems with overlaps $\mathcal S$. We prove exact dimensionality for these image measures, and find a…
We introduce a new type of examples of bounded degree acyclic Borel graphs and study their combinatorial properties in the context of descriptive combinatorics, using a generalization of the determinacy method of Marks. The motivation for…
We prove that coronizations with respect to arbitrary d-regular sets (not necessarily graphs) imply big pieces squared of these (approximating) sets. This is known (and due to David and Semmes in the case of sufficiently large co-dimension,…
We define a number of natural (from geometric and combinatorial points of view) deformation spaces of valuations on finite graphs, and study functions over these deformation spaces. These functions include both direct metric invariants…