Related papers: Long-Range Antiferromagnetic Order in the AKLT Mod…

In this paper, we provide algorithms to rank and unrank certain degree-restricted classes of Cayley trees (spanning trees of the n-vertex complete graph). Specifically, we consider classes of trees that have a given set of leaves or a fixed…

Returning to one of the original generalizations of the AKLT state, we extend prior analysis on the Bethe lattice (or Cayley tree) to a variant with a series of $n$ spin-1 decorations placed on each edge. The recurrence relations derived…

To further develop the statistical inference problem for heterogeneous treatment effects, this paper builds on Breiman's (2001) random forest tree (RFT)and Wager et al.'s (2018) causal tree to parameterize the nonparametric problem using…

In this paper, we provide algorithms to rank, unrank, and randomly generate certain degree-restricted classes of Cayley trees. Specifically, we consider classes of trees that have a given degree sequence or a given multiset of degrees. If…

In this paper, we consider the Ising-Vanniminus model on an arbitrary order Cayley tree. We generalize the results conjectured in [Chinese Journal of Physics, 54 (4), 635-649 (2016)] and [International Journal of Modern Physics,…

The models constructed by Affleck, Kennedy, Lieb, and Tasaki describe a family of quantum antiferromagnets on arbitrary lattices, where the local spin S is an integer multiple M of half the lattice coordination number. The equal time…

Tree-level Feynman diagrams in a cubic scalar theory can be given a metric such that each edge has a length. The space of metric trees is made out of orthants joined where a tree degenerates. Here we restrict to planar trees since each…

A new tree model is introduced based on ordered trees, by distinguishing exactly one child of each node that \emph{has} children. The basic enumeration leads to a cubic equation of the generating function. The extraction of its coefficients…

We prove that every amenable one-ended Cayley graph has an invariant spanning tree of one end. More generally, for any 1-ended amenable unimodular random graph we construct a factor of iid percolation (jointly unimodular subgraph) that is…

We investigate analytically and numerically eigenfunction statistics in a disordered system on a finite Bethe lattice (Cayley tree). We show that the wave function amplitude at the root of a tree is distributed fractally in a large part of…

The staggered magnetization of the Heisenberg antiferromagnet in two dimensions can be systematically approximated by a 1/N expansion. Cancellation between self energy diagrams leads to a Luttinger-like theorem for the ground state. We…

We expound a concise construction of finite groups and groupoids whose Cayley graphs satisfy graded acyclicity requirements. Our acyclicity criteria concern cyclic patterns formed by coset-like configurations w.r.t. subsets of the generator…

The stability of antiferromagnetic long-range order against quenched disorder is considered. A simple model of an antiferromagnet with a spatially varying Neel temperature is shown to possess a nontrivial fixed point corresponding to…

We construct a sequence of finite graphs that weakly converge to a Cayley graph, but there is no labelling of the edges that would converge to the corresponding Cayley diagram. A similar construction is used to give graph sequences that…

We consider nearest-neighbor fertile hard-core models, with three states, on a homogeneous Cayley tree. It is known that there are four type of such models. We investigate all of them and describe translation-invariant and periodic…

We show, under natural conditions, that uniform rooted trees with fixed degree sequence converge after renormalization toward inhomogeneous continuum random trees (ICRT). We also provide a sharp upper-bound for the tail of their heights. We…

There are few exactly solvable lattice models and even fewer solvable quantum lattice models. Here we address the problem of finding the spectrum of the tight-binding model (equivalently, the spectrum of the adjacency matrix) on Cayley…

Strongly regular Cayley graphs with Paley parameters over abelian groups of rank 2 were studied in [J.A Davis, Partial difference sets in p-groups, Arch.Math.63 (1994) 103-110; K.H Leung, S.L. Ma, Partial difference sets with Paley…

The vertices of the Cayley graph of a finitely generated semigroup form a set of sites which can be labeled by elements of a finite alphabet in a manner governed by a nonnegative real interaction matrix, respecting nearest neighbor…

Cayley graphs have a number of useful features: the ability to graphically represent finitely generated group elements and their relations; to name all vertices relative to a point; and the fact that they have a well-defined notion of…