Related papers: The transition constant for arithmetic hyperbolic …
We compute the Cheeger constants of a collection of hyperbolic surfaces corresponding to maximal non-compact arithmetic Fuchsian groups, and to subgroups which are the rotation subgroup of maximal reflection groups. The Cheeger constants…
This paper presents sufficient conditions for Hamiltonian paths and cycles in graphs. Letting $\lambda\left( G\right) $ denote the spectral radius of the adjacency matrix of a graph $G,$ the main results of the paper are: (1) Let $k\geq1,$…
The connective constant of a graph is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. Strict inequalities are proved for connective constants of vertex-transitive graphs. Firstly, the connective…
We identify the finitely many arithmetic lattices $\Gamma$ in the orientation preserving isometry group of hyperbolic $3$-space $\mathbb{H}^3$ generated by an element of order $4$ and and element of order $p\geq 2$. Thus $\Gamma$ has a…
We study the (Ahlfors regular) conformal dimension of the boundary at infinity of Gromov hyperbolic groups which split over elementary subgroups. If such a group is not virtually free, we show that the conformal dimension is equal to the…
Near a quantum critical point (QCP) in a metal, strong Fermion-Fermion interactions mediated by soft collective bosons give rise to two competing phenomena: non-Fermi liquid behavior and superconductivity that deviates from conventional BCS…
Let $\Gamma$ be a Gromov hyperbolic group, endowed with an arbitrary left-invariant hyperbolic metric, quasi-isometric to a word metric. The action of $\Gamma$ on its boundary $\partial\Gamma$ endowed with the Patterson-Sullivan measure…
Measurements of the Nernst signal in the vortex-liquid state of the cuprates to high fields (33 T) reveal that vorticity extends to very high fields even close to the zero-field critical temperature $T_{c0}$. In overdoped $\rm…
A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e. the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study…
We use an analytic criterion for vanishing of exponential damping of correlations developed previously (Piasecki et al, J. Chem. Phys., 133, 164507, 2010) to determine the threshold volume fractions for structural transitions in hard sphere…
Let $X$ be an infinite, locally finite, connected, quasi-transitive graph without loops or multiple edges. A graph height function on $X$ is a map adapted to the graph structure, assigning to every vertex an integer, called height. Bridges…
For any integer $n \geq 5$, we construct an $n$-dimensional $C^1$ vector field exhibiting a robustly transitive singular attractor which is not sectional-hyperbolic. Nevertheless, the attractor is singular-hyperbolic. This provides the…
Standard field theoretic renormalization group is applied to the model of landscape erosion introduced by R. Pastor-Satorras and D. H. Rothman [Phys. Rev. Lett. 80: 4349 (1998); J. Stat. Phys. 93: 477 (1998)] yielding unexpected results:…
By definition, a rigid graph in $\mathbb{R}^d$ (or on a sphere) has a finite number of embeddings up to rigid motions for a given set of edge length constraints. These embeddings are related to the real solutions of an algebraic system.…
We investigate the maximal number $N_h(m)$ of normally hyperbolic limit tori in three-dimensional polynomial vector fields of degree $m$, which extends the classical notion of Hilbert numbers to higher dimensions. Using recent developments…
For a generalized Su-Schrieffer-Heeger model the energy zero is always critical and hyperbolic in the sense that all reduced transfer matrices commute and have their spectrum off the unit circle. Disorder driven topological phase…
We show that if a graph $G$ with $n \geq 3$ vertices can be drawn in the plane such that each of its edges is involved in at most four crossings, then $G$ has at most $6n-12$ edges. This settles a conjecture of Pach, Radoi\v{c}i\'{c},…
We derive improved upper bounds on the number of crossing-free straight-edge spanning cycles (also known as Hamiltonian tours and simple polygonizations) that can be embedded over any specific set of $N$ points in the plane. More…
We formulate the most general gravitational models with constant negative curvature ("hyperbolic gravity") on an arbitrary orientable two-dimensional surface of genus $g$ with $b$ circle boundaries in terms of a $\text{PSL}(2,\mathbb…
In this work, we study a family of random geometric graphs on hyperbolic spaces. In this setting, N points are chosen randomly on a hyperbolic space and any two of them are joined by an edge with probability that depends on their hyperbolic…