arXiv subjects

Zhongyang Li

Publications and source records attributed to Zhongyang Li.

86 records · Page 5Linked to original sources

arXiv2016

Connective constants and height functions for Cayley graphs

The connective constant $μ(G)$ of an infinite transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. In earlier work of Grimmett and Li, a locality theorem was proved for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin. A condition of the theorem was that the graphs support so-called 'unimodular graph height functions'. When the graphs are Cayley graphs of infinite, finitely generated groups, there is a special type of unimodular graph height function termed here a 'group height function'. A necessary and sufficient condition for the existence of a group height function is presented, and may be applied in the context of the bridge constant, and of the locality of connective constants for Cayley graphs. Locality may thereby be established for a variety of infinite groups including those with strictly positive deficiency. It is proved that a large class of Cayley graphs support unimodular graph height functions, that are in addition harmonic on the graph. This implies, for example, the existence of unimodular graph height functions for the Cayley graphs of finitely generated solvable groups. It turns out that graphs with non-unimodular automorphism subgroups also possess graph height functions, but the resulting graph height functions need not be harmonic. Group height functions, as well as the graph height functions of the previous paragraph, are non-constant harmonic functions with linear growth and an additional property of having periodic differences. The existence of such functions on Cayley graphs is a topic of interest beyond their applications in the theory of self-avoiding walks.

math.CO↗

arXiv2016

Critical surface of the hexagonal polygon model

The hexagonal polygon model arises in a natural way via a transformation of the 1-2 model on the hexagonal lattice, and it is related to the high temperature expansion of the Ising model. There are three types of edge, and three corresponding parameters $α,β,γ>0$. By studying the long-range order of a certain two-edge correlation function, it is shown that the parameter space $(0,\infty)^3$ may be divided into subcritical and supercritical regions, separated by critical surfaces satisfying an explicitly known formula. This result complements earlier work on the Ising model and the 1-2 model. The proof uses the Pfaffian representation of Fisher, Kasteleyn, and Temperley for the counts of dimers on planar graphs.

math-ph↗

arXiv2015

Self-avoiding walks and amenability

The connective constant $μ(G)$ of an infinite transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. The relationship between connective constants and amenability is explored in the current work. Various properties of connective constants depend on the existence of so-called 'graph height functions', namely: (i) whether $μ(G)$ is a local function on certain graphs derived from $G$, (ii) the equality of $μ(G)$ and the asymptotic growth rate of bridges, and (iii) whether there exists a terminating algorithm for approximating $μ(G)$ to a given degree of accuracy. In the context of amenable groups, it is proved that the Cayley graphs of infinite, finitely generated, elementary amenable groups support graph height functions, which are in addition harmonic. In contrast, the Cayley graph of the Grigorchuk group, which is amenable but not elementary amenable, does not have a graph height function. In the context of non-amenable, transitive graphs, a lower bound is presented for the connective constant in terms of the spectral bottom of the graph. This is a strengthening of an earlier result of the same authors. Secondly, using a percolation inequality of Benjamini, Nachmias, and Peres, it is explained that the connective constant of a non-amenable, transitive graph with large girth is close to that of a regular tree. Examples are given of non-amenable groups without graph height functions, of which one is the Higman group.

math.GR↗

arXiv2015

The 1-2 model

The current paper is a short review of rigorous results for the 1-2 model. The 1-2 model on the hexagonal lattice is a model of statistical mechanics in which each vertex is constrained to have degree either 1 or 2. It was proposed in a study by Schwartz and Bruck of constrained coding systems, and is strongly connected to the dimer model on a decoration of the lattice, and to an enhanced Ising model and an associated polygon model on the graph derived from the hexagonal lattice by adding a further vertex in the middle of each edge. The general 1-2 model possesses three parameters $a$, $b$, $c$. The fundamental technique is to represent probabilities of interest as ratios of counts of dimer coverings of certain associated graphs, and to apply the Pfaffian method of Kasteleyn, Fisher, and Temperley. Of special interest is the existence (or not) of phase transitions. It turns out that all clusters of the infinite-volume limit are almost surely finite. On the other hand, the existence (with strictly positive probability) of infinite `homogeneous' clusters, containing vertices of given type, depends on the values of the parameters. A further type of phase transition emerges in the study of the two-edge correlation function, and in this case the critical surface may be found explicitly. For instance, when $a \ge b \ge c > 0$, the surface given by $\sqrt a = \sqrt b + \sqrt c$ is critical.

math.PR↗

arXiv2015

Counting self-avoiding walks

The connective constant $μ(G)$ of a graph $G$ is the asymptotic growth rate of the number of self-avoiding walks on $G$ from a given starting vertex. We survey three aspects of the dependence of the connective constant on the underlying graph $G$. Firstly, when $G$ is cubic, we study the effect on $μ(G)$ of the Fisher transformation (that is, the replacement of vertices by triangles). Secondly, we discuss upper and lower bounds for $μ(G)$ when $G$ is regular. Thirdly, we present strict inequalities for the connective constants $μ(G)$ of vertex-transitive graphs $G$, as $G$ varies. As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a non-trivial group element is declared to be a generator. Special prominence is given to open problems.

math.CO↗

arXiv2014

Large-area, Lithography-Free Super Absorbers and Color Filters at Visible Frequencies Using Ultrathin Metallic Films

Plasmonic materials and metamaterials have been widely utilized to achieve spectral transmission, reflection and absorption filters based on localized or delocalized resonances arising from the interaction of photons with nanostructured materials. Realization of visible-frequency, high-performance, large-area, optical filters based on nanoplasmonic materials is rather challenging due to nanofabrication related problems (cost, fabrication imperfection, surface roughness) and optical losses of metals. Here, we propose and demonstrate large-area perfect absorbers and transmission filters that overcome difficulties associated with the nanofabrication using a lithography-free approach. We also utilize and benefit from the optical losses in metals in our optical filter designs. Our resonant optical filter design is based on a modified, asymmetric metal-insulator-metal (MIM) based Fabry-Perot cavity with plasmonic, lossy ultra-thin (~30 nm) metallic films used as the top metallic layer. We demonstrated a narrow bandwidth (~17 nm) super absorber with 97% maximum absorption with a performance comparable to nanostructure/nanoparticle-based super absorbers. We also investigated transmission (color) filters using ultra-thin metallic films, in which different colors can be obtained by controlling the dielectric spacer thickness. With performance parameters of transmission peak intensity reaching 60% and a narrow-band of ~ 40 nm, our color filters exceed the performance of widely studied plasmonic nanohole array based color filters. Proposed asymmetric Fabry-Perot cavities using ultra-thin metallic films could find applications in spectrally selective optical (color and absorber) filters, optoelectronic devices with controlled bandwidth such as narrow-band photodetectors, and light-emitting devices.

physics.optics↗

arXiv2014

Visible-frequency metasurfaces for broadband anomalous reflection and high-efficiency spectrum splitting

Ultrathin metasurfaces have recently emerged as promising materials to enable novel, flat optical components and surface-confined, miniature photonic devices. However, experimental realization of high-performance metasurfaces at visible frequencies has been a significant challenge due to high plasmonic losses and difficulties in high-uniformity nanofabrication. Here, we propose a highly-efficient yet simple metasurface design comprising of single gradient antenna as unit cell. We demonstrate visible broadband (450 - 850 nm) anomalous reflection and spectrum splitting with 85% conversion efficiency. Average power ratio of anomalous reflection to the strongest diffraction was calculated to be ~ 103 and measured to be ~ 10. The anomalous reflected photons and spectrum splitting performance have been visualized using CCD and characterized using angle-resolved measurement setup. Metasurface design proposed here is a clear departure from conventional metasurfaces utilizing multiple, anisotropic resonators, and could enable high-efficiency, broadband metasurfaces for achieving directional emitters, polarization/spectrum splitting surfaces for spectroscopy and photovoltaics.

cond-mat.mes-hall↗

arXiv2014

Strict inequalities for connective constants of transitive graphs

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 constant decreases strictly when the graph is replaced by a non-trivial quotient graph. Secondly, the connective constant increases strictly when a quasi-transitive family of new edges is added. These results have the following implications for Cayley graphs. The connective constant of a Cayley graph decreases strictly when a new relator is added to the group, and increases strictly when a non-trivial group element is declared to be a generator.

math.CO↗

arXiv2013

Bounds on connective constants of regular graphs

Bounds are proved for the connective constant μ of an infinite, connected, Δ-regular graph G. The main result is that μ \ge \sqrt{Δ-1} if G is vertex-transitive and simple. This inequality is proved subject to weaker conditions under which it is sharp.

math.CO↗

arXiv2012

Uniqueness of Infinite Homogeneous Clusters in 1-2 Model

A 1-2 model configuration is a subset of edges of the hexagonal lattice such that each vertex is incident to one or two edges. We prove that for any translation-invariant Gibbs measure of 1-2 model, almost surely the infinite homogeneous cluster is unique.

math.PR↗

arXiv2012

Self-avoiding walks and the Fisher transformation

The Fisher transformation acts on cubic graphs by replacing each vertex by a triangle. We explore the action of the Fisher transformation on the set of self-avoiding walks of a cubic graph. Iteration of the transformation yields a sequence of graphs with common critical exponents, and with connective constants converging geometrically to the golden mean. We consider the application of the Fisher transformation to one of the two classes of vertices of a bipartite cubic graph. The connective constant of the ensuing graph may be expressed in terms of that of the initial graph. When applied to the hexagonal lattice, this identifies a further lattice whose connective constant may be computed rigorously.

math.CO↗

arXiv2012

Spectral Curve of Periodic Fisher Graphs

We study the spectral curves of dimer models on periodic Fisher graphs, obtained from a ferromagnetic Ising model on $\mathbb{Z}^2$. The spectral curve is defined by the zero locus of the determinant of a modified weighted adjacency matrix. We prove that either they are disjoint from the unit torus ($\mathbb{T}^2=\{(z,w):|z|=1,|w|=1\}$) or they intersect $\mathbb{T}^2$ at a single real point.

math.CV↗

arXiv2012

Critical Temperature of Periodic Ising Models

A periodic Ising model is one endowed with interactions that are invariant under translations of members of a full-rank sublattice $\mathfrak{L}$ of $\mathbb{Z}^2$. We give an exact, quantitative description of the critical temperature, defined by the supreme of the temperatures at which the spontaneous magnetization of a periodic, Ising ferromagnets is nonzero, as the solution of a certain algebraic equation, namely, the condition that the spectral curve of the corresponding dimer model on the Fisher graph has a real node on the unit torus. A simple proof for the exponential decay of spin-spin correlations above the critical temperature for the symmetric, periodic Ising ferromagnet, as well as the exponential decay of the edge-edge correlations for all non-critical edge weights of the dimer model on periodic Fisher graphs, is obtained by our technique.

math-ph↗

arXiv2010

Local Statistics of Realizable Vertex Models

We study planar "vertex" models, which are probability measures on edge subsets of a planar graph, satisfying certain constraints at each vertex, examples including dimer model, and 1-2 model, which we will define. We express the local statistics of a large class of vertex models on a finite hexagonal lattice as a linear combination of the local statistics of dimers on the corresponding Fisher graph, with the help of a generalized holographic algorithm. Using an $n\times n$ torus to approximate the periodic infinite graph, we give an explicit integral formula for the free energy and local statistics for configurations of the vertex model on an infinite bi-periodic graph. As an example, we simulate the 1-2 model by the technique of Glauber dynamics.

math-ph↗