Competitively tight graphs

The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between two distinct vertices $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x,v)$ and $(y,v)$ are arcs of $D$. For any graph $G$, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ of a graph $G$ is the smallest number of such isolated vertices. Computing the competition number of a graph is an NP-hard problem in general and has been one of the important research problems in the study of competition graphs. Opsut [1982] showed that the competition number of a graph $G$ is related to the edge clique cover number $\theta_E(G)$ of the graph $G$ via $\theta_E(G)-|V(G)|+2 \leq k(G) \leq \theta_E(G)$. We first show that for any positive integer $m$ satisfying $2 \leq m \leq |V(G)|$, there exists a graph $G$ with $k(G)=\theta_E(G)-|V(G)|+m$ and characterize a graph $G$ satisfying $k(G)=\theta_E(G)$. We then focus on what we call \emph{competitively tight graphs} $G$ which satisfy the lower bound, i.e., $k(G)=\theta_E(G)-|V(G)|+2$. We completely characterize the competitively tight graphs having at most two triangles. In addition, we provide a new upper bound for the competition number of a graph from which we derive a sufficient condition and a necessary condition for a graph to be competitively tight.Comment: 10 pages, 2 figure

Universality in survivor distributions: Characterising the winners of competitive dynamics

We investigate the survivor distributions of a spatially extended model of competitive dynamics in different geometries. The model consists of a deterministic dynamical system of individual agents at specified nodes, which might or might not survive the predatory dynamics: all stochasticity is brought in by the initial state. Every such initial state leads to a unique and extended pattern of survivors and non-survivors, which is known as an attractor of the dynamics. We show that the number of such attractors grows exponentially with system size, so that their exact characterisation is limited to only very small systems. Given this, we construct an analytical approach based on inhomogeneous mean-field theory to calculate survival probabilities for arbitrary networks. This powerful (albeit approximate) approach shows how universality arises in survivor distributions via a key concept -- the {\it dynamical fugacity}. Remarkably, in the large-mass limit, the survival probability of a node becomes independent of network geometry, and assumes a simple form which depends only on its mass and degree.Comment: 12 pages, 6 figures, 2 table

Statistical mechanics of glass transition in lattice molecule models

Lattice molecule models are proposed in order to study statistical mechanics of glass transition in finite dimensions. Molecules in the models are represented by hard Wang tiles and their density is controlled by a chemical potential. An infinite series of irregular ground states are constructed theoretically. By defining a glass order parameter as a collection of the overlap with each ground state, a thermodynamic transition to a glass phase is found in a stratified Wang tiles model on a cubic lattice.Comment: 18 pages, 8 figure

Hartree-Fock Theory of Skyrmions in Quantum Hall Ferromagnets

We report on a study of the charged-skyrmion or spin-texture excitations which occur in quantum Hall ferromagnets near odd Landau level filling factors. Particle-hole symmetry is used to relate the spin-quantum numbers of charged particle and hole excitations and neutral particle-hole pair excitations. Hartree-Fock theory is used to provide quantitative estimates of the energies of these excitations and their dependence on Zeeman coupling strength, Landau level quantum numbers, and the thicknesses of the two-dimensional electron layers. For the case of $\nu$ near three we suggest the possibility of first order phase transitions with increasing Zeeman coupling strength from a many skyrmion state to one with many maximally spin-polarized quasiparticles.Comment: 26 pages, 10 figure

Coexistence of competing first passage percolation on hyperbolic graphs

We study a natural growth process with competition, which was recently introduced to analyze MDLA, a challenging model for the growth of an aggregate by diffusing particles. The growth process consists of two first-passage percolation processes $\text{FPP}_1$ and $\text{FPP}_\lambda$, spreading with rates $1$ and $\lambda>0$ respectively, on a graph $G$. $\text{FPP}_1$ starts from a single vertex at the origin $o$, while the initial configuration of $\text{FPP}_\lambda$ consists of infinitely many \emph{seeds} distributed according to a product of Bernoulli measures of parameter $\mu>0$ on $V(G)\setminus \{o\}$. $\text{FPP}_1$ starts spreading from time 0, while each seed of $\text{FPP}_\lambda$ only starts spreading after it has been reached by either $\text{FPP}_1$ or $\text{FPP}_\lambda$. A fundamental question in this model, and in growth processes with competition in general, is whether the two processes coexist (i.e., both produce infinite clusters) with positive probability. We show that this is the case when $G$ is vertex transitive, non-amenable and hyperbolic, in particular, for any $\lambda>0$ there is a $\mu_0=\mu_0(G,\lambda)>0$ such that for all $\mu\in(0,\mu_0)$ the two processes coexist with positive probability. This is the first non-trivial instance where coexistence is established for this model. We also show that $\text{FPP}_\lambda$ produces an infinite cluster almost surely for any positive $\lambda,\mu$, establishing fundamental differences with the behavior of such processes on $\mathbb{Z}^d$.Comment: 53 pages, 13 figure

Reversal of the Charge Transfer between Host and Dopant Atoms in Semiconductor Nanocrystals

We present ab initio density functional calculations that show P (Al) dopant atoms in small hydrogen-terminated Si crystals to be negatively (positively) charged. These signs of the dopant charges are reversed relative to the same dopants in bulk Si. We predict this novel reversal of the dopant charge (and electronic character of the doping) to occur at crystal sizes of order 100 Si atoms. We explain it as a result of competition between fundamental principles governing charge transfer in bulk semiconductors and molecules and predict it to occur in nanocrystals of most semiconductors.Comment: 4 pages, 4 figures (3 in color), 2 table