Existence of Dλ-cycles and Dλ-paths

A cycle of C of a graph G is called a Dλ-cycle if every component of G − V(C) has order less than λ. A Dλ-path is defined analogously. In particular, a D1-cycle is a hamiltonian cycle and a D1-path is a hamiltonian path. Necessary conditions and sufficient conditions are derived for graphs to have a Dλ-cycle or Dλ-path. The results are generalizations of theorems in hamiltonian graph theory. Extensions of notions such as vertex degree and adjacency of vertices to subgraphs of order greater than 1 arise in a natural way

Hamilton cycles in graphs and hypergraphs: an extremal perspective

As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page limits, this final version is slightly shorter than the previous arxiv versio

Hamilton cycles in highly connected and expanding graphs

In this paper we prove a sufficient condition for the existence of a Hamilton cycle, which is applicable to a wide variety of graphs, including relatively sparse graphs. In contrast to previous criteria, ours is based on only two properties: one requiring expansion of ``small'' sets, the other ensuring the existence of an edge between any two disjoint ``large'' sets. We also discuss applications in positional games, random graphs and extremal graph theory.Comment: 19 page

Effective hamiltonian approach and the lattice fixed node approximation

We define a numerical scheme that allows to approximate a given Hamiltonian by an effective one, by requiring several constraints determined by exact properties of generic ''short range'' Hamiltonians. In this way the standard lattice fixed node is also improved as far as the variational energy is concerned. The effective Hamiltonian is defined in terms of a guiding function $\psi_G$ and can be solved exactly by Quantum Monte Carlo methods. We argue that, for reasonable $\psi_G$ and away from phase transitions, the long distance, low energy properties are rather independent on the chosen guiding function, thus allowing to remove the well known problem of standard variational Monte Carlo schemes based only on total energy minimizations, and therefore insensitive to long distance low energy properties.Comment: 8 pages, for the proceedings of "The Monte Carlo Method in the Physical Sciences: Celebrating the 50th Anniversary of the Metropolis Algorithm", Los Alamos, June 9-11, 200

Monte Carlo simulations of the directional-ordering transition in the two-dimensional classical and quantum compass model

A comprehensive study of the two-dimensional (2D) compass model on the square lattice is performed for classical and quantum spin degrees of freedom using Monte Carlo and quantum Monte Carlo methods. We employ state-of-the-art implementations using Metropolis, stochastic series expansion and parallel tempering techniques to obtain the critical ordering temperatures and critical exponents. In a pre-investigation we reconsider the classical compass model where we study and contrast the finite-size scaling behavior of ordinary periodic boundary conditions against annealed boundary conditions. It is shown that periodic boundary conditions suffer from extreme finite-size effects which might be caused by closed loop excitations on the torus. These excitations also appear to have severe effects on the Binder parameter. On this footing we report on a systematic Monte Carlo study of the quantum compass model. Our numerical results are at odds with recent literature on the subject which we trace back to neglecting the strong finite-size effects on periodic lattices. The critical temperatures are obtained as $T_\mathrm{c}=0.1464(2)J$ and $T_\mathrm{c}=0.055(1)J$ for the classical and quantum version, respectively, and our data support a transition in the 2D Ising universality class for both cases.Comment: 8 pages, 7 figures, differs slightly from published versio

Optimized local modes for lattice dynamical applications

We present a new scheme for the construction of highly localized lattice Wannier functions. The approach is based on a heuristic criterion for localization and takes the symmetry constraints into account from the start. We compare the local modes thus obtained with those generated by other schemes and find that they also provide a better description of the relevant vibrational subspace.Comment: 6 pages, ReVTeX, plus four postscript files for figure