6,014 research outputs found
Partition Function Expansion on Region-Graphs and Message-Passing Equations
Disordered and frustrated graphical systems are ubiquitous in physics,
biology, and information science. For models on complete graphs or random
graphs, deep understanding has been achieved through the mean-field replica and
cavity methods. But finite-dimensional `real' systems persist to be very
challenging because of the abundance of short loops and strong local
correlations. A statistical mechanics theory is constructed in this paper for
finite-dimensional models based on the mathematical framework of partition
function expansion and the concept of region-graphs. Rigorous expressions for
the free energy and grand free energy are derived. Message-passing equations on
the region-graph, such as belief-propagation and survey-propagation, are also
derived rigorously.Comment: 10 pages including two figures. New theoretical and numerical results
added. Will be published by JSTAT as a lette
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
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