55,837 research outputs found
Fastest mixing Markov chain on graphs with symmetries
We show how to exploit symmetries of a graph to efficiently compute the
fastest mixing Markov chain on the graph (i.e., find the transition
probabilities on the edges to minimize the second-largest eigenvalue modulus of
the transition probability matrix). Exploiting symmetry can lead to significant
reduction in both the number of variables and the size of matrices in the
corresponding semidefinite program, thus enable numerical solution of
large-scale instances that are otherwise computationally infeasible. We obtain
analytic or semi-analytic results for particular classes of graphs, such as
edge-transitive and distance-transitive graphs. We describe two general
approaches for symmetry exploitation, based on orbit theory and
block-diagonalization, respectively. We also establish the connection between
these two approaches.Comment: 39 pages, 15 figure
Exactly Solvable Lattice Models with Crossing Symmetry
We show how to compute the exact partition function for lattice
statistical-mechanical models whose Boltzmann weights obey a special "crossing"
symmetry. The crossing symmetry equates partition functions on different
trivalent graphs, allowing a transformation to a graph where the partition
function is easily computed. The simplest example is counting the number of
nets without ends on the honeycomb lattice, including a weight per branching.
Other examples include an Ising model on the Kagome' lattice with three-spin
interactions, dimers on any graph of corner-sharing triangles, and non-crossing
loops on the honeycomb lattice, where multiple loops on each edge are allowed.
We give several methods for obtaining models with this crossing symmetry, one
utilizing discrete groups and another anyon fusion rules. We also present
results indicating that for models which deviate slightly from having crossing
symmetry, a real-space decimation (renormalization-group-like) procedure
restores the crossing symmetry
In search for a perfect shape of polyhedra: Buffon transformation
For an arbitrary polygon consider a new one by joining the centres of
consecutive edges. Iteration of this procedure leads to a shape which is affine
equivalent to a regular polygon. This regularisation effect is usually ascribed
to Count Buffon (1707-1788). We discuss a natural analogue of this procedure
for 3-dimensional polyhedra, which leads to a new notion of affine -regular
polyhedra. The main result is the proof of existence of star-shaped affine
-regular polyhedra with prescribed combinatorial structure, under partial
symmetry and simpliciality assumptions. The proof is based on deep results from
spectral graph theory due to Colin de Verdiere and Lovasz.Comment: Slightly revised version with added example of pentakis dodecahedro
- …