22 research outputs found
Every planar graph with the Liouville property is amenable
We introduce a strengthening of the notion of transience for planar maps in
order to relax the standard condition of bounded degree appearing in various
results, in particular, the existence of Dirichlet harmonic functions proved by
Benjamini and Schramm. As a corollary we obtain that every planar non-amenable
graph admits Dirichlet harmonic functions
Mixing time of critical Ising model on trees is polynomial in the height
In the heat-bath Glauber dynamics for the Ising model on the lattice,
physicists believe that the spectral gap of the continuous-time chain exhibits
the following behavior. For some critical inverse-temperature , the
inverse-gap is bounded for , polynomial in the surface area
for and exponential in it for . This has
been proved for except at criticality. So far, the only underlying
geometry where the critical behavior has been confirmed is the complete graph.
Recently, the dynamics for the Ising model on a regular tree, also known as the
Bethe lattice, has been intensively studied. The facts that the inverse-gap is
bounded for were
established, where is the critical spin-glass parameter, and the
tree-height plays the role of the surface area.
In this work, we complete the picture for the inverse-gap of the Ising model
on the -ary tree, by showing that it is indeed polynomial in at
criticality. The degree of our polynomial bound does not depend on , and
furthermore, this result holds under any boundary condition. We also obtain
analogous bounds for the mixing-time of the chain. In addition, we study the
near critical behavior, and show that for , the inverse-gap
and mixing-time are both .Comment: 53 pages; 3 figure
Multivariate tight affine frames with a small number of generators
We give a simple and explicit construction of compactly supported affine tight frames with small number of generators, associated to multivariate box splines (with respect to the dilation matrix 2I). Moreover, the same technique applied to the case of bivariate box splines on the four-directions mesh with dilation matrix gives tight frames with at most five generators
Multivariate compactly supported biorthogonal spline wavelets
We study biorthogonal bases of compactly supported wavelets constructed from box splines in \uc2N with any integer dilation factor. For a suitable class of box splines we write explicitly dual low-pass filters of arbitrarily high regularity and indicate how to construct the corresponding high-pass filters (primal and dual)