37,123 research outputs found
Phase Transition for Glauber Dynamics for Independent Sets on Regular Trees
We study the effect of boundary conditions on the relaxation time of the
Glauber dynamics for the hard-core model on the tree. The hard-core model is
defined on the set of independent sets weighted by a parameter ,
called the activity. The Glauber dynamics is the Markov chain that updates a
randomly chosen vertex in each step. On the infinite tree with branching factor
, the hard-core model can be equivalently defined as a broadcasting process
with a parameter which is the positive solution to
, and vertices are occupied with probability
when their parent is unoccupied. This broadcasting process
undergoes a phase transition between the so-called reconstruction and
non-reconstruction regions at . Reconstruction has
been of considerable interest recently since it appears to be intimately
connected to the efficiency of local algorithms on locally tree-like graphs,
such as sparse random graphs. In this paper we show that the relaxation time of
the Glauber dynamics on regular -ary trees of height and
vertices, undergoes a phase transition around the reconstruction threshold. In
particular, we construct a boundary condition for which the relaxation time
slows down at the reconstruction threshold. More precisely, for any , for with any boundary condition, the relaxation time is
and . In contrast, above the reconstruction
threshold we show that for every , for ,
the relaxation time on with any boundary condition is , and we construct a boundary condition where the relaxation time is
On the Reconstruction of Geodesic Subspaces of
We consider the topological and geometric reconstruction of a geodesic
subspace of both from the \v{C}ech and Vietoris-Rips filtrations
on a finite, Hausdorff-close, Euclidean sample. Our reconstruction technique
leverages the intrinsic length metric induced by the geodesics on the subspace.
We consider the distortion and convexity radius as our sampling parameters for
a successful reconstruction. For a geodesic subspace with finite distortion and
positive convexity radius, we guarantee a correct computation of its homotopy
and homology groups from the sample. For geodesic subspaces of ,
we also devise an algorithm to output a homotopy equivalent geometric complex
that has a very small Hausdorff distance to the unknown shape of interest
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