37,123 research outputs found

    Phase Transition for Glauber Dynamics for Independent Sets on Regular Trees

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    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 λ\lambda, 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 bb, the hard-core model can be equivalently defined as a broadcasting process with a parameter ω\omega which is the positive solution to λ=ω(1+ω)b\lambda=\omega(1+\omega)^b, and vertices are occupied with probability ω/(1+ω)\omega/(1+\omega) when their parent is unoccupied. This broadcasting process undergoes a phase transition between the so-called reconstruction and non-reconstruction regions at ωrlnb/b\omega_r\approx \ln{b}/b. 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 bb-ary trees ThT_h of height hh and nn 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 ωlnb/b\omega \le \ln{b}/b, for ThT_h with any boundary condition, the relaxation time is Ω(n)\Omega(n) and O(n1+ob(1))O(n^{1+o_b(1)}). In contrast, above the reconstruction threshold we show that for every δ>0\delta>0, for ω=(1+δ)lnb/b\omega=(1+\delta)\ln{b}/b, the relaxation time on ThT_h with any boundary condition is O(n1+δ+ob(1))O(n^{1+\delta + o_b(1)}), and we construct a boundary condition where the relaxation time is Ω(n1+δ/2ob(1))\Omega(n^{1+\delta/2 - o_b(1)})

    On the Reconstruction of Geodesic Subspaces of RN\mathbb{R}^N

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    We consider the topological and geometric reconstruction of a geodesic subspace of RN\mathbb{R}^N 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 R2\mathbb{R}^2, 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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