Phase Transition for Glauber Dynamics for Independent Sets on Regular Trees

Details Phase Transition for Glauber Dynamics for Independent Sets on Regular Trees Phase Transition for Glauber Dynamics for Independent Sets on Regular Trees

2010-07-14

arxiv.org/abs/1007.2255v1

Mathematics

Probability

Scientific paper

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

Affiliated with

Also associated with

No associations

Rating

Phase Transition for Glauber Dynamics for Independent Sets on Regular Trees does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Phase Transition for Glauber Dynamics for Independent Sets on Regular Trees, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Phase Transition for Glauber Dynamics for Independent Sets on Regular Trees will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-120313