Phase transition for the mixing time of the Glauber dynamics for coloring regular trees
Phase transition for the mixing time of the Glauber dynamics for coloring regular trees
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为普通树木着色的格劳伯动力学混合时间的相变
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发表时间：
2009
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影响因子：
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通讯作者：
Linji Yang
中科院分区：
文献类型：
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作者：
P. Tetali;Juan C. Vera;Eric Vigoda;Linji Yang
证明了具有分枝因子的完全树的随机k-染色的Glauber动力学的混合时间在k=b(1+o(1))/ln(B)处经历了一个相变.我们的主要结果给出了具有n个顶点的完全树上的动力学混合时的几乎精确的界，其中k=cb/ln b的颜色为常数。对于C<sup>1</sup>2<sup>2</sup><<i>O</i>(<i>n</i><sup>1+<i>o</i><sub><i>b</i></sub>(1)</sup>>n</>i>，我们证明了混合时间为≥ln<sup>2</sup>)。另一方面，对于C和1，混合时间经历了减慢，特别地，我们证明了它是<i>O</i>(<i>n</i><sup>1/<i>C+o</i><sub><i>b</i></sub>(1)</sup>ln(2)和Ω(<i>n</i><sup>1/<i>C-o</i><sub><i>b</i></sub>(1)</sup>).临界点C=1很有趣，因为它与Sly最近建立的所谓重建阈值重合(至少达到一阶)。重建阈值最近引起了相当大的兴趣，因为它似乎与某些局部算法的效率密切相关，而这项工作的灵感来自于我们试图在这种特定环境下理解这些联系。
We prove that the mixing time of the Glauber dynamics for random <i>k</i>-colorings of the complete tree with branching factor <i>b</i> undergoes a phase transition at <i>k</i> = <i>b</i>(1+<i>o</i><sub>b</sub>(1))/ln <i>b</i>. Our main result shows nearly sharp bounds on the mixing time of the dynamics on the complete tree with <i>n</i> vertices for <i>k</i> = <i>Cb</i>/ln<i>b</i> colors with constant <i>C</i>. For <i>C</i> ≥ 1 we prove the mixing time is <i>O</i>(<i>n</i><sup>1+<i>o</i><sub><i>b</i></sub>(1)</sup> ln<sup>2</sup> <i>n</i>). On the other side, for <i>C</i> < 1 the mixing time experiences a slowing down, in particular, we prove it is <i>O</i>(<i>n</i><sup>1/<i>C+o</i><sub><i>b</i></sub>(1)</sup> ln<sup>2</sup> <i>n</i>) and Ω(<i>n</i><sup>1/<i>C-o</i><sub><i>b</i></sub>(1)</sup>). The critical point <i>C</i> = 1 is interesting since it coincides (at least up to first order) to the so-called reconstruction threshold which was recently established by Sly. The reconstruction threshold has been of considerable interest recently since it appears to have close connections to the efficiency of certain local algorithms, and this work was inspired by our attempt to understand these connections in this particular setting.
普通树木着色的芒硝动力学混合时间
影响因子：
1
作者：
Goldberg L
通讯作者：
Goldberg L