Phase Transition for the Mixing Time of the Glauber Dynamics for Coloring Regular Trees

Abstract

We prove that the mixing time of the Glauber dynamics for random k-colorings of the complete tree with branching factor b undergoes a phase transition at k = b(1+o b (1))/ ln b. Our main result shows nearly sharp bounds on the mixing time of the dynamics on the complete tree with n vertices for k = Cb/ ln b colors with constant C. For C ≥ 1 we prove the mixing time is O(n 1+o b (1) ln 2 n). On the other side, for C < 1 the mixing time experiences a slowing down, in particular, we prove it is O(n 1/C+o b (1) ln 2 n) andThe critical point C = 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.

Randomly coloring constant degree graphs

“…For planar graphs with maximum degree Δ, Hayes et al [10] were able to achieve polynomial mixing time for k ≪Δ, in particular, they showed polynomial mixing time when k > 100Δ/log Δ. More recently, Tetali et al [24] have shown that on the complete tree the mixing time of the Glauber dynamics has a phase transition at k ≈︁Δ /log Δ.…”

The mixing time of Glauber dynamics for coloring regular trees

“…They also generalized the setting for the upper bounds to nonregular trees. Very recently, Tetali, Vera, Vigoda, and Yang [20] have pinned down the constant in the exponent and located the phase transition between "nearly linear" (i.e, n 1+o(b) ) and superlinear mixing time.…”

Randomly coloring planar graphs with fewer colors than the maximum degree

“…This was resolved recently by Lucier and Molloy and Goldberg et al who showed that for constant Δ and constant k , the mixing time is polynomial. (See also for further improvements regarding the mixing time of the Glauber dynamics on the complete tree. )…”

“…Subsequent to the initial publication of this work, the threshold for extremality of the free measure in the tree was established at . More recent work of Tetali et al shows that the mixing time of the Glauber dynamics on the complete tree undergoes a phase transition at (up to first order terms) the reconstruction threshold. More precisely, for constant C > 0, for , on the n ‐vertex ‐ary tree, they prove the mixing time is for and for C < 1.…”