4 research outputs found
Spatial Mixing of Coloring Random Graphs
We study the strong spatial mixing (decay of correlation) property of proper
-colorings of random graph with a fixed . The strong spatial
mixing of coloring and related models have been extensively studied on graphs
with bounded maximum degree. However, for typical classes of graphs with
bounded average degree, such as , an easy counterexample shows that
colorings do not exhibit strong spatial mixing with high probability.
Nevertheless, we show that for with and
sufficiently large , with high probability proper -colorings of
random graph exhibit strong spatial mixing with respect to an
arbitrarily fixed vertex. This is the first strong spatial mixing result for
colorings of graphs with unbounded maximum degree. Our analysis of strong
spatial mixing establishes a block-wise correlation decay instead of the
standard point-wise decay, which may be of interest by itself, especially for
graphs with unbounded degree
Local convergence of random graph colorings
Let be a random graph whose average degree is below the
-colorability threshold. If we sample a -coloring of
uniformly at random, what can we say about the correlations between the colors
assigned to vertices that are far apart? According to a prediction from
statistical physics, for average degrees below the so-called {\em condensation
threshold} , the colors assigned to far away vertices are
asymptotically independent [Krzakala et al.: Proc. National Academy of Sciences
2007]. We prove this conjecture for exceeding a certain constant .
More generally, we investigate the joint distribution of the -colorings that
induces locally on the bounded-depth neighborhoods of any fixed number
of vertices. In addition, we point out an implication on the reconstruction
problem
Local convergence of random graph colorings
Let G(n,m) be a random graph whose average degree d=2m/n is below the k-colorability threshold. If we sample a k-coloring \SIGMA of G(n,m) uniformly at random, what can we say about the correlations between the colors assigned to vertices that are far apart?
According to a prediction from statistical physics, for average degrees below the so-called {\em condensation threshold} \dc, the colors assigned to far away vertices are asymptotically independent [Krzakala et al.: Proc. National Academy of Sciences 2007].
We prove this conjecture for k exceeding a certain constant k_0. More generally, we investigate the joint distribution of the k-colorings that \SIGMA induces locally on the bounded-depth neighborhoods of any fixed number of vertices. In addition, we point out an implication on the reconstruction problem