301 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
Circular Coloring of Random Graphs: Statistical Physics Investigation
Circular coloring is a constraints satisfaction problem where colors are
assigned to nodes in a graph in such a way that every pair of connected nodes
has two consecutive colors (the first color being consecutive to the last). We
study circular coloring of random graphs using the cavity method. We identify
two very interesting properties of this problem. For sufficiently many color
and sufficiently low temperature there is a spontaneous breaking of the
circular symmetry between colors and a phase transition forwards a
ferromagnet-like phase. Our second main result concerns 5-circular coloring of
random 3-regular graphs. While this case is found colorable, we conclude that
the description via one-step replica symmetry breaking is not sufficient. We
observe that simulated annealing is very efficient to find proper colorings for
this case. The 5-circular coloring of 3-regular random graphs thus provides a
first known example of a problem where the ground state energy is known to be
exactly zero yet the space of solutions probably requires a full-step replica
symmetry breaking treatment.Comment: 19 pages, 8 figures, 3 table
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