107 research outputs found
On a Connectivity Threshold for Colorings of Random Graphs and Hypergraphs
Let Omega_q=Omega_q(H) denote the set of proper [q]-colorings of the hypergraph H. Let Gamma_q be the graph with vertex set Omega_q where two vertices are adjacent iff the corresponding colorings differ in exactly one vertex. We show that if H=H_{n,m;k}, k >= 2, the random k-uniform hypergraph with V=[n] and m=dn/k hyperedges then w.h.p. Gamma_q is connected if d is sufficiently large and q >~ (d/log d)^{1/(k-1)}. This is optimal to the first order in d. Furthermore, with a few more colors, we find that the diameter of Gamma_q is O(n) w.h.p, where the hidden constant depends on d. So, with this choice of d,q, the natural Glauber Dynamics Markov Chain on Omega_q is ergodic w.h.p
The condensation transition in random hypergraph 2-coloring
For many random constraint satisfaction problems such as random
satisfiability or random graph or hypergraph coloring, the best current
estimates of the threshold for the existence of solutions are based on the
first and the second moment method. However, in most cases these techniques do
not yield matching upper and lower bounds. Sophisticated but non-rigorous
arguments from statistical mechanics have ascribed this discrepancy to the
existence of a phase transition called condensation that occurs shortly before
the actual threshold for the existence of solutions and that affects the
combinatorial nature of the problem (Krzakala, Montanari, Ricci-Tersenghi,
Semerjian, Zdeborova: PNAS 2007). In this paper we prove for the first time
that a condensation transition exists in a natural random CSP, namely in random
hypergraph 2-coloring. Perhaps surprisingly, we find that the second moment
method breaks down strictly \emph{before} the condensation transition. Our
proof also yields slightly improved bounds on the threshold for random
hypergraph 2-colorability. We expect that our techniques can be extended to
other, related problems such as random k-SAT or random graph k-coloring
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