7 research outputs found
Satisfiability Thresholds for Regular Occupation Problems
In the last two decades the study of random instances of constraint
satisfaction problems (CSPs) has flourished across several disciplines,
including computer science, mathematics and physics. The diversity of the
developed methods, on the rigorous and non-rigorous side, has led to major
advances regarding both the theoretical as well as the applied viewpoints. The
two most popular types of such CSPs are the Erd\H{o}s-R\'enyi and the random
regular CSPs.
Based on a ceteris paribus approach in terms of the density evolution
equations known from statistical physics, we focus on a specific prominent
class of problems of the latter type, the so-called occupation problems. The
regular -in- occupation problems resemble a basis of this class. By now,
out of these CSPs only the satisfiability threshold - the largest degree for
which the problem admits asymptotically a solution - for the -in-
occupation problem has been rigorously established. In the present work we take
a general approach towards a systematic analysis of occupation problems. In
particular, we discover a surprising and explicit connection between the
-in- occupation problem satisfiability threshold and the determination of
contraction coefficients, an important quantity in information theory measuring
the loss of information that occurs when communicating through a noisy channel.
We present methods to facilitate the computation of these coefficients and use
them to establish explicitly the threshold for the -in- occupation
problem for . Based on this result, for general we formulate a
conjecture that pins down the exact value of the corresponding coefficient,
which, if true, is shown to determine the threshold in all these cases
The hitting time of clique factors
In a recent paper, Kahn gave the strongest possible, affirmative, answer to
Shamir's problem, which had been open since the late 1970s: Let and
let be divisible by . Then, in the random -uniform hypergraph process
on vertices, as soon as the last isolated vertex disappears, a perfect
matching emerges. In the present work, we transfer this hitting time result to
the setting of clique factors in the random graph process: At the time that the
last vertex joins a copy of the complete graph , the random graph process
contains a -factor. Our proof draws on a novel sequence of couplings,
extending techniques of Riordan and the first author. An analogous result is
proved for clique factors in the -uniform hypergraph process ()
Charting the replica symmetric phase
Diluted mean-field models are spin systems whose geometry of interactions is induced by a sparse random graph or hypergraph. Such models play an eminent role in the statistical mechanics of disordered systems as well as in combinatorics and computer science. In a path-breaking paper based on the non-rigorous ‘cavity method’, physicists predicted not only the existence of a replica symmetry breaking phase transition in such models but also sketched a detailed picture of the evolution of the Gibbs measure within the replica symmetric phase and its impact on important problems in combinatorics, computer science and physics (Krzakala et al. in Proc Natl Acad Sci 104:10318–10323, 2007). In this paper we rigorise this picture completely for a broad class of models, encompassing the Potts antiferromagnet on the random graph, the k-XORSAT model and the diluted k-spin model for even k. We also prove a conjecture about the detection problem in the stochastic block model that has received considerable attention (Decelle et al. in Phys Rev E 84:066106, 2011)