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Keyword: rigidity threshold ×Clear all

3 claims from 2 papers

1. Computer Science › Constraint Satisfaction and Optimization

   The Asymptotics of the Clustering Transition for Random Constraint Satisfaction Problems

   Budzynski and Semerjian · Journal of Statistical Physics · 2020

   The paper locates the clustering transition of random hypergraph bicoloring and random graph colouring at large k and q, finding one shared constant and relating it to the rigidity threshold.

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   1. UncheckedThe paper characterises a constant γ_d by a functional equation, estimates it numerically at about 0.871, and derives an analytic lower bound of about 0.812.“We characterize $γ_{\rm d}$ via a functional equation, solve the latter numerically to estimate $γ_{\rm d} \approx 0.871$, and obtain an analytic lowerbound $γ_{\rm d} \ge 1 + \ln (2 (\sqrt{2}-1)) \approx 0.812$.”

2. Computer Science › Constraint Satisfaction and Optimization

   Rigid colourings of hypergraphs and contiguity

   Peter and Greenhill · arXiv (Cornell University) · 2018

   The paper determines the rigidity threshold for q-colouring random k-uniform hypergraphs and shows how the solution space changes across it, using a planted model shown to be contiguous with the random one.

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   1. UncheckedBelow the rigidity threshold, almost all vertices of a random hypergraph can be recoloured one vertex at a time within a sequence of valid colourings.“When the edge density is below the threshold, we prove that all but a vanishing proportion of the vertices can be recoloured by a sequence of colourings that recolour only one vertex at a time.”
   2. UncheckedUp to the condensation phase transition, the random colouring model is contiguous with respect to the planted model for colouring random hypergraphs.“As part of our analysis, we show that up to the condensation phase transition, the random colouring model is contiguous with respect to the planted model.”