# On the Potts Antiferromagnet on Random Graphs

@article{CojaOghlan2016OnTP,
title={On the Potts Antiferromagnet on Random Graphs},
author={Amin Coja-Oghlan and Nor Jaafari},
journal={Electron. J. Comb.},
year={2016},
volume={23},
pages={P4.3}
}
• Published 29 February 2016
• Mathematics, Computer Science
• Electron. J. Comb.
Extending a prior result of Contucci et al (Comm. Math. Phys. 2013), we determine the free energy of the Potts antiferromagnet on the Erdos-Renyi random graph at all temperatures for average degrees $d \le (2k-1)\ln k - 2 - k^{-1/2}$. In particular, we show that for this regime of $d$ there does not occur a phase transition.
6 Citations

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#### References

SHOWING 1-10 OF 38 REFERENCES
Antiferromagnetic Potts Model on the Erdős-Rényi Random Graph
• Mathematics, Physics
• 2013
We study the antiferromagnetic Potts model on the Poissonian Erdős-Rényi random graph. By identifying a suitable interpolation structure and an extended variational principle, together with aExpand
The Condensation Phase Transition in Random Graph Coloring
• Physics, Mathematics
• 2016
Based on a non-rigorous formalism called the “cavity method”, physicists have put forward intriguing predictions on phase transitions in diluted mean-field models, in which the geometry ofExpand
A positive temperature phase transition in random hypergraph 2-coloring
• Mathematics, Computer Science
• ArXiv
• 2014
This paper establishes the existence and asymptotic location of this so-called condensation phase transition in the random hypergraph $2$-coloring problem. Expand
Potts glass on random graphs
• Physics
• 2008
We solve the q-state Potts model with anti-ferromagnetic interactions on large random lattices of finite coordination. Due to the frustration induced by the large loops and to the local tree-likeExpand
The Replica Symmetric Solution for Potts Models on d-Regular Graphs
• Mathematics, Physics
• 2012
We establish an explicit formula for the limiting free energy density (log-partition function divided by the number of vertices) for ferromagnetic Potts models on uniformly sparse graph sequencesExpand
Decay of Correlations for the Hardcore Model on the $d$-regular Random Graph
• Mathematics, Physics
• 2014
A key insight from statistical physics about spin systems on random graphs is the central role played by Gibbs measures on trees. We determine the local weak limit of the hardcore model on randomExpand
MCMC sampling colourings and independent sets of G(n, d/n) near uniqueness threshold
This work focuses on the k-colouring model} and the hard-core model with fugacity \lambda when the underlying graph is an instance of Erdos-Renyi random graph G(n,p), where p=d/n and d is fixed and uses the Markov Chain Monte Carlo method for sampling from Gibbs distribution. Expand
Reconstruction on Trees and Spin Glass Transition
• Physics
• 2006
Consider an information source generating a symbol at the root of a tree network whose links correspond to noisy communication channels, and broadcasting it through the network. We study the problemExpand
On the chromatic number of random d-regular graphs
• Mathematics
• 2008
In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d (2k−3)log(k−1), then theExpand
On the chromatic number of a random hypergraph
• Computer Science, Mathematics
• J. Comb. Theory, Ser. B
• 2015
We consider the problem of k-colouring a random r-uniform hypergraph with n vertices and cn edges, where k, r, c remain constant as n ? ∞ . Achlioptas and Naor showed that the chromatic number of aExpand