Abstract
We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the so-called uniqueness regim...
Highlights
Random constraint satisfaction problems have been thoroughly studied in computer science in an effort to analyse the limits of satisfiability algorithms and understand the structure of hard instances
For all integers q ≥ 3 and ∆ ≥ 3, we design approximate sampling algorithms for the q-state Potts model on random ∆-regular graphs, when the parameter B lies in the uniqueness regime of the regular tree, for both the ferromagnetic and antiferromagnetic cases
When restricted to integer values of q, this yields a simple algorithm for the ferromagnetic Potts model on random ∆-regular graphs. 33:4 Sampling the Potts model on Random Regular Graphs
Summary
Random constraint satisfaction problems have been thoroughly studied in computer science in an effort to analyse the limits of satisfiability algorithms and understand the structure of hard instances. For all integers q ≥ 3 and ∆ ≥ 3, we design approximate sampling algorithms for the q-state Potts model on random ∆-regular graphs (regular graphs with n vertices chosen uniformly at random), when the parameter B lies in the uniqueness regime of the regular tree, for both the ferromagnetic and antiferromagnetic cases. When restricted to integer values of q, this yields a simple algorithm for the ferromagnetic Potts model on random ∆-regular graphs To conclude this introductory section, we remark that, for many antiferromagnetic spin systems on random graphs, typical configurations in the Gibbs distribution display absence of long-range correlations even beyond the uniqueness threshold, up to the so-called reconstruction threshold [20, 12]. For the ferromagnetic Potts model on random regular graphs, the structure of typical configurations can be fairly well understood using probabilistic arguments for all temperatures (see, e.g., [5, 11]) and it would be very interesting to exploit this structure for the design of sampling algorithms beyond the uniqueness threshold
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