Sampling in Uniqueness from the Potts and Random-Cluster Models on Random Regular Graphs

The Potts and random-cluster models

Sampling in Potts Model on Sparse Random Graphs

We study the problem of sampling almost uniform proper q-colorings in sparse Erdős-Rényi random graphs G(n, d/n), a research initiated by Dyer, Flaxman, Frieze and Vigoda [2]. We obtain a fully polynomial time almost uniform sampler (FPAUS) for the problem provided q > 3d + 4, improving the current best bound q > 5.5d [6]. Our sampling algorithm works for more generalized models and broader fam...

Models of random regular graphs

This is a survey of results on properties of random regular graphs, together with an exposition of some of the main methods of obtaining these results. Related results on asymptotic enumeration are also presented, as well as various generalisations to random graphs with given degree sequence. A major feature in this area is the small subgraph conditioning method. When applicable, this establish...

On the asymmetry of random regular graphs and random graphs

This paper studies the symmetry of random regular graphs and random graphs. Our main result shows that for all 3 ≤ d ≤ n − 4 the random d-regular graph on n vertices almost surely has no non-trivial automorphisms. This answers an open question of N. Wormald [13].

Random-cluster multihistogram sampling for the q-state Potts model.

Using the random-cluster representation of the q-state Potts models we consider the pooling of data from cluster-update Monte Carlo simulations for different thermal couplings K and number of states per spin q. Proper combination of histograms allows for the evaluation of thermal averages in a broad range of K and q values, including noninteger values of q. Due to restrictions in the sampling p...