In the recent work by Borgs, Chayes, Cohn and Zhao on sparse graph limits (http://arxiv.org/abs/1401.2906), a theory of graph limits is presented where the average degree of the random graph grows, but much more slowly than the number of nodes. For such sequences, the authors show that the different notions of convergence (such as convergence of subcounts, convergence of partition functions, etc) agree. This result has important consequences for statistical physics spin models on these graphs, as it shows that free energies converge to the same limit as for the respective mean-field models. However, the mean-field models can be expected to depend sensitively on the precise properties on the graph sequence, for example its hub structure. Recently, in work of Regts and Schrijver also graph limits of vertex models have been investigated.
We propose to extend the work of Borgs, Chayes, Cohn and Zhao from spin models to vertex models. Further, we plan to investigate the critical behavior of statistical mechanics models, such as the Ising model or Potts models, on such graphs sequences. We are particularly interested in the relation between the statistical mechanics model and the topology of the graph sequence, such as its hub-structure.
|Supervisors||Remco van der Hofstad (TU/e) and Lex Schrijver (UvA)|
|Location||University of Amsterdam (UvA)|