2014
DOI: 10.1007/s00220-014-2080-3
|View full text |Cite
|
Sign up to set email alerts
|

Solution of the Monomer–Dimer Model on Locally Tree-Like Graphs. Rigorous Results

Abstract: We consider the monomer-dimer model on sequences of random graphs locally convergent to trees. We prove that the monomer density converges almost surely, in the thermodynamic limit, to an analytic function of the monomer activity. We characterise this limit as the expectation of the solution of a fixed point distributional equation and we give an explicit expression for the limiting pressure per particle.

Help me understand this report
View preprint versions

Search citation statements

Order By: Relevance

Paper Sections

Select...
4
1

Citation Types

1
23
0

Year Published

2015
2015
2024
2024

Publication Types

Select...
5
1

Relationship

3
3

Authors

Journals

citations
Cited by 18 publications
(24 citation statements)
references
References 19 publications
1
23
0
Order By: Relevance
“…It is interesting to emphasize that the Gaussian representation for the partition function is able to capture the essence of the Heilmann-Lieb recursion relation that is the main tool to solve many monomer-dimer models [16,2]. We show in fact that this recursion relation reduces to integration by parts of the Gaussian measure.…”
Section: Introductionmentioning
confidence: 86%
See 1 more Smart Citation
“…It is interesting to emphasize that the Gaussian representation for the partition function is able to capture the essence of the Heilmann-Lieb recursion relation that is the main tool to solve many monomer-dimer models [16,2]. We show in fact that this recursion relation reduces to integration by parts of the Gaussian measure.…”
Section: Introductionmentioning
confidence: 86%
“…[14] for an overview of matching problems). For a different way of introducing randomness in monomer-dimer systems see [2], where a model on locally tree-like random graphs is solved. The combinatorial problem of perfect matchings on random graphs, already solved in [21,6], corresponds the zero-temperature limit of the latter monomer-dimer model.…”
Section: Introductionmentioning
confidence: 99%
“…Fundamental results were obtained by Heilmann and Lieb, who proved the absence of phase transitions [15] when only the hard-core interaction is taken into account, while the presence of an additional interaction coupling dimers can generate critical behaviours [16]. Monomer-dimers models have been source of a renewed interest in the last years in mathematical physics [1,2,11,13], condensed matter physics [19] and in the applications to computer science [17,22] and social sciences [7,10]. The presence of an interaction beyond the hard-core one that couples different dimers is fundamental for the applications where phase transitions are observed [7,10].…”
Section: Introductionmentioning
confidence: 99%
“…no matter the positions of the two monomers on the boundaries, where α (2) ij depends only of the positions of the two monomers and of the system size L. Consequently, the partition function of the dimer model with two boundary monomers reads…”
Section: General Casementioning
confidence: 99%
“…For the general monomer-dimer problem (cf. Appendix C for a definition) there is no exact solution except in 1d, on the complete and locally tree-like graphs [2] or scale free networks [148]. We can also mention that the matrix transfer method was used to express the partition function of the model [104,91] and a very efficient method based on variational corner transfer matrix has been found by Baxter [8], leading to precise approximations of thermodynamic quantities of the model.…”
Section: Introductionmentioning
confidence: 99%