# Cycle structure of the random exchange model on various graphs

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

The candidate will investigate random interchange models and will attempt to show the occurrence of a phase transition to a phase that contains large permutation cycles.

In general, this is motivated by the quantum Heisenberg ferromagnet of condensed matter theory; indeed, the transition to large cycles in a model with a particular cycle weight factor is proven to corresponds to a magnetic transition and its rigorous proof would be a major breakthrough.

However, there are several models that are simplifying this in its various aspects.

In particular, starting with a seminal work by Oded Schramm, we have a good understanding of the above mentioned transition in its easiest form---the model on a compete graph with uniform cycle weight.

The candidate will study the transition for random interchange models on several other relevant graphs (Hamming graph, the hypercube, subsets of d-dimensional lattice) with various cycle weight functions.

Very recently, it was shown that in the case of the complete graph, where the transition for uniform cycle weight random interchange model is closely linked with the transition of a simpler percolation model, the random interchange model with the cycle weight factor of Heisenberg type is in a similar way linked with a random cluster model (a generalization of a percolation model).

However, it turns out that the transition for random cluster model is settled only in the case of complete graph.

Hence, the first task for the candidate is to understand the transition for random cluster model on other useful graphs like Hamming graph and hypercube, the graphs that are by their geometric structure closer to the underlying physics motivation. Hopefully, this will allow to proceed to the more complex case of the transition for random interchange models.

In his studies, the candidate is using a combination of mathematical tools from probability theory and combinatorics.

In general, this is motivated by the quantum Heisenberg ferromagnet of condensed matter theory; indeed, the transition to large cycles in a model with a particular cycle weight factor is proven to corresponds to a magnetic transition and its rigorous proof would be a major breakthrough.

However, there are several models that are simplifying this in its various aspects.

In particular, starting with a seminal work by Oded Schramm, we have a good understanding of the above mentioned transition in its easiest form---the model on a compete graph with uniform cycle weight.

The candidate will study the transition for random interchange models on several other relevant graphs (Hamming graph, the hypercube, subsets of d-dimensional lattice) with various cycle weight functions.

Very recently, it was shown that in the case of the complete graph, where the transition for uniform cycle weight random interchange model is closely linked with the transition of a simpler percolation model, the random interchange model with the cycle weight factor of Heisenberg type is in a similar way linked with a random cluster model (a generalization of a percolation model).

However, it turns out that the transition for random cluster model is settled only in the case of complete graph.

Hence, the first task for the candidate is to understand the transition for random cluster model on other useful graphs like Hamming graph and hypercube, the graphs that are by their geometric structure closer to the underlying physics motivation. Hopefully, this will allow to proceed to the more complex case of the transition for random interchange models.

In his studies, the candidate is using a combination of mathematical tools from probability theory and combinatorics.

## People |
## ORCID iD |

Roman Kotecky (Primary Supervisor) | |

Darion Mayes (Student) |

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/N509796/1 | 01/10/2016 | 30/09/2021 | |||

1935605 | Studentship | EP/N509796/1 | 02/10/2017 | 30/06/2021 | Darion Mayes |

Description | Initially, we desired to investigate a phase transition in the weighted random interchange process for the hypercube, with the belief that it was related to a phase transition for the random cluster model on the same family of graphs. It transpired that this phase transition had not been investigated. We have succeeded in establishing a phase transition for the random cluster model for a set of parameters related to another physical model. This gives more general bounds that are exponentially better than previous results. We would still like to get more detailed information about this transition, as well as generalise its argument to other families of graphs, such as the Hamming graph. |

Exploitation Route | The established transition gives results on the hypercube random cluster model that motivate investigation on further graphs, and agree with those expected for the random interchange model. This motivates continuing investigation into both models, with applications to statistical mechanics models and the theory of quantum spin systems. |

Sectors | Other |