A combinatorial study of Ising model on maps – IsOMa

The goal of this project is to study the combinatorial properties of maps (that are graphs embedded on a surface) endowed with an Ising model. The combinatorial study of maps (without an Ising model) has been a very active field of research in the last two decades and is now fairly well-undersood. However, the situation is much different when one considers maps endowed with an Ising model (or more generally with a statistical physics model). We are planning to address this problem from many perspectives: enumerative and bijective combinatorics, asymptotic enumeration and connections with stochastic processes.

We expect to make significant contributions on this field along the following three main objectives of the project:
- Continue the development of the bijective combinatorics for maps and related objects. In particular, develop new types of bijections with walks in the quarter plane and with fighting fish. Explain combinatorially rationality schemes and rational parametrizations obtains for Ising-weighted maps on surfaces of higher genus.
- Extend to Ising-weighted maps the study of stochastic processes introduced for maps without matter. Along the way, develop results for multivariate asymptotic enumeration.
- Use the connection between maps and fighting fish to obtain two-way interactions between those models. Exploit the combinatorial methods originally developed in the study of maps to get new results on fighting fish, and conversely, use the combinatorial properties of fighting fish to shed new light on the statistical properties of maps with matter.

All the members of the consortium are specialists in maps combinatorics with complementary expertise, which will allow them to carry out sucessfully this project. The PI has expertise both in combinatorics and in probability theory, which will be key to collaborate efficiently with all the members of the project.